R(η). Recently, schemes based on single-photon interference have been proposed to improve the key rate to Quantum key distribution — the establishment of information-theoretically secure keys based on quantum physics — is mainly limited by its practical performance, which is characterised by the dependence of the key rate on the channel transmittance  Measurement-device-independent QKD schemes suffer from a trade-off between ease of implementation  and high rates (quadratic improvement in rate). Here, the authors propose a protocol which offers both simple implementation and strong performances. A core advantage of QKD compared to other quantum communication tasks is that it is ready to implement with current commercially available off-the-shelf optical devices. However, two major characteristics of QKD—its practical security and key-rate performance—limit its real-life implementation. The key generation speed suffers heavily from transmission loss in the optical channel. Fundamentally, the asymptotic key rate for point-to-point QKD schemes is upper bounded by the repeaterless rate-transmittance bounds5, which are approximately linear functions of the transmittance, R ≤ O(η). For example, when η is small, the PLOB repeaterless rate-transmittance bound5 is about 1.44η. Quantum repeaters8 have been proposed as a radical solution to this problem. Unfortunately, none of the quantum repeater proposals is easy to implement in the near term.Quantum key distribution (QKD)9. A typical QKD system can be divided into three parts: source, channel, and measurement. The security of the channel has been well addressed in the security proofs for QKD12. The source is relatively simple and can be well characterised13. In contrast, the measurement device is complicated and difficult to calibrate. Moreover, an adversary could manipulate the measurement device by sending unexpected signals15. To solve this implementation security problem, measurement-device-independent quantum key distribution (MDI-QKD) schemes have been proposed to close the detection loopholes once and for all16. Various experimental systems have been successfully demonstrated20, with extension to a communication network21.In real-life use, the deviation of the realistic behaviour of physical devices from their ideal ones gives rise to critical issues in practical security. There are many quantum attacks that can take advantage of the loopholes introduced by device imperfections15.A generic MDI-QKD setup is shown in Fig. 16, in which Alice and Bob both encode a ‘dual-rail’ qubit into a single-photon subspace on two polarization modes, unfortunately, cannot overcome this bound. Later, alternative schemes were proposed23 in which the qubit is encoded into two optical time bins. We refer to schemes of this type as two-mode MDI-QKD, in the sense that the single-side key information is encoded in the relative phase of the coherent states in the two orthogonal optical modes, i.e., second-quantized electromagnetic fields. To correlate Alice’s and Bob’s encoded information in a two-mode scheme, a successful two-photon interference measurement is required. If either Alice or Bob’s emitted photon is lost in transmission, there will be no conclusive detection result. For example, in the time-bin encoding scheme23 shown in Fig. A1 and A2 and Bob emitting B1 and B2. Only when both the interference between modes A1 and B1 and that between A2 and B2 yield successful detection can Alice restore Bob’s raw key information. Thus, successful interference requires a coincidence detection. Due to this coincidence-detection requirement, rounds with only a single detection are discarded, resulting in a relatively low key generation rate—one that is a linear function of the transmittance, O(η). From the perspective of practical implementation, however, coincidence detection also has certain merits. This approach can ensure stable optical interference, while Alice and Bob need only to stabilise the relative phases between the two modes.Strictly speaking, MDI-QKD is not a point-to-point scheme, as there is an interference site between Alice and Bob. Consequently, it is not necessarily limited by the repeaterless rate-transmittance bound. Nevertheless, the original MDI-QKD scheme24, illustrated in Fig. 26 (known as phase-matching QKD) or the intensity27 (known as sending-or-not-sending TF-QKD) of coherent states. In this work, we refer to these twin-field-type schemes as one-mode MDI-QKD schemes for a conceptual comparison to the traditional two-mode MDI-QKD schemes, since the single-side information in these schemes is encoded into a single-optical mode in each round. We remark that the single-optical-mode encoding MDI-QKD scheme was first proposed in ref. 28 as “MDI-B92” scheme. Similar to the Duan-Lukin-Cirac-Zoller-type repeater design29, such one-mode schemes use single-photon interference instead of coincidence detection, hence yielding a quadratic improvement in key rate compared to two-mode schemes26. As a result, they can overcome the point-to-point linear key-rate bound5. Unfortunately, one-mode schemes are more challenging to implement due to the unstable optical interference resulting from the lack of global phase references. For example, in the phase-matching QKD (PM-QKD) scheme25, the key information is encoded into the global phase of Alice’s and Bob’s coherent states. The phases of the coherent states generated by two remote and independent lasers need to be matched at the measurement site. A small phase drift or fluctuation caused by the lasers and/or channels is hazardous for key generation.Coincidence detection is the essential factor that prevents MDI-QKD from overcoming the linear key-rate bound. To eliminate this requirement, a new type of MDI-QKD scheme called twin-field quantum key distribution (TF-QKD) based on encoding information into a single-optical mode have been proposed−15 s), optical length (~200 nm) and light frequency (~100 kHz). Recently, experimentalists have made great efforts to demonstrate high-performance in one-mode schemes, utilising high-end technologies to perform a precise control operation to stabilise the global phase by locking the frequency and phase of the coherent states37. However, this increases the experimental difficulty and undermines the applicability of one-mode schemes in real life.At first glance, it seems that we cannot simultaneously enjoy the advantages of one-mode schemes  and two-mode schemes , due to an intrinsic trade-off between the information-encoding efficiency and robustness. On the one hand, the relative information among different optical modes is more difficult to retrieve when the channel loss is large. On the other hand, the global phase of a coherent state is not as stable as the relative phase between two coherent states travelling through the same quantum channel. In a typical 200-km fibre with a telecommunication frequency of 1550 nm, the phase of a coherent state is susceptible to small fluctuations in the optical transmission time  during data postprocessing to optimise the system performance.In this work, we propose a mode-pairing MDI-QKD scheme that aims to offer both—simple implementation and high performance. Hereafter, we refer to this scheme as the mode-pairing scheme for simplicity. By observing that the majority of detection events are single-clicks and are discard in the two-mode MDI-QKD schemes, we try to recycle the discarded single-click in the mode-pairing scheme. To do that, the coherent states in the transmitted modes are initially prepared independently with randomly encoded information. Based on the fact that the two detection events used to read out the encoded information do not need to occur at two predetermined locations, the key is extracted from two paired detection events rather than coincidence detection, as shown in Fig. 40 and discrete phase randomisation here. A complete description of the mode-pairing scheme is given in the Methods’ subsection “Mode-pairing scheme with decoy states”.In the mode-pairing scheme, Alice and Bob first prepare coherent states with independently and randomly chosen intensities and phases in each emitted optical mode. These coherent states are sent to the untrusted measurement site, Charlie. Based on Charlie’s announced measurement results, Alice and Bob pair the optical modes with successful detection and determine the key bits and bases for each mode pair locally. They then sift the bases and generate secure key bits via postprocessing. The scheme is introduced in Box Z-pair data, since they have a much lower quantum bit error rate X-pair data. The encoding of the mode-pairing scheme in Box 23. If Alice’s two paired optical modes {Ai, Aj} are assigned to the Z-basis, then the state of the two optical modes is either κa is the encoded key information and κ. In the other case, in which the two optical modes {Ai, Aj} are assigned to the X-basis, we can rewrite their two independent random phases In the mode-pairing scheme, we mainly consider the keys generated from the mentclass2pt{minimϕa becomes a global random phase on the pulse pair, while ϕa and π). By definition, we have X-pair state can be written as,μa ∈ {0, μ}. When θ = 0 or π/2, Alice emits X-basis or Y-basis states, respectively, as used in the time-bin encoding MDI-QKD scheme23.In this way, the phase Z-pair state in Eq.  phase randomisation, the emitted Z- and X-pair states can be regarded as a mixture of photon number states40. Then, Alice and Bob can estimate the detections caused by the pairs where they both emit single photons and use them to generate secure keys, in a manner similar to traditional two-mode schemes. Therefore, the security of the mode-pairing scheme is similar to that of two-mode schemes. Nevertheless, the mode-pairing scheme in Box Ai} are independent and identically distributed (i.i.d.). Therefore, the information encoded in different optical modes is completely decoupled.The emitted states in different optical modes {Based on the postselection of clicked signals, different optical modes are paired afterwards. The relative information between the two modes is then converted into raw key data.We remark that in either the e in Eq.  or the Xe in Eq. , there i42. It is the untrusted measurement site that determines the location of successful detection and thereby affects the pairing setting. The ‘dual-rail’ qubits encoded on the single photons are ‘postselected’ on the basis of this detection. By virtual of the independence of the optical modes, the information encoded in the ‘postselected’ qubits cannot be revealed from other optical pulses.In the mode-pairing scheme, the key information is determined not in the state preparation step, but by the detection location, sharing some similarities with the differential-phase-shifting QKD scheme27 also uses a Z-basis time-bin encoding, whereby either Alice or Bob emits an optical mode to generate key bits. The state preparation of the mode-pairing scheme shares similarities with the SNS-TFQKD scheme. However, the information of the mode-pairing scheme is encoded into the relative information between the two optical modes. As a result, the basis-sifting and key mapping of the mode-pairing scheme follow different logic originated from the time-bin encoding MDI-QKD scheme23. Note that in the SNS scheme, bits 0 and 1 are highly biased in the Z basis, whereas in the mode-pairing scheme, they are evenly distributed.For another comparison, the sending-or-not-sending (SNS) TF-QKD schemeX-basis error rate fairly estimates the Z-basis phase error rate for the signals for which Alice and Bob both emit single photons.A critical issue in the security analysis of the mode-pairing scheme is to maintain the flexibility to determine in which two optical modes to perform the overall photon number measurement until Charlie announces the detection results. Note that, in the original two-mode QKD schemes, the encoders can always be assumed to perform an overall photon number measurement and post-select the single-photon components as good ‘dual-rail’ qubits before they emit their signals to Charlie. In the mode-pairing scheme, however, this is not viable because the optical pulse pair, for which the single-photon component is defined, is postselected based on Charlie’s detection announcement. To solve this problem, we introduce source replacement for the random