Subquadratic Time Research Articles

On the Complexity of Strongly Connected Components in Directed Hypergraphs

We study the complexity of some algorithmic problems on directed hypergraphs and their strongly connected components (SCCs). The main contribution is an almost linear time algorithm computing the terminal strongly connected components (i.e. SCCs which do not reach any components but themselves). "Almost linear" here means that the complexity of the algorithm is linear in the size of the hypergraph up to a factor alpha(n), where alpha is the inverse of Ackermann function, and n is the number of vertices. Our motivation to study this problem arises from a recent application of directed hypergraphs to computational tropical geometry. We also discuss the problem of computing all SCCs. We establish a superlinear lower bound on the size of the transitive reduction of the reachability relation in directed hypergraphs, showing that it is combinatorially more complex than in directed graphs. Besides, we prove a linear time reduction from the well-studied problem of finding all minimal sets among a given family to the problem of computing the SCCs. Only subquadratic time algorithms are known for the former problem. These results strongly suggest that the problem of computing the SCCs is harder in directed hypergraphs than in directed graphs.

Necklaces, Convolutions, and X+Y

We give subquadratic algorithms that, given two necklaces each with n beads at arbitrary positions, compute the optimal rotation of the necklaces to best align the beads. Here alignment is measured according to the ℓ p norm of the vector of distances between pairs of beads from opposite necklaces in the best perfect matching. We show surprisingly different results for p=1, p even, and p=∞. For p even, we reduce the problem to standard convolution, while for p=∞ and p=1, we reduce the problem to (min,+) convolution and $(\operatorname {median},+)$ convolution. Then we solve the latter two convolution problems in subquadratic time, which are interesting results in their own right. These results shed some light on the classic sorting X+Y problem, because the convolutions can be viewed as computing order statistics on the antidiagonals of the X+Y matrix. All of our algorithms run in o(n 2) time, whereas the obvious algorithms for these problems run in Θ(n 2) time.

A linear algorithm for 3-letter longest common weakly increasing subsequence

The problem of finding a longest weakly increasing common subsequence (LCWIS) of two sequences is a variant of the popular longest common subsequence (LCS) problem. While there are no known methods to find LCS in truly sub-quadratic time, there are faster algorithms to compute LCWIS if the alphabet size is small enough. We present a linear-time algorithm finding LCWIS over 3-letter alphabet. Up to now, the fastest known algorithm was O(min{m+nlogn,mloglogm}), where m≥n denote lengths of the sequences.

Fast multi-sequence shift-register synthesis with the Euclidean algorithm

Feng and Tzeng's generalization of the Extended Euclidean Algorithmsynthesizes the shortest-length linear feedback shift-register for $s \geq 1$ sequences, where each sequence has the same length $n$. In this contribution, it is shown that Feng and Tzeng's algorithm which solves this multi-sequence shift-register problem has time complexity $\mathcal{O}(sn^2)$. An acceleration based on the Divide and Conquer strategy is proposed and it is proven that subquadratic time complexity is achieved.

Sub-quadratic time and linear space data structures for permutation matching in binary strings

Given a pattern P of length n and a text T of length m, the permutation matching problem asks whether any permutation of P occurs in T. Indexing a string for permutation matching seems to be quite hard in spite of the existence of a simple non-indexed solution. In this paper, we devise several o ( n 2 ) time data structures for a binary string capable of answering permutation queries in O ( m ) time. In particular, we first present two O ( n 2 / log n ) time data structures and then improve the data structure construction time to O ( n 2 / log 2 n ) . The space complexity of the data structures remains linear.

On Approximate Jumbled Pattern Matching in Strings

Given a string s, the Parikh vector of s, denoted p(s), counts the multiplicity of each character in s. Searching for a match of a Parikh vector q in the text s requires finding a substring t of s with p(t)=q. This can be viewed as the task of finding a jumbled (permuted) version of a query pattern, hence the term Jumbled Pattern Matching. We present several algorithms for the approximate version of the problem: Given a string s and two Parikh vectors u,v (the query bounds), find all maximal occurrences in s of some Parikh vector q such that u≤q≤v. This definition encompasses several natural versions of approximate Parikh vector search. We present an algorithm solving this problem in sub-linear expected time using a wavelet tree of s, which can be computed in time O(n) in a preprocessing phase. We then discuss a Scrabble-like variation of the problem, in which a weight function on the letters of s is given and one has to find all occurrences in s of a substring t with maximum weight having Parikh vector p(t)≤v. For the case of a binary alphabet, we present an algorithm which solves the decision version of the Approximate Jumbled Pattern Matching problem in constant time, by indexing the string in subquadratic time.

