Yet TEs play a substantial role in genome evolution and are themselves an important subject of study. Repeat annotation, based on counting occurrences of k-mer counting and indexing of large sequence sets. Unlike previous methods, Tallymer is based on enhanced suffix arrays. This gives a much larger flexibility concerning the choice of the k-mer size. Tallymer can process large data sizes of several billion bases. We used it in a variety of applications to study the genomes of maize and other plant species. In particular, Tallymer was used to index a set of whole genome shotgun sequences from maize (B73) . We analyzed k-mer frequencies for a wide range of k. At this low genome coverage (≈ 0.45×) highly repetitive 20-mers constituted 44% of the genome but represented only 1% of all possible k-mers. Similar low-complexity was seen in the repeat fractions of sorghum and rice. When applying our method to other maize data sets, High-C0t derived sequences showed the greatest enrichment for low-copy sequences. Among annotated TEs, the most highly repetitive were of the Ty3/gypsy class of retrotransposons, followed by the Ty1/copia class, and DNA transposons. Among expressed sequence tags (EST), a notable fraction contained high-copy k-mers, suggesting that transposons are still active in maize. Retrotransposons in Mo17 and McC cultivars were readily detected using the B73 20-mer frequency index, indicating their conservation despite extensive rearrangement across cultivars. Among one hundred annotated bacterial artificial chromosomes (BACs), k-mer frequency could be used to detect transposon-encoded genes with 92% sensitivity, compared to 96% using alignment-based repeat masking, while both methods showed 92% specificity.Here we introduce the Tallymer software, a flexible and memory-efficient collection of programs for .The Tallymer software was effective in a variety of applications to aid genome annotation in maize, despite limitations imposed by the relatively low coverage of sequence available. For more information on the software, see  Repetitive elements abound in the genomes of higher organisms. Tandem repeats, simple sequence repeats, long terminal repeats (LTRs), segmental duplications, and transposable elements (TEs) are among those types commonly found in eukaryotic species. The biological role of these entities in genome evolution has been documented ,2, but fFor anticipated plant genome projects and those currently underway, effective and rapid annotation of their many repeats has acquired a new urgency. For example, the maize genome is estimated to be 60–70% repetitive, mainly in the form of retrotransposons that proliferated in the last 2 to 6 million years -5. ExtenRepeat identification strategies fall under two broad categories: de novo detection and similarity-based detection. RepeatMasker , one of De novo methods address many of these concerns. RECON  currentlk-mers. K-mer frequency based methods were originally used in whole genome shotgun assemblers , ZMMBBb0284N04 [Genbank: AC157977], ZMMBBb0614J24 [Genbank: AC157487], and ZMMBBb0448F23 [Genbank: AC160211]. A polymerase chain reaction was used on template ZMMBBb0382K21 to give a unique 1864 bp sequence [Genbank: AC163004] that served to fill a physical gap between ZMMBBb0483G05 and ZMMBBb0284N04. Overlap between clone sequences allowed assembly of a complete supercontig of 453,421 bp.This study used publicly available datasets, including expressed sequence tags (ESTs), gene-enriched genome fractions, representative whole genome sequences, and repeat libraries, as summarized in Table Zea mays L. ssp. mays), generated by the Joint Genome Institute were downloaded from the NCBI Trace Archive on February 13, 2006 and clipped to remove vector sequences using cross match  into equal sized non-overlapping subintervals at some suitable distance Δ. We obtain the intervalsbe the minimum and maximum possible λmin = -mΔ and (n - 1)Δ ≤ λmax <nΔ. Then for each integer j ∈  we computesuch that λ = jΔ. That is, Ωk, M is the fraction of sequences in M falling into the jth subinterval. Ωk, M is called λ-distribution ratio for M.where k-mers in S. That is, it is derived from the distribution of occ for all k-mers w in S. Now let δS, k be a table such that for all i ≥ 1, δS, k[i] is the number of different k-mers occurring exactly i times in S. For example, δS, k by one.According to these equations we can efficiently compute k-mers are computed by hashing methods. However, these only work, if the number 4k of possible k-mers is considerably smaller than n. This does not hold for our application, where k ranges from 10 to 500. We have developed a different approach based on enhanced suffix arrays  for a range of values of k. This is useful when determining the optimal value for k.Traditionally, occurrence counts for x arrays . The spaTo explain our approach we begin with the concept of enhanced suffix arrays, as introduced in .S consists of r sequences. To process S we concatenate all sequences in S into a long string denoted 1,...,$r-1 between the r sequences. Additionally we add a sentinel character $r following the last sequence in the concatenation. Obviously, k-mers as S. That is, we can compute k-mer counts based on Suppose n + 1. i] denotes the character at position i in i ≤ n - 1. For i ≤ j, i..j] denotes the substring of S starting with the character at position i and ending with the character at position j. For i > j, i..j] denotes the empty string. A substring of h, 0 ≤ h ≤ n, h = h..n - 1] denotes the h-th non-empty suffix of h in Suppose that 1 < ... <$r. This character order induces an order on all nonempty suffixes of n, specifying the lexicographic order of the n + 1 non-empty suffixes of The key to our method is to lexicographically sort the suffixes of n. For each h, 1 ≤ h ≤ n, lcp[h] is the length of the longest common prefix of r is the largest suffix in the lexicographic order, r. Hence we always have lcp[n] = 0. In the following, the combination of the suffix array and the lcp-table is called enhanced suffix array.The lcp-table lcp is an array of integers in the range 0 to k-mers. An interval [i..j], 0 ≤ i <j ≤ n, is an lcp-interval of lcp-value ℓ ifThe notion of lcp-intervals, introduced in  is centri] < ℓ,1. lcp[h] ≥ ℓ for all h, i + 1 ≤ h ≤ j,2. lcp[h = 0 or lcp[h] = ℓ for at least one h satisfying i + 1 ≤ h ≤ j,3. j = n or lcp[j + 1] < ℓ.4. i..j] of lcp-value ℓ. We say that an ℓ-interval [i..j] represents the substring i]..suf[i] + ℓ - 1] of We will also use the shorthand ℓ-interval for an lcp-interval [i..i], 0 ≤ i ≤ n is a singleton interval. We say that a singleton interval [i..i] represents the suffix i]..n - 1] of m-interval [l..r] is said to be embedded in an ℓ-interval [i..j] if it is a subinterval of [i..j]  and m > ℓ. Note that we cannot have both i = l and r = j because m > ℓ. A singleton interval [l..l] is said to be embedded in an ℓ-interval [i..j], if i ≤ l ≤ j. If an interval [l..r] is embedded in [i..j], then [i..j] is the interval enclosing [l..r]. If [i..j] encloses [l..r] and there is no interval embedded in [i..j] that also encloses [l..r], then [l..r] is called a child interval of [i..j].An interval [n]. An important property of the lcp-interval tree is the fact that it implicitly stores the number of occurrences of all substrings of i..j] represents a string occurring j - i + 1 times in The parent-child relationship of the intervals constitutes a conceptual  tree which we call the lcp-interval tree of the suffix array. The leaves of the tree are the singleton intervals and the internal nodes of the tree are the lcp-intervals. In particular, the root of this tree is the 0-interval  as follows: Let d be the lcp-value of the parent node of [i..i]. Then i]..suf[i] + k - 1] is a k-mer occurring exactly once in • We process a singleton interval  + k ≤ n,2. suf[i]+d..suf[i] + k - 1] does not contain the symbol N.3. k, max{d + 1, kmin} ≤ k ≤ min{k' - 1, kmax}, we increment δS, k[k' is the minimum value greater than d such that either suf[i] + k' = n or i] + k'] = N.As a consequence, for all i..j] different from [0..n] as follows: Let d be the lcp-value of the parent of [i..j]. Then j]..suf[j] + k - 1] is a k-mer occurring j - i + 1 times in d <k ≤ ℓ. As a consequence, for all k, max{d + 1, kmin} ≤ k ≤ min{ℓ, kmax}, we increment δS, k[j - i + 1] by one.• We process an ℓ-interval  + d..suf[i] + k - 1] does not contain the symbol N. Rather than checking this condition character by character, we use information about ranges of Ns, preprocessed from the input sequence. That is, we store, in sorted order, the first position of each run of consecutive Ns in i, 0 ≤ i ≤ n - 1, i] ≠ N, we can determine the smallest position i' > i such that i' = n or i'] = N using a binary search. Let q be the number of these runs. Then this method takes on the order of log2q time and requires extra space proportional to q.The algorithm to enumerate the lcp-intervals and singleton intervals, given the enhanced suffix array, runs in linear time, see  for detakmax occurring more than once as substrings in kmax, which results in a space requirement proportional to kmax.Due to property (4), the enhanced suffix array does not need to be represented in main memory. At any time, we only need to store two consecutive entries of table suf and lcp. Hence the space requirement is dominated by the stack needed for the bottom-up traversal of the  lcp-interval tree. Our specific application only requires to store nodes representing strings of length shorter than τ be the number of values i satisfying δS, k[i] > 0 for some k ∈ . Then this data structure (e.g. hash table) requires space proportional to τ ≤ n.Besides random access to the sequence, we also need random access to a data structure for accumulating the occurrence counts. Let n + kmax + q and the running time is proportional to n log2q. While the running time does not depend on kmax, the space requirement is linearly dependent on kmax. Since kmax and q are both much smaller than n, they can be neglected. As a result, we obtain an algorithm that is linear in running time and space requirement. In contrast, the hashing methods run in time proportional to n + 4k time and space proportional to 4k for some fixed value of k. That is, their running time and space requirement grows exponentially with k. As a consequence, these methods can only be applied for small values of k. These values are fixed, in contrast to our method which allows the computation for a range of values of k.The overall space requirement of the algorithm is proportional to k-mer counts  is dominated by the suffix array construction. This is especially true for the space requirement. To get an idea of whether our method can be applied to large sequence sets, we have to consider the space requirement of the suffix array constructions in more detail. The most space efficient suffix array construction requires (n⌈log2 n⌉)/8 bytes per input symbol plus 2n/8 bytes for representing the sequence. Given a 32-bit computer with 4 gigabytes (232 - 1 bytes) of main memory, n has to satisfy the inequality (2n + n⌈log2 n⌉)/8 ≤ 232 - 1. That is, the sequence length is limited to 1 gigabyte. Since we want to process considerably larger sequences, we developed a divide-and-conquer approach. This cuts the sequence The analysis above shows that the running time and space requirement of our algorithm for computing occ, pos) where pos is a position in the corresponding section and occ is the number of occurrences of pos..pos + k - 1] in this section. Thus each pair encodes a k-mer, and the pairs are stored on a file in the order they are delivered by the algorithm, namely in lexicographic order with respect to the k-mer. This property allows to combine the appropriate values in a merging process, which works as follows: Suppose we are given, say m, files storing the occurrence counts for the m sections of m - 1 pairwise merge-operations. The merging steps only require to have the entire sequence n/4 bytes suffices for representing the sequence. That is, the merging procedure allows to process sequences of length up to 16 gigabytes on a 32-bit computer.Processing each section by the algorithm described above results in occurrence counts for each section. More precisely, for each section we enumerate pairs  of a sequence v is determined relative to the k-mer content of S. Since we want to compute λ for a fixed set S and many different sequences v, it makes sense to preprocess S into a k-mer frequency index S, occmin, occmax) storing all k-mers occurring between occmin and occmax times in S. Here occmin and occmax are user defined positive numbers. Constraining the indexed k-mers by user specified values occmin and occmax is relevant in applications where we are interested in k-mers occurring rarely  or frequently . Given the index S, occmin, occmax), we want to solve the following tasks:In contrast to the occurrence ratios, the average frequency w of length k, determine if it occurs in the index.(1) For each possible sequence k-mer occurring in the index, determine the number of its occurrences in the index.