Subquadratic Time Research Articles

Two-Sided Convexity Testing with Certificates

We revisit the problem of property testing for convex position for point sets in ℝ𝑑. Our results draw from previous ideas of Czumaj, Sohler, and Ziegler (2000). First, their testing algorithm is redesigned and its analysis is revised for correctness. Second, its functionality is expanded by (i) exhibiting both negative and positive certificates along with the convexity determination, and (ii) significantly extending the input range for moderate and higher dimensions.The behavior of the randomized tester on input set 𝑃 ⊂ ℝ𝑑 is as follows: (i) if 𝑃 is in convex position, it accepts; (ii) if 𝑃 is far from convex position, with probability at least 2/3, it rejects and outputs a (𝑑 +2)-point witness of non-convexity as a negative certificate; (iii) if 𝑃 is close to convex position, with probability at least 2/3, it accepts and outputs a subset in convex position that is a suitable approximation of the largest subset in convex position. The algorithm examines a sublinear number of points and runs in subquadratic time for every fixed dimension 𝑑.

Compact and Low-Latency FPGA-Based Number Theoretic Transform Architecture for CRYSTALS Kyber Postquantum Cryptography Scheme

In the modern era of the Internet of Things (IoT), especially with the rapid development of quantum computers, the implementation of postquantum cryptography algorithms in numerous terminals allows them to defend against potential future quantum attack threats. Lattice-based cryptography can withstand quantum computing attacks, making it a viable substitute for the currently prevalent classical public-key cryptography technique. However, the algorithm’s significant time complexity places a substantial computational burden on the already resource-limited chip in the IoT terminal. In lattice-based cryptography algorithms, the polynomial multiplication on the finite field is well known as the most time-consuming process. Therefore, investigations into efficient methods for calculating polynomial multiplication are essential for adopting these quantum-resistant lattice-based algorithms on a low-profile IoT terminal. Number theoretic transform (NTT), a variant of fast Fourier transform (FFT), is a technique widely employed to accelerate polynomial multiplication on the finite field to achieve a subquadratic time complexity. This study presents an efficient FPGA-based implementation of number theoretic transform for the CRYSTAL Kyber, a lattice-based public-key cryptography algorithm. Our hybrid design, which supports both forward and inverse NTT, is able run at high frequencies up to 417 MHz on a low-profile Artix7-XC7A100T and achieve a low latency of 1.10μs while achieving state-of-the-art hardware efficiency, consuming only 541-LUTs, 680 FFs, and four 18 Kb BRAMs. This is made possible thanks to the newly proposed multilevel pipeline butterfly unit architecture in combination with employing an effective coefficient accessing pattern.

Graphs cannot be indexed in polynomial time for sub-quadratic time string matching, unless SETH fails

The string matching problem on a node-labeled graph G=(V,E) asks whether a given pattern string P equals the concatenation of node labels of some path in G. This is a basic primitive in various problems in bioinformatics, graph databases, or networks, and it was recently proven that solving it in time O(|E|1−ϵ|P|) or O(|E||P|1−ϵ) is hard under OVH, the Orthogonal Vectors Hypothesis, and thus under SETH, the Strong Exponential Time Hypothesis (Equi et al., 2019 [11]). We consider its indexed version, where the graph is indexed to support string queries. We show that, under OVH, no polynomial-time index of the graph performed in time O(|E|α) can support querying P in time O(|P|+|E|δ|P|β), with either δ<1 or β<1.We present our techniques as a general framework, introducing the notion of linear independent-components (lic) reduction, from which we derive our result. This allows us to also translate the quadratic conditional lower bound of Backurs and Indyk (2015) [48] for the problem of matching a query string inside a text, under edit distance, into an analogous tight quadratic lower bound for its indexed version. This improves the recent result of Cohen-Addad, Feuilloley and Starikovskaya (2019) [49], with a slightly different boundary condition. We also apply our technique to obtain the first quadratic indexing lower bounds for Fréchet distance and rooted unlabeled subtree-isomorphism queries.