OPTIMAL PAIRWISE ALIGNMENT OF FIXED PROTEIN STRUCTURES IN SUBQUADRATIC TIME

The problem of finding an optimal structural alignment for a pair of superimposed proteins is often amenable to the Smith-Waterman dynamic programming algorithm, which runs in time proportional to the product of lengths of the sequences being aligned. While the quadratic running time is acceptable for computing a single alignment of two fixed protein structures, the time complexity becomes a bottleneck when running the Smith-Waterman routine multiple times in order to find a globally optimal superposition and alignment of the input proteins. We present a subquadratic running time algorithm capable of computing an alignment that optimizes one of the most widely used measures of protein structure similarity, defined as the number of pairs of residues in two proteins that can be superimposed under a predefined distance cutoff. The algorithm presented in this article can be used to significantly improve the speed-accuracy tradeoff in a number of popular protein structure alignment methods.

Asymptotically efficient algorithms for skyline probabilities of uncertain data

Skyline computation is widely used in multicriteria decision making. As research in uncertain databases draws increasing attention, skyline queries with uncertain data have also been studied. Some earlier work focused on probabilistic skylines with a given threshold; Atallah and Qi [2009] studied the problem to compute skyline probabilities for all instances of uncertain objects without the use of thresholds, and proposed an algorithm with subquadratic time complexity. In this work, we propose a new algorithm for computing all skyline probabilities that is asymptotically faster: worst-case O(n √ n log n ) time and O(n) space for 2D data; O ( n 2−1/d log d −1 n ) time and O(n log d −2 n ) space for d -dimensional data. Furthermore, we study the online version of the problem: Given any query point p (unknown until the query time), return the probability that no instance in the given data set dominates p . We propose an algorithm for answering such an online query for d -dimensional data in O(n 1−1/ d log d −1 n ) time after preprocessing the data in O(n 2−1/d log d −1 ) time and space.

LCS approximation via embedding into locally non-repetitive strings

A classical measure of similarity between strings is the length of the longest common subsequence (LCS) between the two given strings. The search for efficient algorithms for finding the LCS has been going on for more than three decades. To date, all known algorithms may take quadratic time (shaved by logarithmic factors) to find large LCS. In this paper, the problem of approximating LCS is studied, while focusing on the hard inputs for this problem, namely, approximating LCS of near-linear size in strings over a relatively large alphabet (of size at least n ϵ for some constant ϵ > 0 , where n is the length of the string). We show that, any given string over a relatively large alphabet can be embedded into a locally non-repetitive string. This embedding has a negligible additive distortion for strings that are not too dissimilar in terms of the edit distance. We also show that LCS can be efficiently approximated in locally-non-repetitive strings. Our new method (the embedding together with the approximation algorithm) gives a strictly sub-quadratic time algorithm (i.e., of complexity O ( n 2 - ϵ ) for some constant ϵ ) which can find common subsequences of linear (and near linear) size that cannot be detected efficiently by the existing tools.

Bidirectional best hit r-window gene clusters

BackgroundConserved gene clusters are groups of genes that are located close to one another in the genomes of several species. They tend to code for proteins that have a functional interaction. The identification of conserved gene clusters is an important step towards understanding genome evolution and predicting gene function.ResultsIn this paper, we propose a novel pairwise gene cluster model that combines the notion of bidirectional best hits with the r-window model introduced in 2003 by Durand and Sankoff. The bidirectional best hit (BBH) constraint removes the need to specify the minimum number of shared genes in the r-window model and improves the relevance of the results. We design a subquadratic time algorithm to compute the set of BBH r-window gene clusters efficiently.ConclusionWe apply our cluster model to the comparative analysis of E. coli K-12 and B. subtilis and perform an extensive comparison between our new model and the gene teams model developed by Bergeron et al. As compared to the gene teams model, our new cluster model has a slightly lower recall but a higher precision at all levels of recall when the results were ranked using statistical tests. An analysis of the most significant BBH r-window gene cluster show that they correspond to known operons.