(2) For each k-mers. Instead of deriving occurrence ratios by incrementing some table δS, k, the same enumeration process can determine the k-mers w satisfying the constraint occmin ≤ occ ≤ occmax. If this is satisfied, the k-mer is stored on a file. As the k-mers are enumerated in lexicographic order, the index S, occmin, occmax) is simply a sequence of lexico-graphically ordered k-mers stored on a file. Whenever this is needed it can be mapped into main memory.In the previous section we have shown how to compute occurrence ratios by enumerating S, occmin, occmax), we directly store each k-mer together with the occurrence count, if this is required. As a k-mer is a string over an alphabet of size 4, it can be stored in k log2 4 = 2k bits. This, of course restricts the choice of k to small values. But since the optimal values for k are rather small  and (2) can be solved by a binary search over the index. Each step of the binary search requires to compare 2k bits. As these are represented by integers of the machine word size ω, a single comparison takes time proportional to 2 r comparisons to determine if a k-mer occurs in the index. If it occurs, then the occurrence count can be determined in constant time by a single table lookup. Altogether, the search for a k-mer takes time proportional to 2 r. For example, on a 32-bit computer, it takes on the order of 2 log2 r steps to perform the binary search when k = 20.Let To put it together, our index differs in several aspects from other indexing approaches employed in sequence analysis . As a consequence, the querying times for our index are extremely short.• Second, we directly store each tallymer-occratio), for generating a k-mer index , and for searching a k-mer index . The software-distribution is complemented by a program to construct enhanced suffix arrays. The 32-bit version of this program can handle sequences of size about 150 million bp per gigabytes of main memory. For example, to handle the 0.45 × WGS data set  in this amount of main memory, we have to split it into eight sections each of approximately 136 million bp. The program tallymer-occratio provides flexible options to specify the range of k-mer sizes considered and to tailor the values output. For example, one can specify that values are output as ratios instead of absolute counts. Comparison of k-mer frequencies of two sequence sets is not directly supported by the software, but can easily be done as follows: create an index of the first sequence set and query it with the second, and vice versa.The Tallymer software  providestallymer-search takes multiple fasta files as input. It generates all k-mers not containing any N from the given files, and matches them against the index in forward and/or reverse direction. Currently only exact matches are supported. An extension to degenerate matches would require a combination of index traversal and dynamic programming techniques, as, for example, implemented in Vmatch [The program n Vmatch  for the k-mer frequencies for large values of k, we have a wide latitude for selection of a k-mer size. Figure k-mer uniqueness ratio for the 0.45 × WGS for k in the range 10 to 500. As k approaches 500, the curve reaches unity. The information content of the k-mer set increases at a very fast rate from k = 10 to k = 20. Beyond this point, increasing k does not significantly increase the number of unique k-mers, but does decrease the overall resolution of the k-mer set. The inflection point on this curve is likely to change for other genomes and other sources of survey sequence, but for our WGS reference, k = 20 is optimal.Because our method allows us to compute k-mer frequencies is premised on the availability of an unbiased sequence set reflecting the overall repeat character of the genome in question. To test whether the 0.45 × WGS meets this criterion, we computed the corresponding 20-mer frequency index, i.e. S is the 0.45 × WGS set and k = 20. It contains 456,445,768 different 20-mers. There are 1,041,350,089 positions at which a 20-mer occurs. We screened seven publicly available maize sequence sets  maize whole genome sequences, (B) maize repeats, and (c) maize gene enrichment sequences. The seven sequence sets are known to have differing degrees of repetitiveness and should therefore provide a means to verify our method. For example, we expect gene enriched sequence to be less repetitive than RepII sequences (DNA transposons), and RepII sequences to be less repetitive than RepI sequences (TEs).-57)). Though the λ-distribution for RepII also appears to be bimodal, we were unable to find significantly different repeat populations among the DNA transposon derivatives and superfamilies.The results of this analysis are shown in Figure C0t method succeeds in concentrating unique sequence without much leakage of the highly repetitive content evident to some extent in the ESTs, and to a far greater extent in the methyl-filtered library. This difference observed between the High-C0t and methyl-filtered libraries was previously observed in [The three sequence sets delivered by gene enrichment methods shown in Figure erved in .i, in a BAC was masked if the logarithm of the absolute frequency of the 20-mer starting at position i achieves some threshold. Figure We used a previously published set of 100 maize BAC sequences that had been chosen at random to be representative of the whole genome . A positWe compared these results to masking based on the curated MIPS REcat repeat library. This library includes repeats from many sources including annotated TEs from the BACs used in this analysis. Thus application of this library can be regarded as a 'gold-standard' for detection of TEs within these BACs. Indeed, masking using the MIPS REcat library, resulted in a total repeat coverage of 80.8% . This exceeds the masking rate of 67% originally reported for this set , suggestIn contrast, 81% of the MIPS REcat-masked positions are also masked by our method. This was achieved at the absolute frequency threshold of 0.3 Figure . At the We next compared the two repeat detection methods for their ability to discriminate TEs at the level of predicted genes. The set of 100 BACs were annotated using FGENESH and resulting predictions were classified as presumptive genes or as TEs using a similarity-based search (see Section "Methods"). Of the 2504 predicted genes, 359 (14%) were screened out as showing no evidence of homology to NCBI GenPept peptides. Of the remainder, 1842 (86%) were classified as TE while the remaining 303 (14%) were classified as presumptive genes. For these latter two classes the percent of coding sequence masked was calculated based on either RepeatMasker data or on constituent 20-mer frequencies at various thresholds. For each method of masking, receiver operating characteristic (ROC) curves were used to define a threshold of masking that best discriminated TEs from presumptive genes. Area under the curve (AUC), sensitivity, and specificity were used to compare efficacies .k-mers depends on which absolute frequency threshold is used, we first optimized the threshold by comparing all ROC curves for absolute frequency thresholds between 0 and 4.0 at 0.1 increments. The threshold of 0.8 gave the maximum area under the curve (AUC), with a value of 0.961 . For masking using REcat the AUC was only slightly higher, 0.962 . Comparison of these two ROC curves, shown in Figure Because percent masking using k-mer frequencies as found within a low coverage whole genome shotgun sequence. Furthermore, the two methods showed a similar false-positive rate, each having specificity near 92%.Table While both RepeatMasker and our method masked the majority of RepI retroelements, some low copy TEs escaped masking under our method based on average frequencies. As shown in our analysis of the 100 pilot BACs, low-copy DNA transposons, may be annotated as such by curated repeat databases, but missed by the counting approach used here. In the context of directed sequence finishing, low-copy repeats are often as in need of characterization as protein coding genes. Leaving them unmasked in maize is actually in the best interests of the project. But the average frequency threshold must be chosen carefully: more permissive thresholds will lead to finishing TE-like elements, and more strict thresholds may mask high-copy gene families. To use this method optimally, a balance must be struck with respect to the genome in question, and the thresholds need to be adjusted according to the annotation requirements.k-mer frequencies. In another set of experiments, we analyzed a 453 kb portion of maize chromosome 8 (assembled in-house as described in Section "Sequence data sets") and display a portion of its annotation from position 67,000 to 165,000 in Figure ab initio gene predictions classified as either putative protein-coding genes (blue) or transposable elements (red). The Global k-mer Frequency (GKF) track visualizes the absolute frequency across the region. Frequencies in this histogram were generated by querying each overlapping 20-mer in a 5' to 3' direction against the WGS index. Note that some regions contain k-mers occurring on the order of 1000 times in the maize WGS set. In the GKF track, TE-like genes often correspond to regions of high absolute frequency, while genes similar to known proteins reside in regions of smaller absolute frequencies.The validated WGS index can be used to annotate any portion of the genome with respect to its component k-mer frequencies are computed with respect to the region itself, independent of the WGS index, a significantly different picture emerges. The Local k-mer Frequency (LKF) track represents a self-analysis: the frequencies of overlapping 20-mers in the query are determined relative to the 453 kb region alone. Even in a local context many TEs are present at high copy number . But the LKF highlights local features in a way GKF cannot. This region contains four locally duplicated genes related to Selenium-binding proteins. Figure If the scope of the experiment is narrowed, however, and the k-mer frequencies to annotate sequence with copy-number information, we have found that the frequency information contained therein are themselves biologically informative, illuminating cross species differences in repetitive content. For example, Figure ρS, k, 100·ρS, k, 100·ρS, k, 100·ρS, k, 100·ρS, k  is the ratio of k-mers occurring between q and q' times in S. The multiple occurrence ratio q, q') is similarly defined, but takes the number of occurrences of a k-mer into account. See section "Methods" for details.Beyond employing ) Figure  and 100·) Figure , where kFor example, in the case of maize, there are 1,041,350,089 positions at which a 20 mer occurs. There are 456,445,768 different 20-mers of which 378,556,535 are found only once, while the most highly represented sequences exists 47,933 times.k-mers. Maize demonstrates the most substantial growth in repetitive content, a finding attributable to its known TE expansion [k-mer frequency method provides a novel way of quantifying this important evolutionary event.The multiple occurrence ratios represented in Figure xpansion -5. Thus,k-mer sizes greater than 50 base pairs impractical. However, genome assemblies obviously do not suffer from these read length restrictions. Currently, there are very few plant assemblies. Though a number of projects are underway, only four genomes have been published: arabidopsis [k-mer uniqueness ratio for these genomes, as a function of k. We tested k-mer sizes from k = 10 to k = 500. With k approaching 500, the uniqueness ratio converges to 1. More repetitive genomes such as rice and poplar converge at a much slower rate than arabidopsis.Read lengths in whole genome shotgun sequencing projects limit this sort of analysis. Since Sanger reads average around 700 base pairs in length, most repetitive elements will be truncated at the 5' and 3' ends, making experiments with bidopsis , rice [3bidopsis ,40, poplbidopsis , and grabidopsis . In Figuk-mer frequencies cannot be used to annotate repeats across species (data not shown) owing to divergent TE families. But within a species, our method seems to work across cultivars regardless of which subtype is considered. For maize, our standard 0.45 × WGS sequence set is derived from B73, the cultivar currently being sequenced. Figure k-mers in these TEs have undergone little divergence.We performed a number of experiments demonstrating that k-mer frequencies allowing for annotating large repetitive plant genomes. As we have demonstrated, our method is useful as an alternative or supplementary form of repeat annotation in novel genomes. The novelty of genomes being sequenced is precisely the reason why this approach has value. By definition, new genomes lack comprehensive repeat libraries, and the construction of de novo libraries is often hampered by the paucity of sequence available at a project's outset. In the absence of such libraries, survey sequences in the form of Sanger WGS, 454 [k-mer frequency indices for use across multiple subtypes.We have described a method based on WGS, 454 , Solexa WGS, 454 , or SOLIWGS, 454  reads cak-mer frequency approach to be completely independent of any manual annotation, we apply a binary decision framework: given a numerical threshold, sequence is designated as either repeat or non-repeat. Though binary designations are sufficient for sequence masking, we believe that the k-mer frequencies (and measures we derive from it) are best used in combination with other methods designed for careful repeat annotation as demonstrated in the accompanying publication [In designing the lication .k-mers, and to retrieve their occurrence counts from the index. The algorithms are implemented in the Tallymer software. A distribution of the software and the Perl-scripts post processing the output is available from the Tallymer website [To apply our methods to large sequence sets, we have developed fast and memory efficient algorithms to compute occurrence ratios, to index  website .SK developed the algorithms and implemented the Tallymer software. AN analyzed sequence data sets with respect to Tallymer output. JS performed annotation of maize BACs, their visualization, and ROC analysis. DW conceived this study. All authors contributed to writing the article.