phases in the coherent states to purify them as ancillary qudits and define an indirect overall photon number measurement on them. The source-replacement procedure can be found in the Methods’ subsection “Source replacement of the encoding state”. Conditioned on the indirect overall photon number measurement result to be single-photon states, the In Supplementary Note Box 1 Mode-pairing schemeState preparation: In the i-th round , Alice prepares a coherent state Ai with an intensity μ} and a phase π). Similarly, Bob randomly chooses Bi.Measurement: Alice and Bob send modes Ai and Bi to Charlie, who performs single-photon interference measurements. Charlie announces the click patterns for both detectors L and R.N rounds. Then, they postprocess the data as follows.Alice and Bob repeat the above two steps for Mode pairing: For all rounds with successful detection, in which one and only one of the two detectors clicks, Alice and Bob apply a strategy of grouping two clicked rounds as a pair. The encoded phases and intensities in these two rounds form a data pair. A simple pairing strategy is introduced in Box Basis sifting: Based on the intensities of the two grouped rounds indexed by i and j, Alice labels the ‘basis’ of the data pair as Z if the intensities are  or , as X if the intensities are , or as ‘0’ if the intensities are . Bob sets the basis using the same method. Alice and Bob announce the basis of each data pair; if they both announce the basis X or Z, they maintain the data pairs, whereas otherwise, the data pairs are discarded.Key mapping: For each Z-basis pair (Z-pair for simplicity) at locations i and j, Alice sets her key as κa = 0 if κa = 1 if X-basis pair (X-pair for simplicity) at locations i and j, the key is extracted from the relative phase κb and determines θb. The difference in the key mapping for Z-pairs is that, Bob sets the raw key bit κb as 0 if κb = 1 if X-pairs, if Charlie’s detection announcement is  or , Bob keeps the bit κb; otherwise, if Charlie’s announcement is  or , Bob flips κb. For the X-pairs, Alice and Bob announce the alignment angles θa and θb. If θa = θb, then the data pairs are kept; otherwise, the data pairs are discarded.Parameter estimation: Alice and Bob estimate the fraction of clicked signals q and the corresponding phase error rate Z-pairs where Alice and Bob both emit a single photon at locations i and j, using the data of the Z-pairs and X-pairs. They also estimate the quantum bit error rate Eμ, μ),Z, Eμ, μ),Z. The details are shown in the simple pairing strategy in Box i-th round is denoted by a Boolean variable Ci that indicates whether the detection is successful. That is, Ci = 1 implies that either the detector L or R clicks. Otherwise, there is no click or double clicks.Here, we consider a simple pairing strategy in which Alice pairs adjacent detection pulses together if the time interval between them is not too large (≤rp (i.e. the average number of pairs generated per pulse). We assume that Alice and Bob choose intensities 0 and μ with equal probability, maximising the number of successful pairs in the Z basis. With a typical QKD channel model, the pairing rate rp is calculated as shown in the Methods’ subsection “Mode-pairing-efficiency calculation”,p is the probability that the emitted pulses result in a click event, given approximately by ηsμ. Here, ηs and η denote the channel transmittance from Alice to Charlie and the total transmittance from Alice to Bob, respectively. When the channel is symmetric for Alice and Bob, we have p in a pure-loss channel is given in Supplementary Note rp and the detection probability p can be directly obtained by experimentation.To check the efficiency of this pairing strategy, let us calculate the pairing rate rp. Now, let us check the scaling of rp with the channel transmittance in the symmetric-channel case. If the local phase reference is sufficiently stable, then the maximal interval can be set to l → +∞. In this case,μ = O(1), as evaluated in Supplementary Note l = 1; then,The raw key rate mainly depends on the pairing rate 23. Now, if we consider a finite value of l, the dependence of rp on η will be decided by how the denominator of the first term in Eq. (p[1 − (1−p)l], depends on p ≈ ηsμ. When pl ≫ 1, rp scales with p linearly, hence pl ≪ 1, it scales with p2, resulting in rp = O(η). Around pl = 1, there will be a performance transition from rp = O(η).In this case, the experimental requirements for the mode-pairing scheme are close to those of the existing time-bin MDI-QKD schemem in Eq. , p[1 − , q is the fraction of Z-pairs caused by single-photon-pair states ρ in which both Alice and Bob send single-photon states in the two paired modes, ρ, f is the error-correction efficiency, and Eμ, μ),Z and the phase error 44. A detailed estimation procedure for q and The key rate of the mode-pairing scheme, as rigorously analysed in the Supplementary Note X-pair sifting condition θa = θb is impossible to fulfil exactly. This results in insufficient data for X-basis error rate estimation. To solve this problem, one can apply discrete phase randomisation45 such that θa and θb are chosen from a discrete set. We expect the discretisation effect to be negligible when the number of discrete phases is reasonably large, such as D = 16, similar to the situation in previous works on one-mode MDI-QKD46.During the key mapping step in Box l of the mode-pairing scheme as a value between 1 and 1 × 106, aiming to illustrate the dependence of the key rate on l. We also compare the key rate of the mode-pairing scheme with those of a typical two-mode scheme, time-bin encoding MDI-QKD23, and two one-mode schemes — PM-QKD46 and SNS-TFQKD47. The simulation results are shown in Fig. l = 2000. Here, we compare the asymptotic key-rate performance of all the schemes under the scenario of one-way local-operation and classical communication. The simulation formulas for these schemes are listed in Supplementary Note 49 show that the key-rate performance of SNS-TFQKD can be further improved by introducing the two-way classical communication51. We will leave the advanced key distillation for future studies.Based on the above analysis, we simulate the asymptotic performance of the mode-pairing scheme under a typical symmetric quantum-channel model, using practical experimental parameter settings. We assign the maximal pairing interval l = 1, show a performance comparable to that of the original two-mode scheme. These two schemes have the same scaling property, i.e., R = O(η). The deviation is caused by an extra sifting factor in the mode-pairing scheme as a result of independent encoding. When the maximal pairing interval l is increased to 1 × 103, the key rate is significantly enhanced by 3 orders of magnitude compared to the l = 1 case, making it able to surpass the linear key-rate bound. If we further increase l above 1 × 105, then the mode-pairing scheme has a similar key rate to PM-QKD and SNS-TFQKD and a scaling property given by l. When l falls within the range of 1 to 1 × 106, the key rate of the mode-pairing scheme lies between the two extreme cases of O(η) and As shown in Fig. n in Eq. .32). Hence, the coherence time of the laser is around 333 μs. In practice, the frequency fluctuation of the lasers will affect the stabilization of the phase. To test the feasibility of the mode-pairing scheme, we perform an interference experiment using a commercial optical communication system with a repetition rate of 625 MHz. The experiment detail is shown in Supplementary Note l = 3000 ~ 4000. If we apply the state-of-the-art optical communication system with the repetition rate of 4 GHz37, we can realize a pairing interval over l = 20000. As an extra remark, our current discussion on the implementation of the mode-pairing scheme is based on the multiplexing of optical time-bin modes. Nonetheless, the proposed mode-pairing design is generic for the multiplexing of other optical degrees of freedom. For example, we can introduce frequency multiplexing. The optical modes with different frequencies are first prepared and interfered independently, i.e., only the pulses with the same frequency will be interfered. After the announcement of detection results, Alice and Bob then pair the locations with different frequencies during the post-processing. This can be used to increase the effective maximal pairing interval to an even larger value without the global phase locking. From Fig. l = 1 × 104 remains η is smaller than 30 dB, corresponding to a communication distance of 300 km. The asymptotic key rate of the mode-pairing scheme is 3 to 5 orders of magnitude higher than that of the two-mode scheme. We remark that the decoherence effect caused by the optical-fibre channel is negligible compared to the laser coherence time. When the fibre length is around 500 km, the velocity of phase drift in the fibre is <10 rad/ms32, which can be calibrated using strong laser pulses without the need for real-time feedback control. As a result, the value of l depends only upon the local phase reference and not the communication distance.In typical optical experiments, the typical line width of a common commercial laser is 3 kHz  to be  to simplify the notations.The main idea of the security proof for the mode-pairing scheme is to introduce an entanglement-based scheme and reduce the security of the scheme to that of a traditional two-mode MDI-QKD scheme. To realise this, we perform a systematic source-replacement procedureD phases, evenly distributed in [0, 2π). We expect the corresponding correction term in the security analysis due to the discretisation effect to be negligible46. Second, in the security proof, we modify the phase encoding and postprocessing procedures, as shown in Table A1 and A2 based on two random phases X-basis processing, she calculates the relative phase difference θa in the range of [0, π) and a raw key bit κa. We modify these procedures as follows: in addition to the two random phases A1 and A2, respectively. During the X-basis processing, she calculates the relative phase difference For convenience in the security proof, we slightly modify the scheme described in Box d = D) j1 in the first round. Based on the values of With the modification above, Alice further generates a random bit In Fig. A1. The controlled-phase gate is defined asa† and a are the creation and annihilation operators, respectively, of mode A1. Alice also applies a controlled-phase gate A1.Here Alice applies a controlled-phase gate Z-basis and X-basis postprocessing, respectively.In the entanglement-based mode-pairing scheme, Alice and Bob generate the composite encoding state M on A1 and A2. The construction of M is described in Supplementary Note M, Alice (same for Bob) is able to determine at which two locations to perform the global photon number measurement after Charlie’s announcement. With this measurement, Alice and Bob can further reduce the encoding state to a two-mode scheme. The detailed security proof is provided in Supplementary Note A key point in our security proof is that we replace the random phases and register them into purified systems mentclass2pt{minimν to estimate the parameters q11 and State preparation: In the i-th round , Alice prepares a coherent state Ai with an intensity ν, μ} (0 < ν < μ < 1) and a phase Bi.Measurement: , Alice and Bob establish a strategy for grouping two clicked rounds as a pair. A specific pairing strategy is introduced in Box Basis sifting: Based on the intensities of two grouped rounds, Alice labels the ‘basis’ of the data pair as: Z if one of the intensities is 0 and the other is nonzero;X if both of the intensities are the same and nonzero; orZ and X bases; or‘0’ if the intensities are , which will be reserved for decoy estimation of both the ‘discard’ when both intensities are nonzero and not equal.X, Z, ‘0’, or ‘discard’) and the sum of the intensities i, j. If the announced bases are the same and no ‘discard’ state occurs, they record the pair basis and maintain the data pairs; if one of the announced bases is ‘0’ and the other one is X(Z), they record the pair basis as X(Z) and keep the data pairs; if both of the announced bases are ‘0’, they record the pair basis as ‘0’ and maintain the data pairs; and otherwise, they discard the data. See also Table See also Table Key mapping:  that corresponds to the simple mode-pairing strategy in Box p during each round, and the maximal pairing interval l.We calculate the expected pairing number Fk, Rk) of the k-th pair, Alice and Bob also record the starting location Sk, which indicates the location at which the first successful detection signal occurs during the pairing procedure for the k-th pair. If the second successful detection signal Rk is found within the next l locations, then Fk = Sk; otherwise, Fk will be larger than Sk. Let Gk ≔ Sk+1 − Sk denote a random variable that reflects the location gap between the k-th and (k + 1)-th starting pulses. Then the expected pairing number per pulse is given byGk. First, we split it into two parts,Hk: = Rk − Sk and For calculation convenience, we assume that in addition to the front and rear locations (It is easy to show that Hk is more complex. Suppose that we already know the expectation value The calculation of the pulse interval 59, who consider a similar MDI-QKD protocol that match the clicked data to generate key information. Under the assumption that the single-photon distributions in all the Charlie’s successful detection events are independent and identically distributed, the authors simulate the performance of the protocol and show its ability to break the repeaterless rate-transmittance bound.After we submitted our work for reviewing, we became aware of a relevant work by Xie et al.Supplementary Information