CarpeDiem: Optimizing the Viterbi Algorithm and Applications to Supervised Sequential Learning

The growth of information available to learning systems and the increasing complexity of learning tasks determine the need for devising algorithms that scale well with respect to all learning parameters. In the context of supervised sequential learning, the Viterbi algorithm plays a fundamental role, by allowing the evaluation of the best (most probable) sequence of labels with a time complexity linear in the number of time events, and quadratic in the number of labels. In this paper we propose CarpeDiem, a novel algorithm allowing the evaluation of the best possible sequence of labels with a sub-quadratic time complexity. We provide theoretical grounding together with solid empirical results supporting two chief facts. CarpeDiem always finds the optimal solution requiring, in most cases, only a small fraction of the time taken by the Viterbi algorithm; meantime, CarpeDiem is never asymptotically worse than the Viterbi algorithm, thus confirming it as a sound replacement.

All maximal independent sets and dynamic dominance for sparse graphs

We describe algorithms, based on Avis and Fukuda's reverse search paradigm, for listing all maximal independent sets in a sparse graph in polynomial time and delay per output. For bounded degree graphs, our algorithms take constant time per set generated; for minor-closed graph families, the time is O ( n ) per set, and for more general sparse graph families we achieve subquadratic time per set. We also describe new data structures for maintaining a dynamic vertex set S in a sparse or minor-closed graph family, and querying the number of vertices not dominated by S ; for minor-closed graph families the time per update is constant, while it is sublinear for any sparse graph family. We can also maintain a dynamic vertex set in an arbitrary m -edge graph and test the independence of the maintained set in time O (√m) per update. We use the domination data structures as part of our enumeration algorithms.

GUARD PLACEMENT FOR MAXIMIZING L-VISIBILITY EXTERIOR TO A CONVEX POLYGON

Two points a and b are said to be L-visible among a set of polygonal obstacles if the length of the shortest path from a to b avoiding these obstacles is no more than L. For a given convex polygon P with n vertices, Gewali et al.1 addressed the guard placement problem on the boundary of P that covers the maximum area outside to the polygon under L-visibility with P as obstacle. Their proposed algorithm runs in O(n) time if [Formula: see text], where π(P) denotes the perimeter of P. They conjectured that if [Formula: see text], then the problem can be solved in subquadratic time. In this paper, we settle the conjecture in the affirmative sense, by proposing an easy to implement linear time algorithm for any arbitrary value of L.

New dynamic construction techniques for M-tree

Since its introduction in 1997, the M-tree became a respected metric access method (MAM), while remaining, together with its descendants, still the only database-friendly MAM, that is, a dynamic structure persistent in paged index. Although there have been many other MAMs developed over the last decade, most of them require either static or expensive indexing. By contrast, the dynamic M-tree construction allows us to index very large databases in subquadratic time, and simultaneously the index can be maintained up-to-date (i.e., supports arbitrary insertions/deletions). In this article we propose two new techniques improving dynamic insertions in M-tree—the forced reinsertion strategies and so-called hybrid-way leaf selection. Both of the techniques preserve logarithmic asymptotic complexity of a single insertion, while they aim to produce more compact M-tree hierarchies (which leads to faster query processing). In particular, the former technique reuses the well-known principle of forced reinsertions, where the new insertion algorithm tries to re-insert the content of an M-tree leaf that is about to split in order to avoid that split. The latter technique constitutes an efficiency-scalable selection of suitable leaf node wherein a new object has to be inserted. In the experiments we show that the proposed techniques bring a clear improvement (speeding up both indexing and query processing) and also provide a tuning tool for indexing vs. querying efficiency trade-off. Moreover, a combination of the new techniques exhibits a synergic effect resulting in the best strategy for dynamic M-tree construction proposed so far.

Semi-local longest common subsequences in subquadratic time

For two strings a, b of lengths m, n, respectively, the longest common subsequence (LCS) problem consists in comparing a and b by computing the length of their LCS. In this paper, we define a generalisation, called “the all semi-local LCS problem”, where each string is compared against all substrings of the other string, and all prefixes of each string are compared against all suffixes of the other string. An explicit representation of the output lengths is of size Θ ( ( m + n ) 2 ) . We show that the output can be represented implicitly by a geometric data structure of size O ( m + n ) , allowing efficient queries of the individual output lengths. The currently best all string–substring LCS algorithm by Alves et al., based on previous work by Schmidt, can be adapted to produce the output in this form. We also develop the first all semi-local LCS algorithm, running in time o ( m n ) when m and n are reasonably close. Compared to a number of previous results, our approach presents an improvement in algorithm functionality, output representation efficiency, and/or running time.