Correction to: npj Digital Medicine 10.1038/s41746-018-0031-7, Published online: 27 June 2018χ2, two-sided, p > 0.05)”. This has been changed to “The 18–24 year old are over-represented in the all user treatment population, while the 50–64 year old are underrepresented in both tracked and all user population, p-values were <0.05 for age groups and gender.” This has been corrected in the HTML and PDF version of the Article.In the original version of the published Article, there was an error in the caption to Table 1 which stated “None of the differences are statistically significant (


Medicine, Tong Yu's degree appeared incorrectly and should be Master of Law (ML), not PhD.In the article, “Socioeconomic status and self-rated health in China: Findings from a cross-sectional study”,


Correction to:npj Digital Medicine 10.1038/s41746-019-0079-z, Published online 29 January 2019In the original version of the published Article, there was a sentence fragment prior to the last sentence under the subheading “Reproducibility and digital medicine” which stated “Some journals, such as PLOS ONE, have made an early”. To improve clarity, the sentence fragment has been removed. This has been corrected in the HTML and PDF version of the Article.


We prove among other things that every m-hyperconvex domain admits an exhaustion function that is negative, smooth, strictly m-subharmonic, and has bounded m-Hessian measure.We study the geometry of  In this paper, we shall study the geometric properties of these domains. Let us now give a thorough background on the motivation behind this paper. It all starts with the following theorem:The geometry of the underlying space is usually essential when studying a given problem in analysis. The starting point of this paper is the model presented by Caffarelli et al.  in 1985 Assume that regular at every boundary point strong barrier at every point weak barrier at every point exhaustion function that is negative and subharmonic, i.e., there exists a non-constant function Jensen boundary w.r.t. the Jensen measures generated by the cone of functions that are continuous on entclass1pt{minimaThe idea of a regular boundary point can be traced back to 1911 and 1912 with the works of Zaremba \documentJensen measure with barycenterw.r.t.Jensen boundary w.r.t. Let The Jensen boundary is another name for the Choquet boundary w.r.t. a given class of Jensen measures. For further information see e.g., , 48.There are many different spaces of Jensen measures introduced throughout the literature. Caution is advised.The most important tool in working with Jensen measures is the Edwards’ duality theorem that origins from . We onlyg be a real-valued lower semicontinuous function defined on (Edwards’ Theorem). Let We end this section with a convergence result.n, let Assume that The Banach–Alaoglu theorem says that the space of probability measures defined on m-hyperconvex domains , but itIn Example m-subharmonic function if, and only if, k-hyperconvex, sincek-subharmonic exhaustion function. In particular, we get that for For a given integer 11.13 in ). FurtheWe shall need the following elementary lemma. For completeness we include a proof.Let For The aim of this section is to prove the following theorem, especially property (3).m-hyperconvex domains in m-hyperconvex in If m-hyperconvex in The domain m-subharmonic on The domain m-hyperconvex, then m-hyperconvex.If Assume that Part (1) For each m-hyperconvex domain m-hyperconvex in Part (2) This part is concluded by defining a negative exhaustion function byPart (3) The proof of this part is inspired by [pired by . First,  Theorem  we have strictlym-subharmonic exhaustion function for m-subharmonic functions on defined M-test. To see that m-subharmonic, note that if m-subharmonic, we have that m-subharmonic on j.Next, we shall construct a continuous Part (4) The idea of the proof of this part is from [m-hyperconvex. Let m-subharmonic exhaustion function for j, take a smooth function m-subharmonic for c such thatm-subharmonic function defined on m-subharmonic, and  is from . By the A2.4. in ). Hence,m-hyperconvex domain, and To see that (1) holds, first note that it is clear that m-hyperconvex domain it can be proved as in part (3) of Theorem Assume that B-regular and hyperconvex domains within the Caffarelli–Nirenberg–Spruck model. More precisely, in Theorem In this section, we shall investigate the geometry of the corresponding notions of m-hyperconvex in the sense of Definition m-subharmonic;m-subharmonic; andFor every Assume that The implications z that is m-subharmonic. Since there exists a constant  Theorem , we knowm-hyperconvex, and that z, andm-hyperconvex.  Theorem , it folla 2.3 in , impliesBefore we can start with the proof of Theorem m-hyperconvex domain in Let f to a continuous function on f. Since m-hyperconvex domain then by Theorem j, we have thatAssume that  Theorem  gives usm-hyperconvex, then we have a similar result as in Corollary f to If We end this section by proving Theorem m-subharmonic;m-subharmonic, and such that Assume that f be a continuous function on z.m-subharmonic, and by the generalized Walsh theorem .g defined on a neighborhood of  Theorem . \documeAn immediate consequence of Theorem Let z, and let z, that is m-subharmonic, and defined in some neighborhood of z, that is m-subharmonic:Let The purpose of this section is to prove the implication m-Hessian measure of u is defined byAssume that m-Hessian measure is well-defined for much more general functions than needed in this section. For further information see e.g., [The ee e.g., .m-hyperconvex domain in m-hyperconvex domain admits an exhaustion function that is smooth, and strictly m-subharmonic. Our method is that of approximation. Therefore, we first need to prove a suitable approximation theorem. Theorem m. In connection with Theorem m-subharmonic. Thereafter, they prove that approximation is possible. Whereas we prove that smooth approximation is always possible on an m-hyperconvex domain, i.e., there should only exist a negative exhaustion function. Thereafter, we prove the existence of a negative and smooth exhaustion function that is strictly m-subharmonic, and has bounded m-Hessian measure. We believe that Theorem For a bounded  Cegrell . If the m-hyperconvex domain in m-subharmonic function u defined on Assume that Before proving Theorem u, v be smooth m-subharmonic functions in m-subharmonic function Let Now to the proof of Theorem m-subharmonic.By Theorem We want to prove that for any Step 1. Fix a constant m-subharmonic functions decreasing to bu. Out of this sequence pick one function, h is the unique harmonic function on h can be obtain as (see [ as (see )\documen Theorem , we haveStep 2. Let We start by taking m-subharmonic functions decreasing to W such thatW and Step 1, it follows that Take an open set Step 3. Now if proof of .u is a negative m-subharmonic function defined on To finish the proof of this theorem, assume that 1.7.1 in  implies  Then by , there eWe shall end this paper by proving the implication m-hyperconvex domain in m-subharmonic, and has bounded m-Hessian measure.Assume that m-subharmonic, and belongs to n, we arrive atm-subharmonic. Let Theorem


In the original version of the article “A network-based conditional genetic association analysis of the human metabolome” by Y.A. Tsepilov et al. , there w


This research proposes the first one-out-of-two quantum oblivious transfer (QOT) scheme that does not have a two-level structure and is not subject to Lo’s no-go theorem. Instead, the proposed scheme is a simple and efficient approach based on nonorthogonal states. The nonorthogonality causes one of a pair of messages to be unable to be measured to achieve the irreversible goal of discarding a message, resulting in a one-out-of-two selection effect. The proposed QOT protocol is therefore built directly on quantum resources rather than on a two-level structure in which two classical keys must first be created using quantum resources  and then a one-out-of-two protocol is built from there. Furthermore, the proposed protocol allows Alice and Bob to test each other’s loyalty by comparing measurement results. In addition, the relationship with the no-go theorem is discussed in detail; this relationship is often overlooked in other studies. A security analysis demonstrates that the proposed protocol is secure against both external and internal attacks. In addition, an efficiency analysis shows that the proposed protocol is more efficient than other, two-level-structured protocols. In the all-or-nothing OT protocol, a sender Alice wants to send a secret message, m ∈ {0, 1}, to a receiver Bob who has only a 50% probability of receiving m. He will either learn the message m with 100% reliability or learn nothing about m. At the end of all-or-nothing OT, Alice remains oblivious as to whether Bob received the message m. Following the proposal of this protocol, Even et al.2 presented one-out-of-two OT (or it can be abbreviated as 1–2 OT), in which Alice transfers two messages, m0 and m1, to Bob, and he can choose only one of them and will have no idea what the other message is. When the one-out-of-two OT protocol is complete, Alice learns nothing about which message Bob selected. In 1988, Crépeau3 presented a method for building a one-out-of-two OT protocol by using p-all-or-nothing OT, in which the receiver has a probability p of receiving the message m, called Crépeau’s reduction. The receiver builds two key sets to represent his choice, key0 and key1, one of which he learns with 100% certainty and the other of which he learns with 0% certainty. Based on Bob’s choice j ∈ {0, 1}, he asks Alice to encrypt her messages m0 and m1 using keyj and mj under this two-level-structured method.Oblivious transfer (OT) is an important branch of cryptography with many useful and important applications, such as secure computation, bit commitment, remote coin-flipping, and digital contract signing, for which OT protocols are the cryptographic primitives. The two most commonly used OT protocols are the all-or-nothing protocol and the one-out-of-two protocol. All-or-nothing OT was first introduced by Rabin4. However, Shor showed that a quantum algorithm5 can be used to break the RSA cryptosystem in polynomial time, which means that such protocols may be unsafe against quantum algorithms. In 1984, Bennett and Brassard proposed the first quantum key distribution protocol6, called BB84, thereby initiating the study of quantum cryptography. Researchers later showed that BB84 is unconditionally secure10 both in theory and in implementation by achieving a one-time pad. The security of quantum cryptography is based on physical laws, unlike that of classical cryptography, which is based on mathematical complexity. This physical basis allows quantum cryptography to easily achieve many goals that were difficult or unthinkable in the past, including unconditional security.Classical OT protocols are almost all based on the RSA cryptosystem6, researchers have been designing quantum oblivious transfer (QOT) protocols using quantum properties. Crépeau and Kilian11 proposed the first all-or-nothing QOT scheme in 1988, and Bennett et al.12 proposed the first one-out-of-two QOT scheme protected by