We derive a variety of results on this problem. The problem is strongly NP-hard even on graphs of treewidth 2, and it does not allow a polynomial time approximation scheme (unless P=NP). Furthermore, we discuss the performance of a simple greedy algorithm, and we construct and analyze a number of parameterized and exact algorithms.An instance of the non-preemptive tree packing problem consists of an undirected graph  The tree packing problem of Nash-Williams. For a given undirected connected graph w(e) of these spanning trees. For example, consider Fig. G is a complete graph on three vertices, and Williams  considermentclass2pt{minimNash-Williams  derives w(e) time units. The objective now is to activate the edges in such a way that the graph remains connected for the longest possible overall time. Figure The tree packing problem may also be interpreted as a scheduling problem: Every edge entclass1pt{minimaentclass1pt{minimaThe non-preemptive version of tree packing. We consider a non-preemptive variant of the above tree packing problem, where the execution of edges must not be preempted: Every edge e is activated at some time point non-preemptive tree packing (N-TreePack for short), and the optimal objective value for a graph In the example in Fig. Related literature. The non-preemptive tree packing problem can be understood as a topic with connections to several different areas. First of all, it can be understood as a non-preemptive scheduling problem equipped with a structurally rich global constraint. In scheduling it is the conventional case to assume non-preemption of the scheduled tasks . This sf Pinedo  the comptime-varying graphs (TVG). These are graphs over time, where every edge is equipped with the information at which time it is available. Time-varying graphs are used to model networks, where change in the network topology is an inherent part of the system. This includes for example delay-tolerant networks where information links often fail, like satellite networks, vehicular or passenger networks.Our model is also strongly related to the area of constant connected  TVGs anonnected , or to eonnected . Our proFinally, our problem is a modification of the classic tree packing problem  the greedy algorithm fails to find an optimal solution. We show by means of a gap-reduction that (unless P=NP) problem N-TreePack does not allow a polynomial time approximation algorithm with worst case ratio strictly better than 7/6; this of course excludes the existence of a PTAS.With respect to polynomial time approximation, we introduce a simple greedy algorithm which has a worst case guarantee of N-TreePack where both the treewidth and the maximum edge weight are bounded by a constant k allows an FPT-algorithm whose running time is linear in |E|.  Furthermore we design an exact algorithm for N-TreePack whose  running time is bounded by Finally, we derive a number of FPT-results in the area of parameterized complexity. The special case of Organization of the paper. Section N-TreePack, and Sect. a, b] denotes the time slot starting at a and ending at b. For an integer t-th time slot or time slot t. Every graph edge cutinduced subgraph of G by v from G, we obtain the graph We write  by West .N-TreePack is a weighted graph , where schedule for instance  is a map e to its activation time e, the activity interval of e is t-th time slot, and we let V with all the edges that are active in the t-th time slot. Finally, we define the objective valueAn instance for problem G is connected does not necessarily have to be one continuous time interval. Formally, if t, where We clarify a few points in this definition: First, the time during which Throughout the paper, we investigate the (parameterized) complexity of non-preemptive tree packing with respect to different graph classes and structural graph parameters.treewidth of a graph is a popular parameter in the area of parameterized algorithms and measures in some sense how close the graph is to a tree. It can be defined as the minimum integer G is subgraph of a k-tree . For graph Q], , ..., Graph Since graph Graph Since The induction yields for Q], , ..., Q that start at an even multiple of Q) and fully-connected during the time intervals , , ..., Q that start at an odd multiple of Q).The graph Q, 2Q], , ..., Q], , ..., n groups with sum Q, we get a solution for the instance of 3-Partition.Next, consider the subgraph 3-Partition instance has a solution, then we build a schedule Vice versa, if the Hamilton-3-regProblem Instance: A bipartite, 3-regular graph Question: Does The reduction is done in two steps. The first step transforms an instance Hamilton-3-reg into a new 4-regular graph H with the properties described in Lemma H from Lemma N-TreePack.In this section, we show that it is NP-hard to decide whether there exists a schedule of objective value at least 7. The reduction is from the following version of the Hamilton cycle problem; see Akiyama, Nishizeki & Saito Problem Hentclass1pt{minimaentclass1pt{minimaHamilton-3-reg as input and outputs a 4-regular bipartite graph H together with an edge special edge), so that the following holds: (i)Hamilton-3-reg, then the new graph H contains a Hamilton cycle that traverses the special edge If (ii)Hamilton-3-reg, then the new graph H has no Hamilton path starting in vertex u.If There is a polynomial time algorithm that takes an instance H then consists out of the following parts: There are two copies of H is bipartite and 4-regular, so it remains to prove claims (i) and (ii).The reduction is sketched in Fig. C in H using the edge u, and traverse all the vertices of H which uses the edge For claim (i), assume that H has a Hamilton path u, we have to prove that P. Note that if P necessarily switches between P first encounters P switches between P which lie in For claim (ii), we prove the contrapositive. Assume that H be a 4-regular bipartite graph as described in Lemma H, and let G, w) of N-TreePack from H. An informal sketch of this instance  is depicted in Fig. G has the vertex setG has the following edges and edge weights:U and Z, the graph G has exactly the same edges as the graph H. For every such edge e, we set Between the vertex sets i-th gadget of type L and has edges and edge weights as depicted in Fig. For every i-th gadget of type R and has edges and edge weights as depicted in Fig. For every gadgetC.Finally, the two edges Now let G, w) allows some schedule i-th gadget of type L will consist of either one or two connected components under schedule fully-connected during the t-th time slot. If there are two connected components  be the instance described above. Every schedule of objective value 7 for  satisfies the following. (i)R is fully-connected during time slots 2 and 6, and semi-connected during each of the remaining five time slots.A gadget of type (ii)For each (iii)C is fully-connected during time slot 4, and semi-connected during each of the remaining six time slots.The gadget Let  be the corresponding N-TreePack instance. If H contains a Hamilton cycle which uses the special edge, then Let the graph entclass1pt{minimaW using R are fully-connected during time slots 2 and 6, and semi-connected otherwise.All gadgets of type L are fully-connected during time slots 1, 3, 5, and 7, and semi-connected otherwise.All gadgets of type C is fully-connected during time slot 4, and semi-connected otherwise.The gadget All edges All edges All edges All edges It is easy to see that a schedule t, which edges are scheduled and which gadgets are fully connected during the the t-th time slot H form a matching of H during the time slots 1,2,3,5,6, and 7, and form a Hamilton path of H during the time slot 4. By checking each of the cases Let mentclass2pt{minimH and the special edge G, w) be the corresponding N-TreePack instance. If H contains a Hamilton path starting at vertex u.Let the graph entclass1pt{minimav, which are both in E(H) and active during the t-th time slot. The strategy of the proof will be to repeatedly deduce some conditions for So assume there exists a schedule t-th time slot both the gadget C are semi-connected. But of course at least one edge incident to t-th time slot. Hence First, recall Lemma k vertices is acyclic, has k edges, and therefore has k connected components. But because R and the gadget C are semi-connected during time slot 1, this implies that every single gadget of type L is actually fully-connected during time slot 1. The same argument holds for time slot 7. In total, together with Lemma L is fully-connected during the t-th time slot, if and only if Next, consider the graph L are semi-connected during time slot 4. For the sake of contradiction, assume L and R are semi-connected. We also have for all H starting at Finally, we claim By combining Lemmas N-TreePack it is strongly NP-hard to decide whether there exists a schedule of objective value at least 7. For All edge weights in the above reduction are in the set N-TreePack is strongly NP-hard, even if all edge weights are in Problem L. Figure Edges of weight 7 only show up in the gadget of type N-TreePack instances with optimal objective value 6 and N-TreePack instances with optimal objective value 7, we also get the following in approximability result.As it is NP-hard to distinguish between N-TreePack with worst case guarantee better than 7/6. Unless P=NP, there is no polynomial time approximation algorithm for In Sect. N-TreePack on a graph with m edges, it can be decided in For an instance of E into set Let G after contracting the edge set G after contracting the edge set F is acyclic in graph The matroid F is acyclic in graph The matroid The matroid Recall that a base of a matroid is a maximal independent set. By construction, the bases of F is a base of F induces a spanning tree of G.If a set F is a base of F induces a spanning tree of G.If a set F is a base of F induces a spanning tree of G.If a set Now observe that scheduling an edge of weight 1 is equivalent to choosing in which of the three time slots 1,2 or 3 it is scheduled. Scheduling an edge 2 is equivalent to always schedule it in time slot 2 and choose whether it is scheduled in time slot 1 or 3. From these facts together with Lemma By Lemma mentclass2pt{minimlgorithm . \documeBy a similar (but simpler) argument we can also decide in polynomial time whether t. By t.We introduce a greedy algorithm that maintains connectivity by always activating edges of the largest possible weight. Formally, we let c components, we turn it into a tree by adding a maximum weight subset A exists, the Greedy algorithm terminates. Now the Greedy algorithm starts its work by initializing n vertices and for every For every graph For the positive result, we consider the time slot mentclass2pt{minimn vertices with weights For the negative result, we consider the complete graph e Palmer ) says th(i)G is a cactus graph.(ii)N-TreePack instance  to optimality.For every choice For every connected graph N-TreePack instance into smaller instances that do not interact with each other, it is sufficient to prove the statement for two-connected cactus graphs. Hence we will assume that G is a cycle on n vertices, and we let G, w) equals We first show that (i) implies (ii). Since cut-vertices split an H on the four vertices In order to show that (ii) implies (i), we first consider the graph G does contain a subdivision w(e) of the edges H; if an edge G, w) satisfies Now let N-TreePack is fixed parameter tractable with respect to various parameters. As the problem is already NP-hard even for highly restricted graphs, we can only hope for positive parameterized complexity results for quite restricitive graph parameters, or when additionally the edge weights are restricted. Note that Theorems N-TreePack, even if either the treewidth or the edge weights are bounded by a constant.In this section, we show that problem N-TreePack is fixed parameter tractable with respect to t the treewidth of the input graph.Problem t, there is a vertex t . As every edge incident to v has weight at most T. We introduce Boolean variables As the graph  theorem  implies N-TreePack is solvable in exponential time On input graphs T all graphs E|! cases. Each case is easily checked and evaluated in By Lemma N-TreePack is fixed parameter tractable with respect to the size k of a feedback edge set. There is a kernel with Problem Note that the input graph mentclass2pt{minimNow the kernelization procedure is clear: As long as N-TreePack is tractable on instances that in a certain sense are close to the preemptive tree packing problem of Nash-Williams.The last theorem of this section shows that problem G, w) be an instance of N-TreePack on m edges, so that k edges have weight at most k. Then an optimal solution can be found in Let  We have shown that