Addition machines, automatic functions and open problems of Floyd and Knuth

Floyd and Knuth investigated in 1990 register machines which can add, subtract and compare integers as primitive operations. They asked whether their current bound on the number of registers for multiplying and dividing fast (running in time linear in the size of the input) can be improved and whether one can output fast the powers of two summing up to a positive integer in subquadratic time. Both questions are answered positively. Furthermore, it is shown that every function computed by only one register is automatic and that the automatic functions with one input can be computed with four registers in linear time; automatic functions with a larger number of inputs can be computed with 5 registers in linear time. There is a nonautomatic function with one input which can be computed with two registers in linear time.

Circular Silhouette and a Fast Algorithm.

Circular data clustering has recently been solved exactly in sub-quadratic time. However, the solution requires a given number of clusters; methods for choosing this number on linear data are inapplicable to circular data. To fill this gap, we introduce the circular silhouette to measure cluster quality and a fast algorithm to calculate the average silhouette width. The algorithm runs in linear time to the number of points on sorted data, instead of quadratic time by the silhouette definition. Empirically, it is over 3000 times faster than by silhouette definition on 1,000,000 circular data points in five clusters. On simulated datasets, the algorithm returned correct numbers of clusters. We identified clusters on round genomes of human mitochondria and bacteria. On sunspot activity data, we found changed solar-cycle patterns over the past two centuries. Using the circular silhouette not only eliminates the subjective selection of number of clusters, but is also scalable to big circular and periodic data abundant in science, engineering, and medicine.

Efficient computation of Riemann–Roch spaces for plane curves with ordinary singularities

We revisit the seminal Brill–Noether algorithm for plane curves with ordinary singularities. Our new approach takes advantage of fast algorithms for polynomials and structured matrices. We design a new probabilistic algorithm of Las Vegas type that computes a Riemann–Roch space in expected sub-quadratic time.

Testing Polynomials for Vanishing on Cartesian Products of Planar Point Sets: Collinearity Testing and Related Problems

We present subquadratic algorithms, in the algebraic decision-tree model of computation, for detecting whether there exists a triple of points, belonging to three respective sets A, B, and C of points in the plane, that satisfy a certain polynomial equation or two equations. The best known instance of such a problem is testing for the existence of a collinear triple of points in \(A\times B\times C\), a classical 3SUM-hard problem that has so far defied any attempt to obtain a subquadratic solution, whether in the (uniform) real RAM model, or in the algebraic decision-tree model. While we are still unable to solve this problem, in full generality, in subquadratic time, we obtain such a solution, in the algebraic decision-tree model, that uses roughly \(O(n^{28/15})\) constant-degree polynomial sign tests, for the special case where two of the sets lie on two respective one-dimensional curves and the third is placed arbitrarily in the plane. Our technique is fairly general, and applies to many other problems where we seek a triple that satisfies a single polynomial equation, e.g., determining whether \(A\times B\times C\) contains a triple spanning a unit-area triangle. This result extends recent work by Barba et al. (2017) and by Chan (2018), where all three sets A, B, and C are assumed to be one-dimensional. As a second application of our technique, we again have three n-point sets A, B, and C in the plane, and we want to determine whether there exists a triple \((a,b,c) \in A\times B\times C\) that simultaneously satisfies two independent real polynomial equations. For example, this is the setup when testing for collinearity in the complex plane, when each of the sets A, B, C lies on some constant-degree algebraic curve. We show that problems of this kind can be solved with roughly \(O(n^{24/13})\) constant-degree polynomial sign tests.

Algorithms for Colinear Chaining with Overlaps and Gap Costs.