a quantum error-correcting code in 1992. In 1994, Crépeau13 presented a one-out-of-two QOT scheme based on quantum bit commitment (QBC), which guarantees security under the assumption that Bob cannot delay the quantum measurement. In 1995, Yao14 further proved that this protocol is secure against coherent measurement if QBC is secure. However, in 1997, Lo15 doubted that all one-sided two-party computations  but only one of the two parties is allowed to learn the result) may be insecure, including one-out-of-two QOT (the function f in one-out-of-two QOT is a selector). This was called Lo’s no-go theorem, and because of the computational equivalent12 to two OTs, this theorem has caused extreme difficulties in the development of QOT research.Since the proposal of BB8416 presented an interesting communication method analogous to one-out-of-two QOT with a 50% probability of completing the communication. They17 then improved the security of their protocol against entangled pair attacks in 2003. Moreover, in 2006, He and Wang18 proposed a secure all-or-nothing QOT scheme using four entangled states, which, as a result, was no longer subject to Lo’s no-go theorem15. Consequently, He claimed that Lo’s no-go theorem15 did not truly cover all QOT conditions. Thereafter, He19 demonstrated that a one-out-of-two QOT scheme built on all-or-nothing QOT protocol using Crépeau’s reduction3 also is not subject to Lo’s no-go theorem15. The key is that the receiver inputs his choice before the sender inputs her messages m0 and m1, causing the functions f of the one-out-of-two protocol and Lo’s no-go theorem15 to be different.Recent studies have, however, proposed various methods of avoiding Lo’s no-go theorem. In 2002, Shimizu and Imoto19, researchers have been designing new one-out-of-two QOT schemes19. In 2007, Wei Yang et al.20 presented a one-out-of-two QOT scheme using tripartite entangled states based on He’s proof19 and also showed that this scheme is not covered by the cheating strategy of Lo’s no-go theorem15. Li Yang21 presented an all-or-nothing QOT scheme using nonorthogonal states, similar to B9222, and used it as a basis for constructing a one-out-of-two QOT scheme in 2013. Subsequently, Yu-Guang Yang and his research team, as part of a research effort that began in 2014, have proposed several QOT protocols. They have been testing various schemes for building one-out-of-two QOT protocols using He’s proof19. In 2014, they23 proposed all-or-nothing and one-out-of-two QOT protocols based on an untrusted third party. In 2015, they24 developed an all-or-nothing QOT protocol by analyzing the probability of the qubit state distribution, which led them to propose a method of testing the loyalty of the sender and then to build a one-out-of-two QOT protocol on this basis. They25 also designed a one-out-of-two QOT scheme with a two-level structure using BB846 and reduced it to B9222 for an all-or-nothing QOT scheme. In addition, they26 attempted to use Bell states to achieve the same effect as B9222 for one-out-of-two QOT. Furthermore, in 2017, they27 proposed a method of using any two nonorthogonal states by cooperatively measuring the qubit sequence and then built a one-out-of-n QOT scheme using this method.Following He’s proof27 all have two-level structures, in which two classical keys are created using an all-or-nothing QOT protocol and then a one-out-of-two QOT protocol is built on top. The two-level structure is clearly inefficient, because many quantum resources are consumed for all-or-nothing QOT instead of being used to transfer the message. In addition, this structure reduces the elasticity and diversity of protocol design because such designs can only follow He’s proof19 with minor revisions to the details of the all-or-nothing QOT scheme. In our opinion, He’s proof19 not only revealed a different function f, which is not subject to Lo’s no-go theorem15, based on a two-level structure but also provided a new approach in the sense that if any protocol can achieve the same effect as that of f in He’s proof19, then it is also covered by He’s proof19. In this work, the first one-out-of-two QOT protocol is proposed that is directly based only on the properties of quantum resources, namely, nonorthogonal states, rather than a two-level structure, while also being covered by He’s proof19. The key to our protocol is that Bob’s choice is made before Alice inputs her messages m0 and m1. The property of nonorthogonality ensures that one of the two messages cannot be measured and thus maintains obliviousness, thereby achieving the same effect as that of f in He’s proof19. Therefore, our protocol is not only secure (and not subject to Lo’s no-go theorem) but can achieve greater efficiency than protocols27 that are based on a two-level structure.However, these protocols15 and He’s proof19, and its security and efficiency analyses. The preliminaries introduce the properties of quantum machines and define some notation. Then, the basic idea of the proposed protocol is introduced before the details of the protocol itself, which are described in the subsequent section. Moreover, the relationship among Lo’s no-go theorem15, He’s proof19 and the proposed protocol is discussed in the subsection titled “Resisting Lo’s cheating strategy15”. Finally, security and efficiency analyses are presented in the last two subsections.This section consists of six subsections, including the preliminaries, the basic idea of our protocol, the proposed protocol itself, its relationship with Lo’s no-go theoremThis subsection introduces the basic definitions of concepts relevant to quantum machines, such as quantum bits, superposition, entanglement, gates, and operations, as well as some properties of quantum machines.The classical information carrier is called a “bit”. The quantum information carrier is called a “quantum bit”, or a “qubit”. A qubit collapses to certain states of a basis when it is measured. Two bases are commonly used: the Z-basis and the X-basis. The Z-basis is defined as ϕ〉 will collapse to |0〉 and |1〉 with probabilities of Superposition refers to the phenomenon that a qubit can simultaneously exist in both the |0〉 and |1〉 states; i.e., +〉 is measured, as in Eq. 12 or |11〉12, where the subscript indicates the qubit order. As a result, in this case, it is possible to immediately learn the states of two qubits when only one is measured. Einstein referred to this as “spooky action at a distance”.Another important property, entanglement is the phenomenon that qubits cannot exist singly. There are four common entangled states, called Bell states, as shown in Eq. UU* = U*U = I) are regarded as gates in quantum computers. There are four common operations, represented by operators called Pauli matrices, which are denoted by {I, X, Y, Z}, as shown in Eq. I and Z cannot be distinguished in the Z-basis, and Y and Z cannot be distinguished in the X-basis. A single qubit cannot be observed using all four operations, which means that some information is ignored; this is a key element of the proposed protocol. For example, after a Y gate, the state |0〉 becomes −|1〉; i.e., a result of |1〉 will be obtained when the qubit is measured. This negative amplitude is called a global phase and cannot be measured. Another important gate is the Hadamard gate, as described in Eq. H gate. The H gate can be used to convert between two different bases (the Z-basis and the X-basis). For example, after an H gate, the state |0〉 (|+〉) H becomes |+〉 (|0〉). Table I, X, Y, Z}.Moreover, unitary operations , he learns m0 unambiguously , M is the number of channel checking qubits, and K is the number of loyalty testing qubits, then the total length of the QOT sequence is N + M + 2K. Bob randomly prepares M qubits from {|0〉, |1〉, |+〉, |−〉} (each qubit is independent) and inserts them into his sequence. Subsequently, he also inserts his N and 2K candidate choice intentions  into his sequence and then sends the sequence to Alice. Let us give a simple example to describe the proposed one-out-of-two QOT protocol without channel checking, suppose that Bob prepares two qubits in |0+〉12 (N) to represent his choices and an additional two qubits in |0+〉34 (2K) for loyalty testing. He then sends these four qubits to Alice.Step 2. Once she receives the sequence from Bob, Alice first checks the channel for an eavesdropper (Eve) and then tests Bob’s loyalty. First, she asks Bob to publish the bases and states that he has created. If the error rate is higher than a given threshold, then an Eve is present on the channel, and Alice and Bob abort their communication; otherwise, Alice goes on to test Bob’s loyalty. She discards the qubits for channel checking and then randomly selects several positions and requests that Bob publish his bases. If different results, i.e., ∉ {|0〉, |+〉}, are measured and the error rate is higher than the given error rate, then Bob is considered dishonest, and she aborts this communication; otherwise, she proceeds to the next step. Following the above example, once Alice receives the ordered sequence |0+0+〉1234, she randomly chooses a qubit for loyalty testing. Suppose that Alice’s random choice is qubit 4; then, she asks Bob to publish the basis of qubit 4, measures it, and compares the published and measured results. If the error rate is higher than the threshold, then Bob is considered as dishonest; after that, qubit 4 is discarded.Step 3. Since the loyalty test may disturb the order of Bob’s choice intentions, Bob must ask Alice to reorder the qubits. It is for this purpose that 2K additional qubits are initially provided to prevent vacancies in the list of choice intentions. In this step, the sequence after reordering represents Bob’s real choices. Following the above example, after the loyalty test, Bob asks Alice to reorder the remaining qubits in the order 21, and the states become |+0〉21, with qubit 3 discarded. The resulting state |+0〉21 represents Bob’s choices, j1 and j0, respectively.Step 4. Alice now inputs her secret messages m0 and m1 through the I, X, Y and Z operations, corresponding to the combinations “00”, “10”, “11” and “01”, respectively. Following the above example, Alice performs Z and X in accordance with her messages “0112” and “1034”, where the subscripts indicate Alice’s classical bit order, on qubits 2 and 1, respectively. This converts the state |+0〉21 into |−1〉21.Step 5. Alice then randomly inserts decoy qubits from {|0〉, |1〉, |+〉, |−〉} into the sequence for channel checking and sends the sequence to Bob.Step 6. When Bob receives the sequence from Alice, he asks Alice to publish the positions and states of the decoy qubits. If the error rate is higher than the channel error rate, they abort this communication and return to step 1. Otherwise, Bob learns the contents of the classical messages by measuring the qubits with the bases he prepared. Following the above example, Bob performs X- and Z-basis measurements to learn the second and first classical messages, “1” and “1” , respectively.Step 7. At the end of this protocol, where Bob has to test Alice’s loyalty to prevent Alice from cheating, she can learn Bob’s choice with the probability 25%, or 29.3% by POVM. Bob chooses some random positions and asks Alice to publish her operations. Bob performs