Graph theory is a useful mathematical structure used to model pairwise relations between sensor nodes in wireless sensor networks. Graph equations are nothing but equations in which the unknown factors are graphs. Many problems and results in graph theory can be formulated in terms of graph equations. In this paper, we solved some graph equations of detour two-distance graphs, detour three-distance graphs, detour antipodal graphs involving with the line graphs. The recent rapid growth in the Internet of things has necessitated the development of new approaches to persistent issues in wireless sensor networks. These issues include minimum obstacles in the end-to-end communication path, location accuracy, latency, and delay, among others. These problems can be mitigated by using the distance graph to create local algorithms, i.e., algorithms with minimum communication rounds. We study distance graph applications in wireless sensor networks with a focus on minimum path obstacles and high localization accuracy.A wireless sensor network (WSN) is a network of tiny wireless sensors that can sense a parameter of interest. The sensed data is forwarded to a base station through the formed ad hoc network of sensor nodes. There are many application areas of WSNs, including M2M communication and the Internet of Things (IoT). WSNs are a fundamental building block of smart homes, smart workplaces, and smart cities, among others. It has a lot of other essential purposes for modern technology, such as scientific research, rescue operations, and scientific discoveries. As sensors are a power constraint tiny devices, energy conservation for extending the network’s lifetime is a challenging issue. A wireless sensor network’s lifetime heavily depends on innovative schemes that mitigate energy consumption. The distance graph is used to form and localize an ad hoc network of sensor nodes so that sensed data can be forwarded to the base station with little energy cost.Therefore, finding the solutions to these graph equations is essential. A lot of research has been done by many researchers during the past fifty years, and their results have made a significant contribution in graph theory.X be a nontrivial finite connected graph. Every graph X with detour distance D defines a metric space. The n-distance graph of X, denoted by u and v are adjacent in u and v in this graph are adjacent if and only if detour n-distance graph of X. If n equals the detour diameter of X, then this graph is called the detour antipodal graph of X, which is denoted by X is said to be a detour self n-distance graph if Let G is the graph whose vertices correspond to the edges of G and wherein two vertices are adjacent in G.The line graph In this work, we consider the detour distance graphs. Specifically, we solve the graph equations involving detour two-distance graphs, detour three-distance graphs, detour antipodal distance graphs, line graphs, and the complement of graphs. In addition, we solve some equations involving two-distance graphs, three-distance graphs, and antipodal distance graphs with the graphs mentioned earlier.X such that n-distance graph equations, which can be found in [Harary, Heode, and Kedlacek first studied the two-distance graph. They investigated the connectedness of two-distance graphs. This graph and the relationship between the two-distance graph and line graph was further studied in ,6,7,8,9.found in . S. K Sifound in ,15,16,17found in ,19,20. Ifound in .Motivated by the results listed above, we solved some graph equations of distance graphs and line graphs. Other graph-theoretic terms and notations that are not explicitly defined here can be found in .G of these equations.First, we consider some graph equations of type The main result of solving graph equations involving detour two-distance graphs we prove in this section is the following:Theorem 1.Let G be a graph. Then,If G is connected, then(a)(b)(c)(d)(e)(f)(g)(h)(i)Lemma 1.Let Proof. Clearly Proposition 1.Let Proof. Clearly Proposition 2.Let G be a graph.If G is connected non-unicyclic, then If G is disconnected, then Proof. G be connected non-unicyclic. Suppose that G is unicyclic, since G is connected, which is a contradiction to our assumption that G is non-unicyclic. Thus, Let G be disconnected. Then, Let □Proposition 3.Let G be a connected unicyclic graph but not a cycle. Then,Proof. G. Let this cycle be v be a vertex in G which is not in the cycle of G. Without loss of generality, we may assume that v is adjacent to Let If G is the graph other than these graphs, then by the argument of part (a), If G, the graph k. Clearly G is the graph obtained from By the structure of By part (c), If G is the graph other than these graphs, then the maximum length among all the cycles in If By part (e), By part (c), G is the graph other than the graph If By part (c), The proof is similar to the proof of part (h), since By part (e), By part (e), Therefore, By part (a) If G is the graph other than these graphs, then by the argument of part (a), If □Combining Propositions 1–3, we get the proof of Theorem 1.The main result on solving graph equations involving three-distance graphs we prove in this section is the following: Theorem 2.Let G be a graph. Then,If G is connected, then(a)(b)(c)(d)(e)(f)(g)To prove the above theorem, we start with the following:Lemma 2.Let Proof. Clearly, Proposition 4.Let Proof. Clearly, Proposition 5.Let G be a graph.If G is connected non-unicyclic, then If G is disconnected, then Proof. G be connected non-unicyclic. Suppose that G is unicyclic, since G is connected, which is a contradiction to our assumption that G is non-unicyclic. Thus, Let G be disconnected. Then, Let □Proposition 6.Let G be a connected unicyclic graph but not a cycle. Then,Proof. G, the graph k. Clearly, By the structure of By part (a), If G is the graph other than these graphs, then the maximum length among all the cycles in If By part (c), By part (a), (f)-The proof is similar to the proof of part (b), since (h):(i)-The proof of part (i) and (j) are similar to the proof of parts (c) and (d), respectively,(j):since □Combining Propositions 4–6, we get the proof of Theorem 2.The main result on solving graph equations involving detour antipodal graphs we prove in this section is the following:Theorem 3.Let G be a graph. Then,Lemma 3.Consider the graph Lemma 4.Let Proof. The proof follows directly from the definition of a detour antipodal graph. □Proposition 7.Let Proof. Clearly Proposition 8.Let G be either connected non-unicyclic or a disconnected graph. Then,Proof. We need to consider the following cases.Case (i): Let G be connected non-unicyclic. Suppose that G is connected and unicyclic, which is a contradiction to our assumption that G is non-unicyclic. Hence, Case (ii): Let G be disconnected. Then, The proof follows from the above cases. □Proposition 9.Let G be a connected unicyclic graph but not a cycle. Then,Proof. (1):G such that G. We claim that G is k. Therefore, Clearly, (2):If G is the graph other than these graphs, then by the argument of part (a), If (3):If G is non-isomorphic to the above-mentioned graph. Then, by the similar argument used in the proof of part (1), we get Now, we assume that (4):By part (3), (5):G and G or By part (1), we have (6):By part (1) and Lemma 3, none of the graphs (7):G. Furthermore, G is unicyclic. Hence, none of G. By part (3), we have G.If (8):If Thus, By part (3), we have (9):If By part (3), we have (10):If G is the graph other than these graphs, then by the argument of part (a), If □Combining Propositions 7–9, we get the proof of Theorem 3.Given a set of wireless sensor nodes and connections, graph theory provides a useful tool to simplify the many moving parts of dynamic systems. In this work, we mainly focused on the study of detour distance graph equations. In particular, we solved some graph equations involving detour two-distance graphs, detour three-distance graphs, detour antipodal graphs and line graphs. This solution is believed to be useful for many researchers and businesses working in wireless sensor networks.


Recently, several advances have been made on seeding methods for fast sequence comparison. We combine two such methods, syncmers and strobemers, in a novel seeding approach for constructing dynamic-sized fuzzy seeds and implement the method in a short-read aligner, strobealign. The seeding is fast to construct and effectively reduces repetitiveness in the seeding step, as shown using a novel metric E-hits. strobealign is several times faster than traditional aligners at similar and sometimes higher accuracy while being both faster and more accurate than more recently proposed aligners for short reads of lengths 150nt and longer. Availability: The online version contains supplementary material available at 10.1186/s13059-022-02831-7. Aligning Illumina sequencing reads to a reference genome is the first step in many analysis pipelines. Due to the fundamental role of short-read alignment in bioinformatics, there has been considerable work in this area. BWA , 2, BowtA pioneering approach was the use of minimizers , 16 as sseed-filter-extend approach, where the seeding and filtering are at the heart of an aligner’s performance. In  denote a k-mer substring at position i in S covering the k positions s. We will consider 0-indexed strings. We let subsequence of a string as a set of ordered letters that can be derived from a string by deleting some or no letters without changing the order of the remaining letters. A substring is a subsequence where all the letters are consecutive. Our fuzzy seeds produced from S are subsequences of S since they consist of two syncmers S. The syncmers are concatenated after their extraction to a string We use h to denote a hash function m and use We use 2-bit encoding to store nucleotides with 00, 01, 10, and 11 representing A, C, G, and T. With this bit encoding, each string is associated with a sequence of bits that can be interpreted as an integer. For example, k-mer subsampling method described in  as E-hits. The connection to read alignment is immediate. A seed of length s is captured in a read of length r if the read starts at any of the G| if the length of the genome. Since every seed is captured with the same probability Let expected hit rate. The difference between E-hits and the expected hit rate is that the denominator is k-mers and 20 for minimizers with density 1/5  as defined in , hence Eentclass1pt{minimain BLAST . The E-vStrobealign has a peak memory usage of about 25–33 Gb for hg38 (Additional file Additional file 1: Supplementary notes and figures.