Colinear chaining has proven to be a powerful heuristic for finding near-optimal alignments of long DNA sequences (e.g., long reads or a genome assembly) to a reference. It is used as an intermediate step in several alignment tools that employ a seed-chain-extend strategy. Despite this popularity, efficient subquadratic time algorithms for the general case where chains support anchor overlaps and gap costs are not currently known. We present algorithms to solve the colinear chaining problem with anchor overlaps and gap costs in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Õ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math> <b>time, where</b> <i>n</i> denotes the count of anchors. The degree of the polylogarithmic factor depends on the type of anchors used (e.g., fixed-length anchors) and the type of precedence an optimal anchor chain is required to satisfy. We also establish the first theoretical connection between colinear chaining cost and edit distance. Specifically, we prove that for a fixed set of anchors under a carefully designed chaining cost function, the optimal "anchored" edit distance equals the optimal colinear chaining cost. The anchored edit distance for two sequences and a set of anchors is only a slight generalization of the standard edit distance. It adds an additional cost of one to an alignment of two matching symbols that are not supported by any anchor. Finally, we demonstrate experimentally that optimal colinear chaining cost under the proposed cost function can be computed orders of magnitude faster than edit distance, and achieves correlation coefficient &gt;0.9 with edit distance for closely as well as distantly related sequences.

Diameter, Eccentricities and Distance Oracle Computations on H-Minor Free Graphs and Graphs of Bounded (Distance) Vapnik–Chervonenkis Dimension

Under the strong exponential-time hypothesis, the diameter of general unweighted graphs cannot be computed in truly subquadratic time (in the size of the input), as shown by Roditty and Williams. Nevertheless there are several graph classes for which this can be done such as bounded-treewidth graphs, interval graphs, and planar graphs, to name a few. We propose to study unweighted graphs of constant distance Vapnik–Chervonenkis (VC)-dimension as a broad generalization of many such classes—where the distance VC-dimension of a graph is defined as the VC-dimension of its ball hypergraph whose hyperedges are the balls of all possible radii and centers in . In particular for any fixed , the class of -minor free graphs has distance VC-dimension at most . Our first main result is a Monte Carlo algorithm that on graphs of distance VC-dimension at most , for any fixed , either computes the diameter or concludes that it is larger than in time , where only depends on and the notation suppresses polylogarithmic factors. We thus obtain a truly subquadratic-time parameterized algorithm for computing the diameter on such graphs. Then as a byproduct of our approach, we get a truly subquadratic-time randomized algorithm for constant diameter computation on all the nowhere dense graph classes. The latter classes include all proper minor-closed graph classes, bounded-degree graphs, and graphs of bounded expansion. Before our work, the only known such algorithm was resulting from an application of Courcelle’s theorem; see Grohe, Kreutzer, and Siebertz [J. ACM, 64 (2017), pp. 1–32]. For any graph of constant distance VC-dimension, we further prove the existence of an exact distance oracle in truly subquadratic space, that answers distance queries in truly sublinear time (in the number of vertices). The latter generalizes prior results on proper minor-closed graph classes to a much larger graph class. Finally, we show how to remove the dependency on for any graph class that excludes a fixed graph as a minor. More generally, our techniques apply to any graph with constant distance VC-dimension and polynomial expansion (or equivalently having strongly sublinear balanced separators). As a result for all such graphs one obtains a truly subquadratic-time deterministic algorithm for computing all the eccentricities, and thus both the diameter and the radius. Our approach can be generalized to the -minor free graphs with bounded positive integer weights. We note that all our algorithms for the diameter problem can be adapted for computing the radius, and more generally all the eccentricities. Our approach is based on the work of Chazelle and Welzl who proved the existence of spanning paths with strongly sublinear stabbing number for every hypergraph of constant VC-dimension. We show how to compute such paths efficiently by combining known algorithms for the stabbing number problem with a clever use of -nets, region decomposition, and other partition techniques.