the operations according to Alice’s announcement, in order to recover those qubits into {|0〉, |+〉}. If the error rate of loyalty testing is higher than the threshold, then Alice is considered as dishonest. Following the above example, Bob asks Alice to publish operation (X), which performs on qubit 1; then, Bob performs X on qubit 1 to recover the state into |0〉.15 provides a cheating strategy for learning all messages in a one-sided two-party secure computation, which doubts that one-out-of-two QOT is insecure. The key to Lo’s cheating strategy15 is requirement (A-i), namely, that “Bob learns f, j) unambiguously”  represents a pair of messages in a one-out-of-two QOT scenario), which leads to a probability of 100% that the selected state will collapse. In addition, the result is obtained after reversible operations. As a result, choices can be made repeatedly to learn all messages, as shown in Eq. G and |ϕ〉 represent two different selected operations, any unitary operation for inputting two messages and any quantum state, respectively. Therefore, once Bob learns the content of a message, he can recover the state G × |ϕ〉, which is the state after Alice’s input, by applying the selected operation k ∈ {|0〉, |1〉}. In this way, Bob can change his choice and learn the contents of all messages by repeating the above process.Lo’s no-go theorem15. Obviously, definition B is a special case of definition A. Definition B describes the case in which Alice inputs her messages first and then Bob inputs his choices. The important point here is that if Bob inputs his choices first and Alice subsequently input her messages, as in definition C (the proposed protocol), this scenario is not equivalent to the function considered in Lo’s proof15. This is because the function becomes f, j) when Bob inputs his choices first, and f, j1) is meaningless with respect to f, j). Therefore, Bob cannot change i from i to i without Alice’s help. Following from the above relation, in the proposed protocol (definition C), the result after Bob’s and Alice’s actions can be expressed asDefinition B corresponds to the one-out-of-two QOT scenario covered by Lo’s proofG ∈ {I, X, Y, Z}. Eq. G. Therefore, Bob cannot perform Lo’s cheating strategy15. As a result, Bob cannot reverse the effects of his inputs without Alice’s help. The condition of Equation f, j) is similar to that of He’s proof30. This proof shows that the order of input of the choices and messages may change the function f, which means that this protocol is not subject to Lo’s no-go theorem. In addition, He30 has extended the concept of his proof19 to the general case; if Alice and Bob interact with each other and Bob cannot eliminate the effects of his operations independently, then the interaction is covered by He’s proof19 and resists Lo’s cheating strategy15.Definition A: ideal one-sided two-party secure computation(A-i) Bob learns f unambiguously.(A-ii) Alice learns nothing about j and f.(A-iii) Bob learns nothing about i more than what logically follows from the values of j and f.Definition B: one-out-of-two OT (Lo’s no-go theorem15)(B-i) Alice inputs i, which is a pair of messages .(B-ii) Bob inputs j = 0 or 1.(B-iii) At the end of the protocol, Bob learns the content of message mj but not of the other message f = m0 and f = m1.(B-iv) Alice does not know which mj Bob received.Definition C: the proposed protocol(C-i) Bob inputs j = 0 or 1 to change the qubit state to {|0〉, |+〉} (Z- or X-basis) in accordance with his choice intention.(C-ii) Alice inputs her messages m0 and m1 using {I, X, Y, Z}.(C-iii) Bob learns the content of either m0 or m1 using the basis (Z- or X-basis) he prepared.Here, Two security conditions are considered in this study: security against external and internal attacks. External attacks involve an eavesdropper, Eve, attempting to steal messages without being detected. Internal attacks involve either Alice or Bob attempting to steal the other’s secret information; i.e., Alice wants to learn Bob’s choices, or Bob wants to learn the contents of both of Alice’s messages.31, Eve will be able to illicitly eavesdrop on their messages. In the proposed protocol, several single qubits ∈ {|0〉, |1〉, |+〉, |−〉} are randomly inserted into the transmitted sequence as decoy qubits for channel checking, as described in steps 1 and 5 of the protocol. The positions and states of these qubits are then published and measured to check whether an Eve is present. If the measured results obtained with the same bases are different and the error rate is higher than the channel error rate, then an Eve is present. Two common external attack strategies are the intercept-and-resend attack and the entangling attack. They are discussed below.Alice and Bob must ensure that the communication channel between them is secure, because without channel checking or reduced frequency6, each qubit has a probability of M. As a result, the security level can be assessed based on the detection rate by legal agents, ξ1, as expressed in Eq. Eve intercepts all qubits during transmission when the sender sends the qubit sequence to the receiver, measures them to obtain the message contents, and then resends those qubits to the receiver. This action should disturb the states of the qubits, including the decoy qubits, because Eve does not know which bases have been prepared by Alice and Bob. According to the detection rate of BB84E〉, and performs a unitary operation Ue on the intercepted qubit to entangle it with her qubit |E〉 during transmission. The unitary operation Ue is defined as shown in Eq. e00〉, |e01〉, |e10〉, and |e11〉 are four states determined by the unitary operation Ue, Ue must satisfy a = d = 1, b = c = 0, and |e00〉 = |e11〉, and as a result, the proposed protocol ensures that no information can be obtained in this way.Eve may instead use a different method that does not disturb the qubit states, namely, the entangling attack. In this attack, she intercepts the transmitted sequence, prepares an ancillary qubit |Internal attacks involve the legal agents Alice and Bob attempting to steal each other’s secret information; i.e., Alice wants to learn Bob’s choices, or Bob wants to learn the contents of all messages sent by Alice. Therefore, two conditions must be discussed, namely, Alice’s and Bob’s cheating strategies.I, X, Y, Z, H} etc., and she has 25% or 29.3% chance to learn Bob’s choices; however, this kind of attack can be always detected in our protocol. The second condition is that Alice has the ability of entanglement. In this condition, Alice has the ability to perform two or more qubit gates, which leads to diverse attacks. However, Bob is also required to have the ability of entanglement to resist attacks from Alice, and a dishonest Alice will be detected by the discussion below.There are two conditions to be discussed. The first condition is that Alice has no ability of entanglement. In this condition, Alice only has the ability to perform a single qubit gate such as {22, because a measurement in the incorrect basis can yield incorrect measurement results that nevertheless help Alice to determine Bob’s initial state. For example, if Bob sends the state 0 to Alice, she has a probability of E1, E2, I − E1 − E2} on Bob’s qubit, whereA dishonest Alice can learn 25% of Bob’s choices, as in B92Then, she can unambiguously distinguish states {|0〉, |+〉} with probabilityξ2 can be decided by the user through the number of qubit D for loyalty testing. Therefore, Bob can detect that Alice is cheating. If Alice does not use POVM, the total detection rate is In other word, the remaining 70.7% of Bob’s choices will be unknown, which means that Alice should randomly create several state ∈ {|0〉, |1〉, |+〉, |−〉} to send to Bob. However, she cannot know Bob’s final measurement results, as he does not publish any information about the bases. In other words, for each bit, he will be unable to correctly decrypt with a probability of I, Z} to escape this testing, and there are two branches: 1. Bob uses Z-basis as his choice; in this case, Alice can always escape the testing; 2. Bob uses X-basis as his choice; in this case, Bob has a 50% to get |+〉 or |−〉. When he gets |+〉, the operation Z cannot restore the state |+〉 to |+〉. Otherwise, when he gets |−〉, operation I cannot restore the state |−〉 to |+〉. Therefore, Bob always has a probability to detect Alice’s dishonesty.For a simple example regarding the detection rate with a remaining qubit, which Alice randomly prepared, Alice prepares a qubit in state |0〉 and guesses Bob’s choice. In this case, she can only publish operation {+〉AB to perform a teleportation attack. In this way, she can pass Alice’s loyalty testing, and then, learn Bob’s choices with 25% or 29.3% chance without being detected. For a simple example to explain the teleportation attack, in step 3, the qubit from Bob after Bob’s loyalty testing is called |φ〉C. In step 4, instead of inputting her secret message into qubit C, Alice creates a Bell state in B to Bob to replace qubit C, and holds qubit A. After that, Alice performs a Bell measurement , BM for short, on qubit C and A, and publishes one of four operations I, X, Y or Z as her secret message according to BM results |00〉CA, |01〉CA, |11〉CA and |10〉CA, respectively. Bob can then perform one of four operations I, Z, X or Y to recover state |0〉B or |+〉B according to a result that Alice published. As shown in Eqs B or |+〉B, because the BM results |00〉CA, |10〉CA, |01〉CA and |11〉CA can always match operations I, Z, X and Y, respectively.A dishonest Alice can prepare Bell states in |ΦA1 to Alice. Under normal conditions, we can know that Alice will do honest behavior. After Alice performs an operation in {I, X, Y, Z} on qubit A1, the entangled state will be Eq. A1 and B1 with one of two bases {IY, XZ} according to Alice’s operations to check Alice’s loyalty.However, without loss of generality, Bob should also have the same entanglement ability as Alice. He can prepare an entangled state in mentclasspt{minimaB to Bob for the teleportation attack. Alice will perform the BM on qubit A1 and A, which leads the entire system to be Eq. ξ3 is as given in Eq. F is the number of detected entangled qubit pairs. Therefore, Alice and Bob can decide security level ξ3, and whether they will continue the protocol according to the detection result. In summary, if the dishonest Alice only has the ability to perform single-qubit operations, then follow this protocol Bob can always have a probability to detect dishonesty one. Moreover, if the dishonest Alice can prepare Bell states or perform teleportation attacks, Alice’s cheating becomes more and more difficult because she has to have the technology to store the qubits received from Bob. However, such a long-term quantum storage technology is still a technical challenge and an open issue today. Even though when the long-term quantum storage technology can be built, the protocol still can intentionally delay the operation time between step 4 and step 7 to prevent these attacks.However, Alice may be dishonest and perform a teleportation attack. Alice creates an entangled state in +〉, as given in Eq. K positions and asks Bob to publish the bases he prepared in step 2. If different measurement results are given, i.e., ∉ {|0〉, |+〉}, Bob is