N = 256, compared with the BP bit-flip-1 (BPF-1) algorithm at The Belief Propagation (BP) algorithm has the advantages of high-speed decoding and low latency. To improve the block error rate (BLER) performance of the BP-based algorithm, the BP flipping algorithm was proposed. However, the BP flipping algorithm attempts numerous useless flippings for improving the BLER performance. To reduce the number of decoding attempts needed without any loss of BLER performance, in this paper a metric is presented to evaluate the likelihood that the bits would correct the BP flipping decoding. Based on this, a BP-Step-Flipping (BPSF) algorithm is proposed which only traces the unreliable bits in the flip set (FS) to flip and skips over the reliable ones. In addition, a threshold  Polar code is the first error-correcting code  which haThe Successive Cancellation (SC) algorithm, which was proposed by Arıkan , is one To improve the original BP decoding performance, many methods to improve the BP-based algorithm have been proposed—for example, the BP list (BPL)  decoder.T (CS-T) [T denotes the size of the CS. By analyzing the behavior of the incorrect decoding results of the bit-flipping BP decoder with CS, a BP decoder with multiple bit-flipping sets (BFSs) and stopping criteria (BP-MF-MC) was proposed [BP flipping algorithms identify and then flip error-prone positions to improve decoding performance. For polar codes, bit-flipping strategies have been applied to both SC and BP decoding. However, the principles of flipping strategies on SC and BP decoding are different due to their different decoding schedules. The key to the BP flipping algorithm is to construct a flip set (FS), which consists of the error-prone bits. FS can accurately indicate the error bits, thus flipping the bits within FS can narrow the search space. Initially, the FS was constructed in SC , which pT (CS-T) , where Tproposed . The CS proposed  with a rosed [Ω) , where t decoder  was propA stepping strategy is proposed. We first analyze the behavior of FS and find that only a few bits in the FS could correct the error frame. Therefore, a concept to evaluate the likelihood of bits correcting the trajectory of BP decoding is presented to judge whether the bits in FS should be flipped or not. The judgement condition determines whether flipping the bits in the FS is helpful in correcting error frames.Based on the stepping strategy, an optimization algorithm for the BP flipping algorithm, the BP step-flipping (BPSF) algorithm, is proposed. The algorithm flips only unreliable bits in FS and steps reliable bits to shrink the number of flipping attempts necessary. Similarly, the stepping strategy is also added into the GBPF-lgorithm ,18 to reIn addition, we notice that some effective flipping bits may be skipped over when the LLR magnitude is small. We further propose the enhanced BPSF-This paper aims to reduce the number of decoding attempts in the BP flipping algorithms. The main contributions are summarized as follows:The remainder of this paper is organized as follows. r, In this paper, we use calligraphic characters, such as N independent copies of binary-input discrete memoryless channels are converted to N split channels with different capacities [After channel combining and channel splitting, pacities . Some ofP , where N is the code length of the polar code and K represents the length of the information bits. Meanwhile, R = K information bits comprise r-bit CRC. The set of information bits and frozen bits are denoted as n-th Kronecker power of Polar code can be represented as n matrix , and F⊗nThe BP decoding is initiated from the received value N sub-channels by channel splitting, defining i = 1, 2, …, N. Let the LLR of A binary input memoryless channel N is represented by an n-stage DFG. We use  to indicate the nodes of the DFG, where i indicates node index and j indicates column index. The leftmost nodes in the DFG mean j = 0, such as the blue and black nodes in the j = n, as shown in the grey node column.The BP algorithm for polar codes is based on a DFG. A polar code with code length j-th received bit. The The classic BP DFG is depicted in  used in . Lj,i anThe maximum number of iterations The bit-flipping strategy is a feasible method with which to improve the performance of BP-based algorithms. Due to the parallelism of BP decoding, there may be more than one bit that could correct the error frame by flipping it. Not only can the real error bits correct the error frames, but there are still other bits that can correct the error frames in the process of the iterative computation of the DFG . HoweverInitially, the CS is used to identify unreliable bits in the SC flipping decoder, which is composed of the first bit index of each Rate-1 node . As showFor the GBPF decoding ,18 algorSpecifically, the GBPF flipping operation can be written asThe oracle-assisted BP (OA-BP) decoder  knows whAlgorithm 1 BP flipping algorithm. 1:Input: 2:Output: 3:4:if CRC, which flips the left message The original flipping strategy used in BPF  is to flSimilarly, inspired by the GBPF-lgorithm ,18, we dThe BPSF-Algorithm 2 EBPSF-2 decoding. 1:Input: T2:Output: 3:4:if CRC algorithm is close to the BPF-T = 39) algorithm in the average number of iterations. For T = 39, the EBPSF-1  outperforms the CA-SCL (L = 4) decoder by 0.13 dB when BLER = N = 512 is depicted in T = 60 and T = 60, the performance of the EGBPSF-1 (L = 2) decoder.When N = 64 is illustrated in T = 32, the EBPSF-2  and (L = 8). N = 256. The EBPSF-2  decoder by 0.02 dB at BLER = The two-bit flipping performance for To reduce the average number of iterations in the BP flipping algorithm, this paper proposes a BP flipping algorithm with the stepping strategy. To narrow the search space, the reliable bits are skipped by the stepping strategy to improve the accuracy of flipping bits. The magnitude of LLRs are used to determine whether the bits are reliable or not, skip over the reliable bits in the FS, and flip the unreliable bits to reduce the average number of iterations. Furthermore, to make the algorithm more robust, we propose the threshold


Nature 10.1038/s41586-022-04893-w Published online 4 July 2022Correction to: y-axis label of Fig. 5, where in “mF” appeared in error as “mV”; the label has been amended in the HTML and PDF versions of the article.In the version of this article initially published, there was an error in the top


Communications Biology 10.1038/s42003-022-03583-7, published online 29 June 2022.Correction to: The original version of this article contained errors in the text.In the original version of the Article, the third paragraph of the Introduction incorrectly stated “LKB1 is also regulated by direct phosphorylation of eleven of the twelve AMPK-related kinases…”. The is now corrected to, “LKB1 also regulates by direct phosphorylation eleven of the twelve AMPK-related kinases…”In the Introduction section, page 2 “precipitates” had been changed to “precipitate”, which is now corrected.In the Results section, page 2, errors were introduced that have now been corrected:Fig. 1dI, e-g now reads Fig. 1di, e-gFig. 1dII, e-g now reads Fig. 1dii, e-gFig. 1dIII, e-g now reads Fig. 1diii, e-gIn the Discussion section page 11, “Further support for this proposal is provided by the rank order by the severity” has been corrected to“Further support for this proposal is provided by the rank order of severity”In the Discussion section, page 11 a comma has been deleted after “while by contrast,” so that the sentence now correctly reads: “while by contrast the peak of the Sustained Phase of both responses remained unaltered”.These corrections have been made in the PDF and HTML versions of the Article.


In the article titled “Role of Baicalin in Anti-Influenza Virus A as a Potent Inducer of IFN-Gamma” , there aThe corrected figure, as approved by the editorial board, is shown below.