On the Planar Piecewise Quadratic 1-Center Problem

In this paper we introduce a minimax model unifying several classes of single facility planar center location problems. We assume that the transportation costs of the demand points to the serving facility are convex functions {Q i }, i=1,…,n, of the planar distance used. Moreover, these functions, when properly transformed, give rise to piecewise quadratic functions of the coordinates of the facility location. In the continuous case, using results on LP-type models by Clarkson (J. ACM 42:488–499, 1995), Matousek et al. (Algorithmica 16:498–516, 1996), and the derandomization technique in Chazelle and Matousek (J. Algorithms 21:579–597, 1996), we claim that the model is solvable deterministically in linear time. We also show that in the separable case, one can get a direct O(nlog n) deterministic algorithm, based on Dyer (Proceedings of the 8th ACM Symposium on Computational Geometry, 1992), to find an optimal solution. In the discrete case, where the location of the center (server) is restricted to some prespecified finite set, we introduce deterministic subquadratic algorithms based on the general parametric approach of Megiddo (J. ACM 30:852–865, 1983), and on properties of upper envelopes of collections of quadratic arcs. We apply our methods to solve and improve the complexity of a number of other location problems in the literature, and solve some new models in linear or subquadratic time complexity.

Improved dense multivariate polynomial factorization algorithms

We present new deterministic and probabilistic algorithms that reduce the factorization of dense polynomials from several variables to one variable. The deterministic algorithm runs in sub-quadratic time in the dense size of the input polynomial, and the probabilistic algorithm is softly optimal when the number of variables is at least three. We also investigate the reduction from several to two variables and improve the quantitative version of Bertini’s irreducibility theorem.

Deterministic Communication in Radio Networks with Large Labels

We study deterministic gossiping in ad hoc radio networks with large node labels. The labels (identifiers) of the nodes come from a domain of size N which may be much larger than the size n of the network (the number of nodes). Most of the work on deterministic communication has been done for the model with small labels which assumes N = O(n). A notable exception is Peleg's paper, where the problem of deterministic communication in ad hoc radio networks with large labels is raised and a deterministic broadcasting algorithm is proposed, which runs in O(n2log n) time for N polynomially large in n. The O(nlog2n)-time deterministic broadcasting algorithm for networks with small labels given by Chrobak et al. implies deterministic O(n log N log n)-time broadcasting and O(n2log2N log n)-time gossiping in networks with large labels. We propose two new deterministic gossiping algorithms for ad hoc radio networks with large labels, which are the first such algorithms with subquadratic time for polynomially large N. More specifically, we propose: a deterministic O(n3/2log2N log n)-time gossiping algorithm for directed networks; and a deterministic O(n log2N log2n)-time gossiping algorithm for undirected networks.

Advances on sorting by reversals

The problem of sorting signed permutations by reversals is inspired by genome rearrangement problems in computational molecular biology. Given two genomes represented as signed permutations of the same elements (e.g. orthologous genes), the problem consists in finding a most parsimonious scenario of reversals that transforms one genome into the other. Following the first polynomial solution of this problem, several improvements, simplifications, generalizations, tutorials or surveys have been published on the subject. While the reversal distance problem—i.e. the problem of computing the minimum number of reversals in a sorting sequence, without giving the sequence itself—seems to be well explored, the problem of giving a scenario realizing the distance still raises some open questions, one of which by Ozery-Flato and Shamir about whether an algorithm with subquadratic time complexity could ever be achieved for solving the problem. We give a positive answer to this question by describing an algorithm of time complexity O ( n 3 / 2 log n ) .

Distance-based detection and prediction of outliers

A distance-based outlier detection method that finds the top outliers in an unlabeled data set and provides a subset of it, called outlier detection solving set, that can be used to predict the outlierness of new unseen objects, is proposed. The solving set includes a sufficient number of points that permits the detection of the top outliers by considering only a subset of all the pairwise distances from the data set. The properties of the solving set are investigated, and algorithms for computing it, with subquadratic time requirements, are proposed. Experiments on synthetic and real data sets to evaluate the effectiveness of the approach are presented. A scaling analysis of the solving set size is performed, and the false positive rate, that is, the fraction of new objects misclassified as outliers using the solving set instead of the overall data set, is shown to be negligible. Finally, to investigate the accuracy in separating outliers from inliers, ROC analysis of the method is accomplished. Results obtained show that using the solving set instead of the data set guarantees a comparable quality of the prediction, but at a lower computational cost.