Algorithms and Complexity on Indexing Founder Graphs

We study the problem of matching a string in a labeled graph. Previous research has shown that unless the Orthogonal Vectors Hypothesis (OVH) is false, one cannot solve this problem in strongly sub-quadratic time, nor index the graph in polynomial time to answer queries efficiently (Equi et al. ICALP 2019, SOFSEM 2021). These conditional lower-bounds cover even deterministic graphs with binary alphabet, but there naturally exist also graph classes that are easy to index: For example, Wheeler graphs (Gagie et al. Theor. Comp. Sci. 2017) cover graphs admitting a Burrows-Wheeler transform -based indexing scheme. However, it is NP-complete to recognize if a graph is a Wheeler graph (Gibney, Thankachan, ESA 2019). We propose an approach to alleviate the construction bottleneck of Wheeler graphs. Rather than starting from an arbitrary graph, we study graphs induced from multiple sequence alignments (MSAs). Elastic degenerate strings (Bernadini et al. SPIRE 2017, ICALP 2019) can be seen as such graphs, and we introduce here their generalization: elastic founder graphs. We first prove that even such induced graphs are hard to index under OVH. Then we introduce two subclasses, repeat-free and semi-repeat-free graphs, that are easy to index. We give a linear time algorithm to construct a repeat-free (non-elastic) founder graph from a gapless MSA, and (parameterized) near-linear time algorithms to construct a semi-repeat-free (repeat-free, respectively) elastic founder graph from general MSA. Finally, we show that repeat-free founder graphs admit a reduction to Wheeler graphs in polynomial time.

Developing Biceps to completely compute in subquadratic time a new generic type of bicluster in dense and sparse matrices

Developing Biceps to completely compute in subquadratic time a new generic type of bicluster in dense and sparse matrices

Efficient Computation of Sequence Mappability

Sequence mappability is an important task in genome resequencing. In the (k, m)-mappability problem, for a given sequence T of length n, the goal is to compute a table whose ith entry is the number of indices j ne i such that the length-m substrings of T starting at positions i and j have at most k mismatches. Previous works on this problem focused on heuristics computing a rough approximation of the result or on the case of k=1. We present several efficient algorithms for the general case of the problem. Our main result is an algorithm that, for k=O(1), works in O(n) space and, with high probability, in O(n cdot min {m^k,log ^k n}) time. Our algorithm requires a careful adaptation of the k-errata trees of Cole et al. [STOC 2004] to avoid multiple counting of pairs of substrings. Our technique can also be applied to solve the all-pairs Hamming distance problem introduced by Crochemore et al. [WABI 2017]. We further develop O(n^2)-time algorithms to compute all (k, m)-mappability tables for a fixed m and all kin {0,ldots ,m} or a fixed k and all min {k,ldots ,n}. Finally, we show that, for k,m = Theta (log n), the (k, m)-mappability problem cannot be solved in strongly subquadratic time unless the Strong Exponential Time Hypothesis fails. This is an improved and extended version of a paper presented at SPIRE 2018.

Completing gene trees without species trees in sub-quadratic time.

As genome-wide reconstruction of phylogenetic trees becomes more widespread, limitations of available data are being appreciated more than ever before. One issue is that phylogenomic datasets are riddled with missing data, and gene trees, in particular, almost always lack representatives from some species otherwise available in the dataset. Since many downstream applications of gene trees require or can benefit from access to complete gene trees, it will be beneficial to algorithmically complete gene trees. Also, gene trees are often unrooted, and rooting them is useful for downstream applications. While completing and rooting a gene tree with respect to a given species tree has been studied, those problems are not studied in depth when we lack such a reference species tree. We study completion of gene trees without a need for a reference species tree. We formulate an optimization problem to complete the gene trees while minimizing their quartet distance to the given set of gene trees. We extend a seminal algorithm by Brodal et al. to solve this problem in quasi-linear time. In simulated studies and on a large empirical data, we show that completion of gene trees using other gene trees is relatively accurate and, unlike the case where a species tree is available, is unbiased. Our method, tripVote, is available at https://github.com/uym2/tripVote. Supplementary data are available at Bioinformatics online.

Addition Machines, Automatic Functions and Open Problems of Floyd and Knuth

Floyd and Knuth investigated in 1990 register machines which can add, subtract and compare integers as primitive operations. They asked whether their current bound on the number of registers for multiplying and dividing fast (running in time linear in the size of the input) can be improved and whether one can output fast the powers of two summing up to a positive integer in subquadratic time. Both questions are answered positively. Furthermore, it is shown that every function computed by only one register is automatic and that automatic functions with one input can be computed with four registers in linear time; automatic functions with a larger number of inputs can be computed with 5 registers in linear time. There is a nonautomatic function with one input which can be computed with two registers in linear time.