Selecting a subset of k-mers in a string in a local manner is a common task in bioinformatics tools for speeding up computation. Arguably the most well-known and common method is the minimizer technique, which selects the ‘lowest-ordered’ k-mer in a sliding window. Recently, it has been shown that minimizers may be a sub-optimal method for selecting subsets of k-mers when mutations are present. There is, however, a lack of understanding behind the theory of why certain methods perform well.a, b, n)-words, and an upper bound for minimizers. As a demonstration of our results, we modified the minimap2 read aligner to use a more conserved k-mer selection method and demonstrate that there is up to an 8.2% relative increase in number of mapped reads. However, we found that the k-mers selected by more conserved methods are also more repetitive, leading to a runtime increase during alignment. We give new insight into how one might use new k-mer selection methods as a reparameterization to optimize for speed and alignment quality.We first theoretically investigate the conservation metric for k-mer selection methods. We derive an exact expression for calculating the conservation of a k-mer selection method. This turns out to be tractable enough for us to prove closed-form expressions for a variety of methods, including (open and closed) syncmers, (supplementary methods are available at https://github.com/bluenote-1577/local-kmer-selection-results. os-minimap2 is a modified version of minimap2 and available at https://github.com/bluenote-1577/os-minimap2.Simulations and Bioinformatics online. We now define a local k-mer selection method.We will sometimes refer to a k-mer selection method as a selection method or just a method when the value of Definition 2.A method f is a q-local method if for every of length after an appropriate shift in the position of the k-mers.q = 1 implies that the method can be defined by examining all k-mers and deciding if each k-mer is selected or not. We will always assume that In other words, a q-local method is just defined on Theorem 1. Let f be a q-local method. If two strings share a region of length , i.e. , then every k-mer in is also in and f(S) (if ignoring the index of the starting position).Proof. Follows easily from the definition of q-locality.□Let Σ be our alphabet. We will be implicitly dealing with nucleotides , one can easily see density is equal to i is not near the edges (the start/end) of S. The issue near the edges is that in Definition 2, a k-mer near the start of S appears in less of the Given a long uniform random string nimizers .2.2.1Definition 5.Given a triple where w, k are integers and is an ordering on the set of all k-mers, a minimizer outputs the smallest k-mer appearing in a -mer, or equivalently a window of w consecutive k-mers.w-local method with a w-window guarantee by examining all windows of w k-mers and selecting k-mers inside this window. k and w are application-dependent parameters that control for density and k-mer size, so there is one free parameter which is the ordering From now on, when we specify the smallest value in a window, ties are broken by letting the leftmost k-mer be the smallest. A minimizer gives rise to a random minimizer to be a minimizer with the ordering defined by a random permutation. charged context.Somewhat surprisingly, the ordering, which can be string-dependent, plays a very important part of minimizer performance. We define a Definition 6Given parameters , a window of consecutive w + 1 k-mers (i.e. a -mer) is a charged context if the smallest k-mer is the first or last k-mer in the window.(Charged contexts). charging appears in A very similar definition of Theorem 2  position of the k-mer selected at the window of w k-mers starting at i. By the above paragraph, the expected number of k-mers is then just Mathematically, let w + 1 consecutive k-mers starting at i is first or last k-mer, or equivalently, if this window is a charged context. If the smallest k-mer is the first k-mer, then iX = i, but w + 1, the probability that the smallest k-mer is the first or last is simply The next key step is to notice that The most well-known class of local k-mer selection methods are minimizer methods, originally appearing in w and k. However, with a specific ordering, one can achieve provably better densities for the same parameters. For example, the miniception  example of a method that is 1-local and has a window guarantee. We also analyze the open syncmer defined in Definition 7Let  be parameters with s < k and . Considering a k-mer as a window of consecutive s-mers, a k-mer is selected if the smallest s-mer appears at position t in the window.(Open syncmer). may not have a window guarantee. Consider the string AAAAA… with t = 2 for any k, s. Remembering the leftmost s-mer is the smallest by convention, the smallest s-mer always occurs at the first position so no k-mers are selected. For t =  1, -words method. The density of this method is W can be constructed, but we will not analyze such methods theoretically since these W are found by an optimization algorithm and thus hard to analyze. We will instead test against more sophisticated W empirically.They offer a simple form for While we have discussed intrinsic features of selection methods such as locality, density and window guarantee, we have not yet discussed the ‘performance’ of selection methods. It turns out that evaluating the performance of a method is subtle and depends on the task at hand. In certain contexts, methods cannot be compared because some tasks such as constructing de Bruijn graphs from minimizers  or countw, the assumption being that a method’s performance is only reliant on the size of the window guarantee. This is not an unreasonable assumption for certain tasks, but it is not applicable to methods without a window guarantee.To compare methods, we will fix the density across methods and evaluate a precise notion of performance which we will define below. In previous studies on minimizers . Let i be a position in S which is away from the edges. Define the event representing whether or not a k-mer at position j is selected by f, and the probability that some k-mer is selected from . We call the probability vector of f.. Define the conservation to be f overlapping the base and (ii) this k-mer is unmutated from S to S and The set Let iX be indicator random variables where iX = 1 if iX are not independent; if S, we get that i in S away from the edges.We now show that 3.3.1i, the k-mers covering i on Given the base i, an unmutated k-mer being one for which all bases are unmutated. These unmutated k-mers are candidates for being in Given some configuration, we will get k in θ. This problem is well-studied; see Chapter 5.3 in From 3.3.2α, we getConditioning on a consecutive unmutated k-mers in S and lE but over x is arbitrary and can be thought of as the starting position of the first unmutated k-mer (the probability does not depend on x). Locality comes into play now; if f is a 1-local method, then the event f. Hence Theorem 3. Let  and f be a 1-local method. Then If f is not 1-local, then follows from the trivial upper bound f is not 1-local, then context dependency problem; if a k-mer is selected in S and the same k-mer is also in If 3.3.3Theorem 4. For i.i.d Bernoulli trials with success probability and For the case , the probability of successes is just Even though the successful runs problem is solved for general parameters, the probability formula is derived by manipulating generating functions and is a bit unruly. For Theorem 5. 1 be a 1-local method and fn be anLet f local method with n > 1 such that the densities of both methods are equal. That is, for a long uniformly random string S, . Then f, by linearity of expectation, we haveProof. Given any local method x is unique in f(S) and doing the same for S, we can condition on Writing S is uniformly random, x. Therefore, the first term is independent of the method.Now notice that if never be true if x is not unique in S by the definition of 1-locality. However, for an n-local method, it is certainly possible for x is not unique in S. Hence, S is large, the theorem follows. □However, for 1-local methods, all 1-local methods have the same expected number of unique k-mers if S is uniformly random. The same idea in the proof can be re-applied to show that the average expected multiplicity over all k-mers (i.e. how many times it repeats in the f(S)) is also higher when using a 1-local method. We explore the consequences of this in Section 5.2.3.The above result show that 1-local methods may cause more repetitive matches than other methods such as minimizers. In fact, the above proof shows that Given that 1-local methods are superior for conservation, which is a measure of sensitivity, it is natural to ask if such methods have an inherent precision trade-off. In the context of alignment, k-mer matching precision is related to spurious k-mer matchings caused by repeat k-mers, as such matches would force an aligner to evaluate several candidate alignments. In general, unique k-mer matches are easier to handle computationally than repeat k-mer matches. It may therefore be more advantageous for a local selection method to select non-repetitive k-mers. We show that indeed, locality is related to repetitive k-mer selection.k, s, t be given and f be either a closed or open syncmer method. To calculate α consecutive k-mers. Breaking these k-mers into s-mers, we get a window of k = 15 and w = 10. To achieve approximately the same density using an open syncmer, we need s = 10, and it is easy to show that this probability is very small. Since all s-mers are distinct and we assume a random ordering on all s-mers, the relative ordering of s-mers in this window is a uniformly random permutation in i in the window. Determining whether or not a k-mer is selected then amounts to analyzing a random permutation’s smallest elements . If , then If f is a closed syncmer method, then Corollary 1. We prove this theorem in We can also count open syncmers. Unfortunately, the number of permutations is only determined as a recurrence relation and the formula is not as nice. Theorem 7 is proved in Now given a permutation Theorem 7 Using parameters k, s, t as defined in Definition 7, let and be the number of permutations in such that for some window the smallest element is . Define . Then (Successful permutations for open syncmers). We define as where the subscript indicates falling factorial, and if t. In t was investigated on the performance of open syncmers, but no explicit conclusion was made about how to choose t. Below we prove a theorem that says that the parameter t should be chosen to be the middle position of a window of One can divide Theorem 8. Let . Then for any valid choice of t.t. By Theorems 3 and 8, we can rigorously justify the optimal value for t to maximize conservation. Thus, t is not actually a free parameter when optimizing for conservation.We prove Theorem 8 in w, k be parameters for a random minimizer method f. To calculate α consecutive k-mers, we now have to look all k-mers in all windows containing any of these α consecutive k-mers. As in the syncmer case, we will assume all k-mers are distinct.Let α consecutive k-mers, we need to also know the ordering of the w‒1 k-mers to the left and to the right of these α k-mers because they are included in some window containing one of these consecutive k-mers. This gives us Given nS satisfying a general condition.Theorem 9 Given parameters with , let n such that for some windowbe the number of permutations in S , the smallest element is one of . Then where and using to mean the falling factorial,(Successful permutations for random minimizers). α consecutive k-mers since the p parameter describes the leftmost position of the first unmutated k-mer covering position i. Therefore, as before, The specific choice of parameters It was shown to us recently  that forα = 1, we get the desired result that the density is f is a 1-local method, which minimizers are not. We give an example that shows how context dependency leads to lower conservation in When We proceed by counting permutations corresponding to some k-mer being chosen by a random minimizer method. We first define a function a, b, n)-words method which selects k-mers based on their prefix. We prove this result in Theorem 10. under the -words method is where if x < 0.We can also derive the probability vector for the previously  is defined in Section 3.1 as the upper bound on probability vectors. For the miniception method, it is not obvious how parameters affect d so we let k0 and w slightly until we get the density close to d. For the words method, we use two choices of W as W4 and W8 -words method, the closest choice of parameters leading to the most similar density is n = 2, 3 which gives density We first fix W4, W8 and the open syncmer methods perform well compared with the other methods. The large drop in conservation between the empirical random minimizer and the upper bound for the random minimizer indicates that context dependency plays a highly non-trivial role in conservation. We note that for d = 1/4, the -words method performs poorly as it does not take advantage of selecting k-mers with non-overlapping prefixes n = 0.The results in These results also suggest that the best methods already achieve W for each density. We note that since minimap2 was designed with minimizers in mind, we expect the benefits of switching k-mer selection methods to be dampened compared with designing an aligner with open syncmers in mind.We now investigate applications to read mapping. We modified minimap2 , a stateseeds to be matched onto the selected k-mers of the reference. Colinear sets of k-mer matches are collected into chains, and then dynamic programming-based alignment is performed to fill gaps between chains. Our modification was to swap out the k-mer selection method, originally random minimizers, to an open syncmer method instead. Our version of minimap2 can be found at https://github.com/bluenote-1577/os-minimap2.Minimap2 aligns reads using a seed-chain-extend procedure. Roughly speaking, this works by first applying a k-mer selection method to all reads and reference genomes. Selected k-mers on the reads are then used as We operate on four sets of real and simulated publicly available long-read datasets:https://tinyurl.com/uhuwvxb8 and their corresponding assemblies.Real microbial PacBio Sequel long-reads from PacBio, available at https://github.com/marbl/CHM13  and the corresponding assembly CHM13.Real human ONT nanopore reads from Simulated RNA (cDNA) ONT long-read data from Trans-NanoSim  in the dataset and compared the mappings for open syncmers versus minimizers. This is shown in For each alignment of a read, minimap2 computes a t = 3, the optimal parameter of t with t increases for open syncmer methods because the vectors t increases; see t increases is suggestive that goodness of chaining is indeed correlated with conservation, even for PacBio reads which have higher rates of indels than substitutions  ONT data, where we can analyze mapping precision due to having a ground truth. We chose to analyze such data because read lengths are shorter, and in After generating a set of reads from a transcriptome, we mapped the reads back to the genome (GRCh38) using the minimap2-x splice option. For every mapping, we take the primary alignment if the mapQ > 0. We designate a mapping successful if the transcript corresponding to the read overlaps the true gene, otherwise we designate it as an error. We only analyze genes found on the primary chromosomes of GRCh38 and ignore genes found on alternate loci.k to create more seed matches; this was also not investigated in .Increasing k-mer matching sensitivity by lowering k and choosing more conserved methods effectively do the same thing by increasing the number of seed hits. Larger versions of this subplot are shown in Open syncmers and minimizers have similar error/success curves as seen in the top-right of k versus using open syncmers, open syncmers allow for new parameterizations. While the quality of the mapping seems independent of lowering d = 1/7, we noticed that syncmers may offer a sizeable error-runtime improvement for small k, see d; this can also be seen in the top-left of d may be necessary in cases where a large number of genomes must be indexed by k-mers and memory is an issue.For the case of fixed density k is fixed. For large k, this is primarily due to indexing speed, which we did not attempt to optimize. For smaller k, this seems to be due to a larger number of repetitive k-mers and spurious matches as mentioned in Section 3.4. Syncmers take longer when t value is unique when Interestingly, we found that for even values of minished . This ismaximize sensitivity, which is related to quality, and minimize the number of repetitive k-mers in the indexed set, which is related to speed. Further research should investigate if this trade-off can be improved using new local-selection methods and how to engineer aligners that take advantage of the trade-off.In summary, we first described a new mathematical framework for understanding k-mer selection methods in the context of conservation, which allowed us to prove results pertaining to upper bounds, optimal parameter choices and closed-form expressions for conservation. We then investigated conservation empirically and then found that augmenting minimap2 with a more conserved method increases alignment sensitivity as predicted. However, such methods may give rise to more repetitive seed matches, increasing the computational time of alignment. To optimize for quality of mapping versus speed, one should A notable result of ours is that the best methods can already achieve > 0.96 of the upper bound for conservation for certain parameters, implying that major improvements in conservation are not possible. However, real genomes do not consist of i.i.d uniform letters. There is a mix of high-complexity and low-complexity regions in genomes, so techniques for better distributed selection of k-mers should be investigated . TheoretAn orthogonally related recent idea are strobemers , which hbtab790_Supplementary_DataClick here for additional data file.


