Sublinear Algorithms Research Articles

Embedding Torus Automata into a Ring of Automata

In this paper, we deal with two-dimensional cellular automata. Our goal is to present a linear space and sub-linear time algorithm to embed a bounded finite torus, considered as a cellular space, into a ring of automata. Such an embedding has to preserve the neighborhood of the cells and to keep the periodicities of the torus. Finally, we generalize this result to torus of greater dimension.

A sublinear algorithm for two-dimensional string matching

A simple algorithm based on the Boyer-Moore idea is presented for two-dimensional string matching. The algorithm examines a strip of columns at a time, and the shift of the pattern is based on a string of several characters on a row. The expected running time is shown to be sublinear for random texts and patterns. The algorithm is easy to implement, and it works well in practice.

Fully dynamic biconnectivity in graphs

We present an algorithm for maintaining the biconnected components of a graph during a sequence of edge insertions and deletions. It requires linear storage and preprocessing time. The amortized running time for insertions and for deletions isO(m 2/3 ), wherem is the number of edges in the graph. Any query of the form ‘Are the verticesu andv biconnected?’ can be answered in timeO(1). This is the first sublinear algorithm for this problem. We can also output all articulation points separating any two vertices efficiently. If the input is a plane graph, the amortized running time for insertions and deletions drops toO(√n logn) and the query time isO(log2 n), wheren is the number of vertices in the graph. The best previously known solution takes timeO(n 2/3 ) per update or query.

A sublinear algorithm for approximate keyword searching

Given a relatively short query stringW of lengthP, a long subject stringA of lengthN, and a thresholdD, theapproximate keyword search problem is to find all substrings ofA that align withW with not more than D insertions, deletions, and mismatches. In typical applications, such as searching a DNA sequence database, the size of the “database”A is much larger than that of the queryW, e.g.,N is on the order of millions or billions andP is a hundred to a thousand. In this paper we present an algorithm that given a precomputedindex of the databaseA, finds rare matches in time that issublinear inN, i.e.,N c for somec<1. The sequenceA must be overa. finite alphabet σ. More precisely, our algorithm requires 0(DN pow(ɛ) logN) expected-time where ɛ=D/P is the maximum number of differences as a percentage of query length, and pow(ɛ) is an increasing and concave function that is 0 when ɛ=0. Thus the algorithm is superior to current O(DN) algorithms when ɛ is small enough to guarantee that pow(ɛ) < 1. As seen in the paper, this is true for a wide range of ɛ, e.g., ɛ. up to 33% for DNA sequences (¦⌆¦=4) and 56% for proteins sequences (¦⌆¦=20). In preliminary practical experiments, the approach gives a 50-to 500-fold improvement over previous algorithms for prolems of interest in molecular biology.

A sublinear parallel algorithm for some dynamic programming problems

This paper presents a sublinear parallel algorithm for dynamic programming problems such as computing an optimal order of matrix multiplications, an optimal binary search tree or an optimal triangulation of polygons. An algorithm was presented in Rytter (1988) which runs in O(log 2 n) time using O( n 6/log n) processors. An algorithm is presented in this paper which runs in O(√ nlog n) time using O( n 3.5/log n) processors on a CREW PRAM. Amongst known sublinear algorithms, this is an improvement by a factor of O( n 2 log n) in terms of processor-time product.

An Exact Sublinear Algorithm for the Max-Flow, Vertex Disjoint Paths and Communication Problems on Random Graphs

This paper describes a randomized algorithm for solving the maximum-flow maximum-cut problem on connected random graphs. The algorithm is very fast—it does not look up most vertices in the graph. Another feature of this algorithm is that it almost surely provides, along with an optimal solution, a proof of optimality of the solution. In addition, the algorithm's solution is, by construction, a collection of vertex-disjoint paths which is maximum. Under a restriction on the graph's density, an optimal solution to the NP-hard communication problem is provided as well, that is, finding a maximum collection of vertex-disjoint paths between sender-receiver pairs of terminals. The algorithm lends itself to a sublogarithmic parallel and distributed implementation. Its effectiveness is demonstrated via extensive empirical study.

An improved parallel algorithm for integer GCD

We present a simple parallel algorithm for computing the greatest common divisor (gcd) of twon-bit integers in the Common version of the CRCW model of computation. The run-time of the algorithm in terms of bit operations isO(n/logn), usingn 1+ɛ processors, where ɛ is any positive constant. This improves on the algorithm of Kannan, Miller, and Rudolph, the only sublinear algorithm known previously, both in run time and in number of processors; they requireO(n log logn/logn),n 2 log2 n, respectively, in the same CRCW model. We give an alternative implementation of our algorithm in the CREW model. Its run-time isO(n log logn/logn), usingn 1+ɛ processors. Both implementations can be modified to yield the extended gcd, within the same complexity bounds.