Fast Diameter Computation within Split Graphs

When can we compute the diameter of a graph in quasi linear time? We address this question for the class of {\em split graphs}, that we observe to be the hardest instances for deciding whether the diameter is at most two. We stress that although the diameter of a non-complete split graph can only be either $2$ or $3$, under the Strong Exponential-Time Hypothesis (SETH) we cannot compute the diameter of an $n$-vertex $m$-edge split graph in less than quadratic time -- in the size $n+m$ of the input. Therefore it is worth to study the complexity of diameter computation on {\em subclasses} of split graphs, in order to better understand the complexity border. Specifically, we consider the split graphs with bounded {\em clique-interval number} and their complements, with the former being a natural variation of the concept of interval number for split graphs that we introduce in this paper. We first discuss the relations between the clique-interval number and other graph invariants such as the classic interval number of graphs, the treewidth, the {\em VC-dimension} and the {\em stabbing number} of a related hypergraph. Then, in part based on these above relations, we almost completely settle the complexity of diameter computation on these subclasses of split graphs: - For the $k$-clique-interval split graphs, we can compute their diameter in truly subquadratic time if $k={\cal O}(1)$, and even in quasi linear time if $k=o(\log{n})$ and in addition a corresponding ordering of the vertices in the clique is given. However, under SETH this cannot be done in truly subquadratic time for any $k = \omega(\log{n})$. - For the {\em complements} of $k$-clique-interval split graphs, we can compute their diameter in truly subquadratic time if $k={\cal O}(1)$, and even in time ${\cal O}(km)$ if a corresponding ordering of the vertices in the stable set is given. Again this latter result is optimal under SETH up to polylogarithmic factors. Our findings raise the question whether a $k$-clique interval ordering can always be computed in quasi linear time. We prove that it is the case for $k=1$ and for some subclasses such as bounded-treewidth split graphs, threshold graphs and comparability split graphs. Finally, we prove that some important subclasses of split graphs -- including the ones mentioned above -- have a bounded clique-interval number.

The diameter of AT‐free graphs

AbstractA graph algorithm is truly subquadratic if it runs in time on connected ‐edge graphs, for some positive . Roditty and Vassilevska Williams (STOC'13) proved that under plausible complexity assumptions, there is no truly subquadratic algorithm for computing the diameter of general graphs. In this study, we present positive and negative results on the existence of such algorithms for computing the diameter on some special graph classes. Specifically, three vertices in a graph form an asteroidal triple (AT) if between any two of them there exists a path that avoids the closed neighbourhood of the third one. We call a graph AT‐free if it does not contain an AT. We first prove that for all ‐edge AT‐free graphs, one can compute all the eccentricities in truly subquadratic time. For the AT‐free bipartite graphs, it can be improved to linear time. Then, we extend our study to several subclasses of chordal graphs—all of them generalizing interval graphs in various ways—as an attempt to understand which of the properties of AT‐free graphs, or natural generalizations of the latter, can help in the design of fast algorithms for the diameter problem on broader graph classes. For instance, for all chordal graphs with a dominating shortest‐path, there is a linear‐time algorithm for computing a diametral pair if the diameter is at least four. However, already for split graphs with a dominating edge, under plausible complexity assumptions, there is no truly subquadratic algorithm for deciding whether the diameter is either 2 or 3.

Approximating Edit Distance in Truly Subquadratic Time: Quantum and MapReduce

The edit distance between two strings is defined as the smallest number of insertions , deletions , and substitutions that need to be made to transform one of the strings to another one. Approximating edit distance in subquadratic time is “one of the biggest unsolved problems in the field of combinatorial pattern matching” [37]. Our main result is a quantum constant approximation algorithm for computing the edit distance in truly subquadratic time. More precisely, we give an quantum algorithm that approximates the edit distance within a factor of 3. We further extend this result to an quantum algorithm that approximates the edit distance within a larger constant factor. Our solutions are based on a framework for approximating edit distance in parallel settings. This framework requires as black box an algorithm that computes the distances of several smaller strings all at once. For a quantum algorithm, we reduce the black box to metric estimation and provide efficient algorithms for approximating it. We further show that this framework enables us to approximate edit distance in distributed settings. To this end, we provide a MapReduce algorithm to approximate edit distance within a factor of , with sublinearly many machines and sublinear memory. Also, our algorithm runs in a logarithmic number of rounds.