dishonest. The detection rate by legal agents, or the security level ξ4, is as given in Eq. A dishonest Bob can prepare entangled qubits of the form |Φ25 based on Crépeau’s reduction3. The protocols of Wei Yang et al.20 and Yu-Guang Yang et al.27 are not considered because that of Wei Yang et al. may not work, whereas the first protocol of Yu-Guang Yang et al.23 involves an untrusted third party, and the second27 is a one-out-of-n QOT protocol for which the resource consumption for one-out-of-two QOT is similar to that of Li Yang’s protocol21. The three protocols considered for comparison25 have two-level structures in which one-out-of-two OT is built on all-or-nothing QOT. However, the probability p of all-or-nothing QOT (where p is the probability of the unambiguous key) is not always 50%. Significant quantum resources are required to build two classical keys (one unambiguous and the other unknown) using all-or-nothing QOT for one-out-of-two OT. In addition, every transmission should include decoy qubits for channel checking. Some protocols may need many transmissions, and many decoy qubits, to complete all-or-nothing QOT. For fairness, the security level ξ1 is ensured to be at least 99.9999% by using 50 decoy qubits for each transmission. Then, the most important indicators are the conversion efficiency between two OT protocols and the number of transmissions (which indirectly affects the number of decoy qubits). The total cost of each protocol, as calculated under the requirement that R message bits are received, is given in Table 24 do not consider any loyalty testing, which means that the sender and receiver may not truly trust each other, whereas the loyalty testing of the third protocol24 requires the consumption of a large number of qubits, making the quantum cost difficult to calculate. Detailed descriptions of the protocols considered for comparison25 are given below.This section presents a performance comparison of the proposed protocol with three modern two-level-structured one-out-of-two QOT protocols21 uses the B9222 protocol as the all-or-nothing QOT protocol on which it is based. Therefore, it requires four qubits on average to obtain an unambiguous key and only one transmission. However, the cited study focused more on the bit-commitment protocol than on the OT protocol, with no further security analysis of the QOT protocol or strategies for detecting eavesdroppers. Therefore, no strategy was provided for loyalty testing between Alice and Bob. In addition, the number of decoy qubits is computed as described above because the original detection strategy of B9222 is less efficient. Therefore, the same detection strategy with decoy qubits is used. As a result, the total quantum cost is 4 × R + 50.This protocol24 also requires at least four qubits on average to obtain an unambiguous key using its all-or-nothing QOT strategy, i.e., R + 2 × 50.This protocol25 reduces BB846 to B9222. It uses the BB846 strategy and the publication of additional state information to allow Bob to generate unambiguous keys as in B9222. It also requires four qubits for the generation of an unambiguous key. In addition, it requires one transmission to complete all-or-nothing QOT. However, it does not include a loyalty testing method for the all-or-nothing QOT stage. Errors may be detected at the application level. The overall cost of the protocol is 4 × R + 50.This protocol25, with a probability p of R + 2 × 50.The proposed protocol is based directly on quantum resources and consumes one qubit for each received bit. The proposed protocol is more efficient than the others25; this is also shown in Fig. 3, which requires a two-level structure.As seen from Table 15. A simple and efficient one-out-of-two QOT protocol with single nonorthogonal qubits, in which one of the two messages is discarded automatically, has been successfully developed based on the most basic properties of quantum machines. Second, the proposed protocol can effectively prevent both external and internal attacks, as proven by a detailed security analysis. Regarding internal attacks, an important feature of the proposed protocol is that loyalty testing is applied to provide security against internal attacks by a dishonest Alice or Bob; the dishonest one can always be detected at the loyalty testing stages with safety parameters ξ2, ξ3 and ξ4, respectively. Finally, the proposed protocol has a lower cost and is more efficient than many traditional protocols based on a two-level structure. In addition, as this protocol uses only a single qubit, it is easily implemented.In conclusion, there three important contributions of the proposed method. First, to the best of our knowledge, the proposed protocol is the first one-out-of-two QOT protocol to be designed directly based on quantum properties without relying on all-or-nothing QOT, and it has been proven to be secure and not subject to Lo’s no-go theorem


Ron,sp) lateral double-diffused metal-oxide-semiconductor transistor (LDMOS) with enhanced dual-gate and partial P-buried layer is proposed and investigated in this paper. On-resistance analytical model for the proposed LDMOS is built to provide an in-depth insight into the relationship between the drift region resistance and the channel region resistance. N-buried layer is introduced under P-well to provide a low-resistance conduction path and reduce the resistance of the channel region significantly. Enhanced dual-gate structure is formed by N-buried layer while avoiding the vertical punch-through breakdown in off-state. Partial P-buried layer with optimized length is adopted under the N-drift region to extend vertical depletion region and relax the electric field peak in off-state, which enhances breakdown voltage (BV) with low drift region resistance. For the LDMOS with enhanced dual-gate and partial P-buried layer, the result shows that Ron,sp is 8.5 mΩ·mm2 while BV is 43 V.An ultra-low specific on-resistance ( Ron,sp) and maximizing off-state breakdown voltage (BV) [Ron,sp vs. BV for LDMOS devices [Ron,sp and the contribution of the channel region cannot be ignored.With the increase of demand for more complex and faster logic function in analog power IC, it is significant to improve the performance of the lateral double-diffused metal-oxide-semiconductor transistor (LDMOS), specially minimizing specific on-resistance (age (BV) –9. Most  devices –20. In o devices . The depRon,sp and high BV.In this work, a novel ultra-low specific on-resistance LDMOS with enhanced dual-gate and partial P-buried layer is investigated. The physical models IMPACT.I, BGN, CONMOB, FLDMOB, SRH, and SRFMOB are used in numerical simulation. On-resistance analytical model is proposed to provide an in-depth insight into the relationship between the drift region resistance and the channel region resistance. Based on the model, N-buried layer and partial P-buried layer are optimized to achieve low Ron,sp and enhance BV, respectively. In the channel region, the enhanced dual-gate is formed by trench gate and highly doped N-buried layer. Compare to conventional dual-gate structure, N-buried layer significantly reduce the resistance of the channel region by provide a low on-resistance conduction path under P-well in the on-state. In the drift region, the partial P-buried layer with high doping concentration is introduced under the N-drift region to enhance BV while maintaining low Ron,sp. The partial P-buried layer helps to reduce the vertical electric field in the off-state without breaking charge balance in the drift region. The key size of the novel device is listed in Table Figure Rd) and the resistance of the channel region (Rc) in series. In the channel region, surface channel conduction path parallels the trench channel conduction path. Thus, Rc is equal to (Rchs + Racc)//(Rcht + Rnb), where Rchs, Racc, Rcht, and Rnb are the resistances of the surface-gate channel, the accumulation region, the trench gate channel, and the N-buried layer, respectively. Based on the proposed on-resistance model, the reduction of Rc would achieve by decreasing Rnb without affecting the other performances, because the other resistances are mainly determinate by the process technology, operation voltage, and threshold voltage. The Rd has been reduced by introducing P-buried layer under N-drift region to enhance the Reduce Surface-field (RESURF) effect in our previous work. In this work, the partial P-buried layer is adopted to improve the BV while maintaining the low Rd.Figure Rc, the N-buried layer with high doping concentration is introduced under P-well. Figure Rc as functions of the doping concentration of the N-buried layer (Nnb) with single-gate and dual-gate. It is indicated that the dual-gate structure helps to reduce Rc compared with the single-gate. When Nnb = Nd = 5.5 × 1016 cm−3, Rc is 110 mΩ. According to the on-resistance model, Rnb is the main contributor to Rc. And then, the Rnb is desired to decrease with the aim of smaller Rc. As shown in Fig. Rc is reduced with Nnb increasing. When Nnb = 1.35 × 1017 cm−3, Rc is reduced to 85 mΩ. However, Fig. Nnb would be limited by punch-through breakdown. Because of adding trench gate, Rc is decreased firstly by 34% with Nnb = Nd = 5.5 × 1016 cm−3. As Nnb increases, Rc continuously decreases. With optimized Nnb = 1.05 × 1017 cm−3, Rc is decreased by 45% at last. When Nnb > 1.05 × 1017 cm−3, punch-through breakdown will happen in P-well. The analytical result of Ron,sp shown in Fig. Aiming at the reduction of Nnb with different doping concentration of P-well (Npwell). Nnb has an effect on not only the Rc, but also the BV. For a given Npwell, BV keeps unchanged at small Nnb, and then decreases with Nnb increasing. When Nnb increases to 1.2 × 1017 cm−3, BV starts to drop with Npwell = 2 × 1017 cm−3. The drop of BV is ascribed to punch-through breakdown in the P-well region as shown in Fig. Npwell, the depletion mainly extends to the drift region, and the punch-through breakdown is avoided without degrading the BV. Although P-well with high doping concentration benefits to avoid the punch-through breakdown, it would enhance the threshold voltage. Thus, Npwell of 2 × 1017 cm−3 is chosen with consideration to threshold voltage and the trade-off between the BV and Ron,sp.Figure Rd and high BV, partial P-buried layer is introduced under the N-drift region. Figure Lpb with different Npb. For a given Npb, as ΔLpb increases, BV increases and then decreases slightly. When ΔLpb = 0.1 μm, Npb = 1 × 1017 cm−3, BV reaches the maximum value 43 V. The insert shows the equipotential contour profile with Npb = 1 × 1017 cm−3. It is indicated that the equipotential contour in the partial P-buried layer structure extends more to substrate with comparison to full P-buried layer. Figure Rd.In order to achieve low Ron,sp as functions of the doping concentration of the P-buried (Npb) for different Nd. For a given Nd, BV has a maximum value with varied Npb, and the maximum of BV increases with the decrease of Nd. However, Ron,sp can be increased as the Nd decreasing. Due to BV required higher than 40 V, the Nd = 5.5 × 1016 cm−3 and Npb = 1 × 1017 cm−3 are chosen. Figure Ron,sp as functions of the thickness of the STI layer (Tsti). Tsti has strong impact on BV and Ron,sp, and it should be designed and optimized carefully as well as our previous work [Tsti < 0.3 μm, the breakdown point under the edge of poly field plate has a high electric field peak. As Tsti increases, the electric field peak is relaxed, and then BV increases. For Tsti = 0.3 μm, BV of 43 V is obtained. For Tsti ≥ 0.3 μm, the electric field peak under the edge of poly field plate is enough low, as a result, the breakdown point transfers to P/N junction under the drain side. As Tsti increases, BV increases and then saturates.Charge balance between N-drift and partial P-buried layer is required to achieve high BV. Figure ous work . For TstRon,sp is 8.5 mΩ·mm2 while BV = 43 V, which is reduced by about 37% compared with our previous work.Figure Rc. Partial P-buried layer is introduced under the N-drift region to enhance BV with keeping charge balance. The fabrication process of the LDMOS in this work is compatible with the existing BCD technology reported in our previous work. The result shows that the Ron,sp of the proposed LDMOS is reduced by 37% at BV of 43 V compared with previous work. With the semiconductor processing technology going to nanometer level, the Ron,sp can reduce further with channel length decrease.A novel ultra-low specific on-resistance LDMOS with enhanced dual-gate and partial P-buried layer is proposed and investigated by numerical simulation in this paper. N-buried layer with high doping concentration is utilized to achieve enhanced dual-gate with reducing