The proposed cost function is optimized using an iterative algorithm derived from the alternative direction method of multipliers. Moreover, we propose improved versions of the proposed algorithms by incorporating a smoothing penalty function into the cost function. We also introduce angular constrained forms of the proposed algorithms by considering a special case as follows. The imperfect error values are constant over all the projection angles, as in the case where the source of ring artifact is the non-uniform sensitivity of the detector. Real data and simulation studies were performed to evaluate the proposed algorithms. Results demonstrate that the proposed algorithms with incorporating smoothing penalty and their angular constrained forms are effective in ring artifact elimination.Ring artifact elimination is one of the popular problems in computed tomography (CT). It appears in the reconstructed image in the form of bright or dark patterns of concentric circles. In this paper, based on the compressed sensing theory, we propose a method for eliminating the ring artifact during the image reconstruction. The proposed method is based on representing the projection data by a sum of two components. The first component contains ideal correct values, while the latter contains imperfect error values causing the ring artifact. We propose to minimize some sparsity-induced norms corresponding to the imperfect error components to effectively eliminate the ring artifact. In particular, we investigate the effect of using different sparse models,  The sinogram data corresponding to the image containing the ring artifact is also contaminated by incorrect measurement values similar to the example shown in Ring artifact is one of the well-known image degradations that restrict the quality of X-ray computed tomography (CT) images . In addiThis paper is organized as follows. The related work is presented in Section 2. The proposed algorithms are detailed in Section 3. After that, we show the results of the experimental studies in Section 4. Finally, the conclusion is provided in 5.Pre-processing methods: remove the vertical stripes from the sinogram before image reconstruction.Post-processing methods: remove the ring artifact from the image after image reconstruction.In-processing methods: remove the ring artifact during image reconstruction.There are numerous methods that have been investigated to suppress the ring artifact. These methods can be classified into two general categories. The first category contains methods based on the modification of hardware, such as those detailed in , 13. TheA technique based on a combination of wavelet decomposition and Fourier filtering was investigated to eliminate the stripes from the sinogram . Also, aℓ0-norm/ℓ1-norm regularization term to remove the ring artifact from the reconstructed images , where Ai denotes the system matrix corresponding to the i-th projection angle. Similarly, we express the projection data as i-th projection angle. Using these notations, the image reconstruction problem in the angular constrained form can be formulated asθ, which originates from the constraint that values of elements in the imperfect vector are same for all the angles.Generally, it is clear from the different causes of ring artifact that the values of the imperfect vector in the projection data may be dependent on the measured angles. In other words, the imperfect values are variables for all the angles. In addition to this general case, we suggest studying a special case where the ring artifact originates from physical factors related to hardware, such as the existence of some imperfect detector bins. Then, the imperfect elements in ℓ1-norm as an example of the sparsity-induced norm ρ > 0 is a positive constant. Using the augmented Lagrangian function in The formulated cost function can be optimized using any iterative algorithm. For example, first-order primal-dual method, alternative direction method of multipliers (ADMM) , and proα > 0 denotes the step size, L is the maximum number of iterations in the GD method, and μ = 1/ρ. It can be solved directly using a simple formula in closed form via the soft thresholding [The update of image sholding  functionAlgorithm 1ℓ1-ring algorithm.Input the projection data α > 0 and ρ > 0).Set initial value for fork = 1, 2, …, Kdo forl = 1, 2, …, Ldo   end    endOutput the reconstructed image i.e.ℓ0-norm, Huber ℓ1-norm and Huber ℓ0-norm. The result of the derivation shows the following. Almost same algorithms can be obtained for the cases of using four different norms, and the only difference among the four cases appears in the form of the proximity operator in Step 2. For clarity, the proximity operators corresponding to the four cases are summarized in ℓ1-norm, ℓ0-norm, Huber ℓ1-norm, and Huber ℓ0-norm by ℓ1-ring, ℓ0-ring, Hℓ1-ring, and Hℓ0-ring algorithms, respectively.We followed the same steps to derive three iterative algorithms corresponding to the remaining three sparsity-induced norms mentioned above, β > 0 denotes the hyper-parameter that controls the strength of smoothing. In this work, we choose to use a smoothing penalty similar to Markov random fields (MRFs) [ℓ1-norm case, followed by explaining how it can be changed to the other three norms. The cost function in the ℓ1-norm case is given byUj denotes the set of eight neighboring pixels around the j-th pixel, and j-th pixel and the We propose to incorporate a smoothing penalty s (MRFs) . This isAlgorithm 2ℓ1-smth-ring algorithm.Input the projection data α > 0, ρ > 0 and Set initial value for fork = 1, 2, …, Kdo forl = 1, 2, …, Ldo   end    endOutput the reconstructed image ℓ1-smth-ring, ℓ0-smth-ring, Hℓ1-smth-ring, and Hℓ0-smth-ring algorithms, respectively. In addition, we call a collection of these four algorithms using the smoothing penalty by improved algorithms to distinguish them from the algorithms without the smoothing.Similarly to the case without the smoothing penalty, the iterative algorithms corresponding to the other three norms mentioned above are given as follows: change the proximity operator in Algorithm 2, as shown in i.e. the formulation of k. Each element λrθ of the imperfect vector r) in every projection angle (θ) based on each elements drθ of the vector This section describes iterative algorithms for the angular constrained form where elements of imperfect error vector have the same values for all the angles, r = 1, 2, …, R dofor θ = 1, 2, …, Θ do for    endendApplying the angular-independency constraint means that the update of r = 1, 2, …, R dofor  θ = 1, 2, …, Θ do for    endendℓ1-ring, cstr-ℓ0-ring, cstr-Hℓ1-ring, and cstr-Hℓ0-ring algorithms) for the case without the smoothing, and  for the case with the smoothing.Using the update of ℓ1-norm or Huber ℓ1-norm is guaranteed for the following reason. These algorithms can be considered as an application of ADMM to the convex minimization subject to the linear constraint. However, the iterative algorithms using ℓ0-norm or Huber ℓ0-norm are not guaranteed to converge due to the non-convex nature of cost functions. However, as we will show in the next section, the algorithms using ℓ0-norm or Huber ℓ0-norm converge to nice approximate solutions in practice. Also, their power to eliminate the ring artifact is much stronger compared to the ℓ1-norm based algorithms. This observation motivated us to keep the ℓ0-norm based algorithms in this paper in spite of the lack of mathematical guarantee in the convergence. We note that similar behaviors, i.e. approximate solution with the ℓ0-norm works better than the exact solution with the ℓ1-norm, have been observed in other application areas of CS.The convergence of iterative algorithm using μm) thick scintillator was used for measuring the projection data. The number of projection data was 1800 over 180 degrees, and each projection data consisted of 1024 × 1024 pixels. From the conventional FBP reconstruction of this data, the central slice of the reconstructed 3D image was contaminated with strong ring artifact. Moreover, in the simulation data, we used a 2D slice obtained from a dataset of screening chest CT images. The dataset includes 68 volumes for various patients scanned using Hitachi CT-W950SR scanner. Each volume is composed of 25 to 31 transaxial slices, where each slice is composed of 320 × 320 (pixels) with a slice thickness of 10 (mm) and a pixel size of 1 × 1 (mm). The 2D chest slice was resized to 512 × 512 (pixels). The simulation studies were carried out with and without adding noise to the projection data. It is known that the statistical noise in the CT measurements follows Poisson distribution [b0 is the number of average photon counts transmitted from the x-ray source, and bi is the detected photons measured after attenuation by the scanned object at detector bin i. Then, the noisy projection data gi at detector bin i can be calculated using Eqs σ, decomposition level L, and wavelet base function W). On the other hand, to correct the image by the M2 method, some parameters need to be adjusted, such as the ratio between the defective and background values R. Also, the size of the smoothing filters that remove the small to medium and large stripes in the sinogram  is required for executing the M2 method. In order to obtain satisfactory results with the M1 and M2 methods, we selected their parameters based on the best image quality in terms of RRMSE values besides visual evaluation. Furthermore, to study the effect of using the standard ℓ2-norm on ring artifact removal, we compared our proposed algorithms with two algorithms using ℓ2-norm for the data fidelity term with and without smoothing penalty. Additionally, the GD method was used for the optimization problem of these two algorithms, which we name ℓ2 and ℓ2-smth algorithms. All the algorithms were implemented using C++ platform on a machine with Intel(R) core(TM) i7–2670QM CPU @ 2.20 GHz processor and 6 GB RAM. The average execution time of the proposed algorithms for every iteration was 6.0 seconds.To evaluate the proposed algorithms, we performed experimental studies using real data in addition to simulation data. In the real data, some experiments for propagation-based phase-contrast imaging were performed by synchrotron radiation X-ray micro-CT , Japan , 36. A sribution . Assume μx and μz are the mean values of images σxz is the correlation coefficient value of image c1 = (k1*s)2 and c2 = (k2*s)2 where k1 = 0.01 and k2 = 0.03 and s is the dynamic range of the pixel