A fully polynomial parameterized algorithm for counting the number of reachable vertices in a digraph

We consider the problem of counting the number of vertices reachable from each vertex in a digraph G, which is equal to computing all the out-degrees of the transitive closure of G. The current (theoretically) fastest algorithms run in quadratic time; however, Borassi has shown that this problem is not solvable in truly subquadratic time unless the Strong Exponential Time Hypothesis fails [Borassi, 2016 [13]]. In this paper, we present an O(f3n)-time exact algorithm, where n is the number of vertices in G and f is the feedback edge number of G. Our algorithm thus runs in truly subquadratic time for digraphs of f=O(n13−ϵ) for any ϵ>0, i.e., the number of edges is n plus O(n13−ϵ), and is fully polynomial fixed parameter tractable, the notion of which was first introduced by Fomin et al. (2018) [22]. We also show that the same result holds for vertex-weighted digraphs, where the task is to compute the total weights of vertices reachable from each vertex.

Improved MPC Algorithms for Edit Distance and Ulam Distance

Edit distance is one of the most fundamental problems in combinatorial optimization to measure the similarity between strings. Ulam distance is a special case of edit distance where no character is allowed to appear more than once in a string. Recent developments have been very fruitful for obtaining fast and parallel algorithms for both edit distance and Ulam distance. In this work, we present an almost optimal MPC (massively parallel computation) algorithm for Ulam distance and improve MPC algorithms for edit distance. Our algorithm for Ulam distance is almost optimal in the sense that (1) the approximation factor of our algorithm is 1+ε1+ε, (2) the round complexity of our algorithm is constant, (3) the total memory of our algorithm is almost linear (~O <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ε</sub> (n)Õε(n)), and (4) the overall running time of our algorithm is almost linear which is the best known for Ulam distance. We also improve the work of Hajiaghayi et al. for edit distance in terms of total memory. The best previously known MPC algorithm for edit distance requires ~O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2x</sup> )Õ(n2x) machines when the memory of each machine is bounded by ~O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1-x</sup> )Õ(n1-x). In this work, we improve the number of machines to ~O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(9/5)x</sup> )Õ(n(9/5)x) while keeping the memory limit intact. Moreover, the round complexity of our algorithm is constant and the total running time of our algorithm is truly subquadratic. However, our improvement comes at the expense of a constant factor in the approximation guarantee of the algorithm. This improvement is inspired by the recent techniques of Boroujeni et al. and Chakraborty et al. for obtaining truly subquadratic time algorithms for edit distance.

An independent condensed nearest neighbor classification technique for medical image retrieval

Nowadays, medical images play a vital role in the clinical diagnosis system, because these images contain a vast amount of medical information. The huge amount of medical images is generated everyday using the digital imaging techniques in medical examination centers and hospitals. The generated data are stored in large database and retrieving the same images is a significant task for better diagnosis. However, manual retrieval of clinical data is a tough and time consuming process due to the large database. The memory requirement and computation complexity of the existing condensed nearest neighbor (CNN) is more, it is solved by proposed independent condensed nearest neighbor (ICNN). The ICNN classification technique is developed to automatically retrieve the medical images from large database. The important features are extracted by using histogram of gradient (HOG) technique. The sub-quadratic time complexity presented in ICNN requires only few iterations to retrieve the query images, which improves the retrieval accuracy of the proposed technique. The experiments are used to test the performance of ICNN method in terms of classification and retrieval presentation. The proposed ICNN method outperforms existing CNN method by achieving specificity of 99.55% in the classification performance.