Nature Communications 10.1038/s41467-018-07613-z; published online 05 December 2018.Correction to: The original version of this Article contained an error in Fig. 4a, in which the “=” sign of the equation was inadvertently replaced with a “-“ sign. This has been corrected in the PDF and HTML versions of the Article.


Scientific Reports 10.1038/s41598-017-08007-9, published online 09 August 2017Correction to: The original version of this Article contained an error in the title of the paper, where the word “mycorrhizal” was incorrectly given as “mychorrizal”. This has now been corrected in the PDF and HTML versions of the Article, as well as the Supplementary Information that accompanies the Article.


Numerical results validate the efficiency of the proposed schemes by showing a received data-rate fairness among UEs and an improvement in the UE’s minimum received data rate.Future-generation radio access networks (RAN) are projected to fulfill the diverse requirements of user equipment (UE) by adopting a heterogeneous network (HetNet) environment. Necessary integration of different radio access technologies (RAT), such as 2G, 3G, 4G, wireless local area network (WLAN), and visible light communication (VLC) is inevitable. Moreover, UEs equipped with diverse requirements will be capable of accessing some or all the RATs. The complex HetNet environment with diverse requirements of UEs will present many challenges. The HetNet is likely to suffer severely from load imbalance among the base stations (BSs) from inheriting the traditional user association scheme such as max-SINR /max-RSSI , unless some sophisticated schemes are invented. In this paper, a novel scheme is devised for a joint-user association for load balancing, where BSs are densely deployed and UEs typically have a certain degree of mobility. Unlike most of the present works, a dynamic network is considered where the position and channel condition of the UEs are not fixed. We develop two complex and distributed association schemes based on  The traffic demand in wireless communication systems has surged over the past few years. This trend will continue in the future because of the growing popularity and heavy usage of wireless devices . Accordiuser association, power control, and resource allocation that need to be addressed [To boost areal spectral efficiency, reuse of radio resources is indispensable. However, the deployment of more macro BSs, forming a dense homogeneous network, is an unattractive solution as the deployment costs, along with the operational and maintenance costs, are very high. An alternative to this homogeneous deployment is to have a heterogeneous one, where many low-powered small cells are deployed within the coverage area of a macro BS. A HetNet cellular network consists of randomly located BSs forming a multi-tier cellular network. Here, each tier differs in the average transmitting power, data rate, and deployment density. The HetNet provides big capacity, high performance, ease of operation, and fast deployment, but with the cost of additional revenue, interference, resource management, and backhaul management . HetNet ddressed . The traddressed , used byddressed . This isddressed , and biaddressed  schemes.ddressed . The UE ddressed , where tThe definition of “load” in a cellular network changes according to the environment. The authors in Refs. ,17,19 coIn this paper, we propose low-complexity online distributed algorithms for cell selection in a HetNet environment, while considering two types of load definitions. The BSs and UEs are treated as bidders and objects, respectively, such as that in online combinatorial auctions. We present an analytical model to show the near-optimal performance of our proposed solutions.The remainder of this paper is organized as follows. Recent surveys ,21 show In the offline user association scheme, a static network environment is provided where the complete setup is assumed to be known previously. In general, a throughput optimization problem is formulated for the user association problem. Authors in Refs. ,41,42,43probabilistic and d-choices-based user association schemes. D-choices-based distributed load-balancing algorithms have been successfully applied to various server selection and channel allocation problems [d number of servers uniformly at random and associates with that server, which has a minimum load. A solution to balls into non-uniform bins was investigated in Ref. [n is the number of bins and d is the number of choices for a ball. A more realistic environment was considered in Ref. [d-choices-based scheme has not been applied to the user association problem in a cellular network.The low complexity and dynamic nature of the distributed algorithm is well-suited for the cell selection problem. In this paper, we proposed two types of online distributed algorithms for load-balancing in a heterogeneous cellular network, such as problems , where a in Ref. , where a in Ref. ,where a K-tier HetNet, and denote the set of tiers as U arrives last. The maximum capacity, as well as maximum transmission power varies from one tier to another. Each BS has a positive integer capacity, also termed the size of a BS. We denote j at tier k, where j ∈ k ∈ j at tier k, i, and BS j at tier k. We consider a  in Ref. ; hence,  in Ref. . The avej of tier k. If the BS schedules for equal resources for each UE in an OFDMA-based MAC protocol [j at tier k. Note that if a scheme can evenly distribute UEs among the BSs, then the minimum data rate will also increase as the per-user resource sharing rate will be high.In a conventional network, the UE will associate with that BS whose protocol ,19,27,39i depends on the total number of UEs attached to and sharing the same BS [We assume that equal time-sharing will be used to schedule the UE, in case a BS is associated with multiple UEs. Hence, the throughput of an UE  same BS ,19,39. WSimilarly, while considering the average data rate of a BS as Note that both Equations  and 5)  5) repreBefore the problem statement is formally defined, we highlight the trade-off problem between throughput and fairness. The authors in Ref.  presentein Refs. ,52 proviin Refs. .i obtains utility i is connected to the BS j of tier k and 0 otherwise. We formulate the user association problem for Q1):Q1 is an integer program, it is generally difficult to find the optimal solution. The problem Q1 can easily be converted to a convex optimization problem by simply making the utility function as a logarithmic utility and relaxing the constraint in Equation  as follows:Taking a utility function perspective, we assume that UE ,j∈B,k∈Kxi,j,k∈{0Q2 maximizes the average UE throughput, which ensures fairness in the network. The problem Q2 has same set of constraints as that of Q1.The problem In the following, we consider a static environment and implement a centralized solution, and later compare with our proposed schemes to show the novelty. j. ith UE. The UE obtains the Algorithm 1 Centralized solution1:Assign UEs to the BSs which have 2:Find the BS with a   3:4:L,Find a BS which has a   5:ifthen6:  7:ifthen  8:    Do not consider this base station in future processing9:    Go to Row 210:end if  11:  Go to Row 412:else13:  14:i whose A  Select an UE 15:ifthen  16:    17:    Go to Row 1418:end if  19:  Assign the Selected UE to BS 20:end if21:Update the 22:Goto Row 2d-choices-based selection, which try to balance the load over the whole network.The above-mentioned scheme works well in a static, idealistic environment where there is no mobility and the channel quality remains intact. A slight change in the j at tier k. Higher values of j. The intuition behind this scheme is that higher i arrives at the system, it checks for the BSs in range and calculates the b be the set of BSs the UE can associate with. For each BS that satisfies the SINR constraint i calculates the following association parameter In this probability scheme, the UE selects a BS in a probabilistic manner upon arrival at the network. Actually, the UE considers two parameters, i then calculates the probability of association. The probability of association with a BS is given by:The value of The UE now has information regarding the probability of association with each BS. Equipped with this information, the UE then selects one BS in a probabilistic way. Note that the BS having higher Algorithm 2 Probability scheme1:i ∈ j ∈ k ∈ Initialize 2:for each UE i in the network do3:  4:  Calculate 5:i with BS j of tier k with probability   Associate UE 6:end forEach UE executes the Algorithm 2 at their respective end. In Step 1, all the UEs in the network initialize their association probability to 0. Cell selection begins at Step 2 and in Step 3, the UE collects the BS information, which satisfies the SINR threshold. These are the potential BSs the UE can attach to. As the BS broadcasts their load information periodically, the UE is able to calculate Equation , is diffd number of bins to associate with, and the expected maximum load is given by Equation  load-balancing in different tiers, and (2) load-balancing in each BS in a particular tier. The network traffic for different RATs must flow through the corresponding service gateway, so care must also be taken to balance the load in the service gateway level. We assumed that all BSs broadcasted their load information periodically. The macro BS, having the highest coverage area, has a special task, in that it accumulates all the load parameters of BSs of different tiers, adds them up tier-wise, and then broadcasts the load value. The broadcast signal by macro BS will have the load value of all tiers in the network.The d tiers uniformly at random, and then chooses a tier from the d number of tiers which has a minimum load. The next step is to choose a BS from the already chosen tier. The UE repeats the above procedure for selecting a BS from the chosen tier. The UE chooses d number of BSs uniformly at random, and selects a BS from the d number of BSs which has a minimum load. This procedure ensures load balancing at two levels: (1) tier-level load-balancing, and (2) BS-level load-balancing. Note that tier-level load-balancing is required to shift the load from a heavily loaded tier to a lightly loaded tier. Algorithm 3, d-choices, depicts the above distributed d-choices scheme by considering The UE performs the load-balancing task in two levels. Algorithm 3 Two-level d-choices scheme.1:ith UE | 2:for all UEs, ido3:  Independently choose a set 4:  5:  find 6:  7:  Independently choose a set 8:  9:  find 10:i with    Associate UE 11:end for12:13:14:15:d-choices schemes in two levels. The UE begins its user association procedure by accumulating the BSs which satisfy the d-choices scheme. The same procedure is applied to choose a BS from the chosen tier in Steps 6–9. Then in Step 10, the UE associates with the chosen BS. As the two load parameters, Algorithm 3 executes the d-choices, we consider a hypothetical scenario where there are four tiers and each tier has 10 servers. 