values.To evaluate image quality achieved by the proposed algorithms, we used two image quality metrics throughout the simulation studies. Suppose that M) index , which cρ and δ) control the degree of the ring artifact removal in the proposed algorithms. All the algorithms were implemented using the same values for these parameters ρ, δ, N, and K for the following reasons: to obtain a fair comparison between the different algorithms and to show the effect of the different sparsity-induced norms on the ring artifact elimination. The parameter values used in our implementation are summarized as follows: L = 2, K = 2, 000, α = 125 × 10−3, ρ = 0.004, δ = 0.092, η = 0.001. Also, the parameter values  and  were used for the correction by M1 and M2 methods, respectively. ℓ2 and ℓ2-smth algorithms for the first study case. The true slice, the corrupted image, the reconstructions using all the proposed algorithms, and the images corrected by using the M1 and M2 methods are shown in Figs In this section, we applied the proposed algorithms to the simulated projection data without statistical noise corresponding to the above-mentioned 2D chest slice. The simulated sinogram consisted of 512 (bins) × 1,000 (angles), where the standard parallel-beam geometry with 360° angular range was assumed. We considered three different cases, each of which corresponds to a different setup of the non-uniform sensitivity bins and consequently a different ring artifact pattern in the image domain. The details are summarized as follows. In the first case, we assumed that some non-adjacent bins have non-uniform sensitivities producing the ring artifact. In the other two cases, some adjacent detector bins have non-uniform sensitivities producing the thick ring artifact of multiple pixel width where the maximum width was set to 3 (pixels). We assumed that the imperfect error value in each detector bin is constant with respect to the angle in the first and second cases, and varied with respect to the angle in the third case. Using the sinograms ℓ2 and ℓ2-smth algorithms. On the other hand, it is clear from the reconstructions shown in Figs ℓ1-ring algorithm succeeds in eliminating most of the ring artifact except few rings located near the center. For example, we can notice that one ring appeared with low contrast in the first study case, while very small number of rings still appeared in the second and third study cases. Meanwhile, the reconstructions by ℓ1-smth-ring algorithm illustrate that the remaining artifacts were smoothed well. The reconstructions by ℓ0-ring algorithm produced ring-free images, though very little new artifacts were introduced in the second study case. Additional improvement with the smoothing penalty was able to efficiently suppress these remaining artifacts. It also produced high-quality image closer to the original image without any loss in the fine details. Furthermore, the reconstruction by Hℓ1-ring and Hℓ0-ring algorithms show that they provide similar results to those in ℓ1-ring and ℓ0-ring algorithms, respectively. However, the latter algorithms eliminate more ring artifacts and produce better results than the former ones. The residual artifacts were disappeared well with Hℓ0-smth-ring algorithm and were smoothed well with Hℓ1-smth-ring algorithm, where they still slightly appeared with low contrast. Moreover, it is notable from Figs From ℓ1-smth-ring and ℓ0-smth-ring algorithms and their angular constrained forms outperformed the other proposed algorithms.For quantitative evaluation, we measured the quantitative metrics (RRMSE and SSIM) between the original reference image and the reconstructed image by each algorithm. The obtained metric values are summarized in Tables Finally, from the above qualitative and quantitative evaluation, the proposed algorithms and their angular constrained forms, especially with incorporating the smoothing penalty, are effective in suppressing the ring artifact. They also achieve higher image quality compared with the other methods implemented in this work. 6 (counts/bin) was added to the sinogram of the first study case in the previous section. Using this sinogram, we reconstructed images by the proposed algorithms. The parameter values in the proposed algorithms were as follows: L = 2, K = 500, α = 5 × 10−3, ρ = 0.001, δ = 0.03 and η = 0.01. Also, ℓ1-smth-ring, 55.0 in ℓ0-smth-ring, 75.0 in Hℓ1-smth-ring, 45.0 in Hℓ0-smth-ring, and 150.0 in the angular constrained forms. Moreover, the parameter values in the M1 and M2 methods were the same as in the case without noise. The true image, the corrupted image, and the reconstructions by the proposed algorithms and the compared methods are shown in Figs We also studied the case where projection data contains statistical noise in addition to the ring artifact. In the noisy data case, Poisson noise corresponding to 10ℓ1-ring and ℓ0-ring algorithms did not succeed in completely eliminating the ring artifact, where it only slightly decreased the intensity of some rings. The reconstructions by Hℓ1-ring and Hℓ0-ring algorithms produced similar results to those in ℓ1-ring and ℓ0-ring algorithms, respectively. Meanwhile, the remaining artifacts were weakened with ℓ1-smth-ring algorithm. Also, they were suppressed with ℓ0-smth-ring, Hℓ1-smth-ring, and Hℓ0-smth-ring, where the artifact slightly appeared with low contrast. It can be notable that the reconstructions by Hℓ1-ring and Hℓ0-ring algorithms contained less noise compared to ℓ1-ring and ℓ0-rings algorithms. The reason for this is the existence of the quadratic part in Huber ℓ1-norm and Huber ℓ0-norm cost functions in Hℓ1-ring and Hℓ0-ring algorithms.From the visual evaluation of the results, we can notice that the M1 and M2 methods produced similar results to those in the previously studied noise-free case. Furthermore, we can observe that the reconstructions by ℓ0-ring and Hℓ0-ring algorithms. They also removed most of the rings except some rings located near the center in both ℓ1-ring and Hℓ1-ring algorithms, although some new artifacts were introduced. Additionally, we can observe that the angular constrained forms of the improved algorithms could effectively remove the artifact and produce ring-free images.Concerning the results of the angular constrained forms of the proposed algorithms, it is clear that the angular constrained forms of the proposed algorithm could eliminate all the ring artifact with ℓ1-ring or Hℓ0-ring algorithms. Consequently, we plan to incorporate a statistical model in addition to the sparsity-induced norms for further improvement of the reconstruction with noisy projection data.To perform quantitative evaluation of the proposed algorithms, RRMSE and SSIM values were measured and presented in the last two columns in For additional evaluation of the proposed algorithms, we implemented the proposed algorithms with real data. We used the projection data corresponding to the central slice of the above-mentioned real data. We implemented the proposed method and the compared methods with this data. The results are shown in ℓ2 and ℓ2-smth algorithms and their angular constrained versions (cstr-ℓ2 and cstr-ℓ2-smth) still included the strong ring artifact. Since the real projection data is corrupted with noise, Hℓ0-ring, Hℓ0-smth-ring, and the angular constraint versions of Hℓ1-ring and Hℓ0-ring algorithms could remove most of the ring artifact well. Moreover, cstr-smth-Hℓ1-ring and cstr-smth-Hℓ0-ring algorithms almost completely removed the ring artifact leading to high-quality reconstructions. Finally, we would like to strengthen the following fact. Using ℓ2-norm cannot suppress the ring artifact, whereas the combination of ℓ1-or ℓ0-norm and ℓ2-norm as in Huber ℓ1- and Huber ℓ0-norms is able to remove the ring artifact. This means that the use of sparsity-induced norm is essential in the ring-artifact elimination problem.The results are summarized as follows. Most of the ring artifacts were removed well with the corrections using M1 and M2 methods. However, some remaining rings were still visible. These existing techniques worked well, but were not complete in this experiment. The reconstructed images with i.e. M1 and M2 methods. In summary, we can conclude that the sparsity-induced norms for the imperfect error components of the projection data in the data fidelity as well as the smoothing penalty can effectively eliminate the ring artifact.In this work, we developed a class of algorithms to eliminate the ring artifact from CT images. The first group of algorithms minimizes different sparsity-induced norms of the imperfect error components of the sinogram. In the second group of algorithms, we improved the first class of algorithms by incorporating the smoothing penalty into the cost function to remove the remaining ring artifact. Furthermore, we also introduced angular constrained forms corresponding to the proposed algorithms and their improved versions. The proposed algorithms are able to correct the ring artifact well during image reconstruction and need neither pre-processing nor post-processing steps. We evaluated the performance of these algorithms through a number of simulation and real data studies. In the simulation, we considered three different cases where different setups of the ring artifact patterns were simulated. Both the qualitative and quantitative evaluation results demonstrated that the improved versions of the proposed algorithms and their angular constrained forms are efficient in ring artifact suppression. They also provide significantly better results than the compared methods, S1 File(RAR)Click here for additional data file.


Nature Communications 10.1038/s41467-022-34710-x, published online 16 November 2022Correction to: The original version of this Article contained an error in Fig. 4, in which in panel (b) the residues in the sequence presented were highlighted with a wrong color coding, i.e., the second “S” in the sequence “NALVSRSKGTSKSNFIPFRDSKLTR” was highlighted in pink instead of black. In addition, the arrows in panel (a) have decreased thickness to improve the visibility of the structure. This has been corrected in both the PDF and HTML versions of the Article.


Nature Communications 10.1038/s41467-022-30856-w, published online 07 June 2022.Correction to: In the original version of this Article, the positions of “Morphine” and “Codeine” and the associated chemical structures in the morphinan pathway in Fig. 1a were swapped. These have been corrected in both the PDF and HTML versions of the Article.