1000 UEs arrive at the system one by one to attach with a server. First, the UE randomly picks d-choices, average Jain’s fairness index, and the total probes made by a UE are given in To evaluate the effect of different d chosen tiers and allocate the UE to the minimum loaded BS, then the scenario will be totally different. This strategy may not ensure a load-balancing scenario. Think of a situation where all the servers of a tier have an equal number of UEs, but the total UE number is less than another unbalanced tier. The balanced tier will never be selected, even though the total number of UEs are less than other tiers. The final outcome will be an unbalanced allocation; hence, this method is not desirable.Apart from selecting the tier with a minimum load, a new scheme could be considered for the evaluation. If we select the most unbalanced tier from the In this section, we give an analytic model of the performance for the online algorithms by considering mong UEs . The utiDefinition 1.The UE utility function Lemma 1.Proof. Our proof is, in essence, similar to the proof given in Ref. . Here, wFor every For N = ∅, we need to show that submodular and monotone.The function i’s utility for BS j belongs to tier k is given as Now, we prove the fact that the UE utility function, Now, we establish the similarity between Equations 15) and (17). Equation ( and (17)The expression Equation  exhibitsEquation , we concTheorem 1.Under the submodularity and monotonicity of Proof. The submodularity and monotonocity of  in Ref. , we can C, where the time complexity for Algorithm 2 will then be of k, in the first level and a BS, j, from that tier in the second level. Both selections of the tier and BS are performed on the number of choices In this section, we compute the time complexity of the different proposed algorithms. The time complexity of Algorithm 2 depends on the number of UEs and BSs. Each UE calculates the D), where D is in kilometers. The path-loss model for pico and femto BS is taken as 37 + 30*logD), where D is in meters. We assume log-normal shadowing with a standard deviation d values in Algorithm 3, we restrict the choices to only two, as the probability for higher d decreases in a real environment [Here, we consider a three-tier HetNet environment consisting of macro, pico, and femto BSs. We develop the HetNet environment and run the simulation on the MATLAB platform. The BSs are uniformly and independently distributed in a 2000 × 2000 mironment . We perf(a) Cumulative distribution function (CDF) of spectrum efficiency(b) Normalized minimum user rate  R be the minimum data rate experienced by the UE(s) in the network. The value R will vary from one scheme to other. Let there be N schemes, so the normalized user rate, The minimum user rate is normalized with the optimum value. The optimum value is the maximum among the minimum user rate on each scheme. Let (c) Normalized sum user rate  S be the sum user rate of the network. The value S will vary from one scheme to other. The normalized sum user rate, The sum user rate of the network is normalized with the optimum value. The optimum value is the maximum among the sum user rate on each scheme. Let (d) Jain’s fairness  max-SINR, we also consider the BS-centric algorithm (BS-centric), proposed in Ref. [max-SINR shows an uneven UE share. However, we can see that a considerable amount of UEs are being offloaded to Pico and Femto BSs in the case of Prob. scheme 1, Prob. scheme 2, and BS-centric. UEs associated with macro BS are being offloaded to both Pico and Femto BSs, which verifies the effectiveness of our proposed probability schemes. The d-choices schemes, on the other hand, focus on tier-wise balanced UE allocation. d-choices 1 and d-choices 2 schemes. A similar trend can be seen in non-uniform deployment. More UEs are offloaded to pico and femto BSs.Along with the traditional  in Ref. , as our Prob. scheme 1, Prob. scheme 2, d-choices 1, d-choices 2, and BS-centric schemes perform quite well at spectrum efficiency as compared to the traditional max-SINR scheme. When considering the BS-centric one, our proposed schemes give better service to the majority of UEs—approximately 65%—in terms of spectrum efficiency. In all the schemes (except the max-SINR) a majority of the UEs have a higher spectrum efficiency. Approximately 80% of UEs in all the schemes have higher spectrum efficiency, as compared to max-SINR. This is because the macro BS is highly loaded, and the total resource is divided among the UEs. This, in turn, lowers the spectrum efficiency of the associated UEs. For non-uniform UE deployment, although many UEs are placed around the lower-powered BSs, the outcome is similar to that of uniform deployment. All of our proposed schemes performed better in the mid-segment of the CDF graph, where the spectrum efficiency of the UEs are higher than that of max-SINR. Our proposed scheme, Prob. scheme 1, continuously performs better as compared to BS-centric for non-uniform deployment, as shown in One way to give a better experience to the UEs experiencing low spectrum efficiency is to take the resource from strong UEs and assign it to weak UEs. This task of taking the resource from strong UEs is done through load-balancing. max-SINR is about 0.30, while Prob. scheme 1 and Prob. scheme 2 show improved performance. Prob. scheme 1 shows a 70% improvement, while a similar trend can be seen for Prob. schemes 2. d-choices 1 and d-choices 2 show good results, and the minimum UE rate is higher, as compared to max-SINR. This gain in minimum user rate ensures that our scheme performs well for the cell edge UE, which mainly suffers with low data rate. All our proposed schemes perform better than BS-centric and max-SINR, while Prob. scheme 1 achieves the highest minimum user rate. A similar trend can be seen in non-uniform deployment, as shown in Prob. scheme 1 continues to perform well in non-uniform deployment. The max-SINR scheme, on the other hand, continues to perform poorly. The metric “normalized sum rate”, denoted as max-SINR schemes, the UEs near to the BSs receive higher signal strength. As can be seen in max-SINR. On the other hand, for the non-uniform deployment, as shown in max-SINR performs slightly better. However, the max-SINR can achieve a higher sum user rate while ignoring the user fairness.Another performance metric for evaluation of the network performance is considering the normalized data rate of the UEs. The normalized minimum user rate, max-SINR in uniform deployment is 0.1, which is very low. Unlike max-SINR, our proposed schemes perform better. The fairness index nearly equals to 0.6 for all the proposed schemes. The same outcome can be observed in the non-uniform UE deployment scenario.Jain’s fairness index of UEs’ received data rate in a network shows the overall fairness. The fairness index becomes 1 when all UEs receive the same rate. d-choices load1 and d-choices load2 schemes. Each time the UE encounters a different load environment, it tries to balance the load of the network. For non-uniform UE deployment, the UEs are restricted to moving around the BS. d-choices schemes continue to show good results, and there is slight poor performance by the Prob. schemes, but it is better than max-SINR.User mobility is an unavoidable condition in a network. A network needs to perform well in a mobility environment. We considered Jain’s fairness index for the UE share in different schemes to show our performance on the mobility. A network is said to be balanced in a mobility environment if the fairness index is high in different time instances. We monitored the network for ten time instances and noted the fairness. We adopted a random walk for the UEs. In The spatial distribution of network elements, such as BSs and UEs, plays an important role for the calculation of different network parameters. From an analytical and tractability point of view, many researchers have considered the Poisson point process (PPP), a clustered point process for BS and UE deployment. The authors in Ref.  assumed Prob. and d-choices schemes. In the K-tier HetNet, the user association, conventionally connected to the BS with maximum SINR, results in less association with pico and femtocells. This ultimately leads to load imbalance in the whole network. Unlike the max-SINR, in the Prob. scheme, UE selects the BS in a probabilistic manner upon arrival into the network by considering both SINR and the load of a BS. This technique eventually offloads the UEs toward small cells. Inspired by the “balls and bins” load-balancing problem and the power of two choices, we proposed the d-choices association scheme. The UE can choose d number of BSs and selects the BS which has the lightest load among the chosen BSs. Prob. scheme 2 showed better results in minimum user rate, while d-choices 2 showed good results for tier-wise load-balancing in both uniform and non-uniform kinds of UE deployment. Load-balancing naturally provides the possible inclusion of more numbers of UEs in the network, typically giving better fairness and improving UE experience. These distributed schemes show an impressive result in dynamic environments by reducing the signal overhead between UE and BS, increasing the throughput per UE and network load balance. Future work includes the optimization of small cells in k-tier HetNet, while considering both energy and spectral efficiency with high user mobility. In this paper, we considered a three-tier HetNet environment, proposing user association schemes and dynamic load-balancing through


Nature Communications 10.1038/s41467-019-11373-9, published online 31 July 2019.Correction to: The original version of this Article contained an error in Figure 3. In panel b, the legend embedded in the panel was incorrect and indicated a solid grey line for w/w and a dotted black line for f/f. The correct legend is now provided and shows a dotted black line for w/w and a solid grey line for f/f. This has been corrected in both the PDF and HTML versions of the Article.