Zvi Galil

Efficient algorithms for finding minimum spanning trees in undirected and directed graphs

Recently, Fredman and Tarjan invented a new, especially efficient form of heap (priority queue). Their data structure, theFibonacci heap (or F-heap) supports arbitrary deletion inO(logn) amortized time and other heap operations inO(1) amortized time. In this paper we use F-heaps to obtain fast algorithms for finding minimum spanning trees in undirected and directed graphs. For an undirected graph containingn vertices andm edges, our minimum spanning tree algorithm runs inO(m logβ (m, n)) time, improved fromO(mβ(m, n)) time, whereβ(m, n)=min {i|log(i)n ≦m/n}. Our minimum spanning tree algorithm for directed graphs runs inO(n logn + m) time, improved fromO(n log n +m log log log(m/n+2)n). Both algorithms can be extended to allow a degree constraint at one vertex.

Efficient algorithms for finding maximum matching in graphs

This paper surveys the techniques used for designing the most efficient algorithms for finding a maximum cardinality or weighted matching in (general or bipartite) graphs. It also lists some open problems concerning possible improvements in existing algorithms and the existence of fast parallel algorithms for these problems.

Explicit constructions of linear-sized superconcentrators

In some cases people are interested in certain kinds of networks and want them to be of small size. Usually these kinds of networks are defined by certain requirements on the connectivity between their inputs and outputs. One trick that sometimes yields the desired networks is counting argument. One considers a certain family of graphs of size n and counts how many of these graphs do not satisfy the requirements (the bad ones). In case the number of bad ones is smaller than the number of graphs in the family, the existence of a good graph (the desired network) is established. In fact, in many of the cases one shows that most of the graphs in the family are good. (The fraction of bad graphs tends to zero as n grows.) The weakness of this approach is that it is nonconstructive: al though one knows that many of these graphs are good, he cannot construct one such graph. An (n, m, k) concentrator is a directed acyclic graph (dag) with n inputs, m < n outputs, and at most kn edges, such that for every subset of m inputs there are m vertex disjoint paths going from these m inputs to the outputs. An (n, k) superconentrator (s.c.) is a dag with n inputs, n outputs and at most kn edges, such that for every 1 < r < it and any two sets of r inputs and r outputs there are r vertex disjoint paths connecting the two sets. A family of l inear concentrators [superconcentrators] of density k is a set of (n, m, k + o(l)) concentrators [(n, k + o(l)) s.c.‘s] for 1 <m < n<cn [for I<n<oo]. In (61 Pinsker constructed a family of l inear concentrators of density 29. His construction uses counting argument. Valiant [9] used Pinsker’s linear concentrators to construct a family of l inear s.c.‘s of density 238. By doing so he disproved a conjecture that superconcentrators require more than a linear number of edges (1, p. 450, research problem 12.371. Pippenger [7] has discovered a direct way to construct a family of l inear s.c.‘s of density 39. His construction used a certain type of graphs. the existence of which is

Sparsification—a technique for speeding up dynamic graph algorithms

We provide data strutures that maintain a graph as edges are inserted and deleted, and keep track of the following properties with the following times: minimum spanning forests, graph connectivity, graph 2-edge connectivity, and bipartiteness in time<italic>O</italic>(<italic>n</italic><supscrpt>1/2</supscrpt>) per change; 3-edge connectivity, in time <italic>O</italic>(<italic>n</italic><supscrpt>2/3</supscrpt>) per change; 4-edge connectivity, in time <italic>O</italic>(<italic>n</italic>α(<italic>n</italic>)) per change; <italic>k</italic>-edge connectivity for constant <italic>k</italic>, in time <italic>O</italic>(<italic>n</italic>log<italic>n</italic>) per change;2-vertex connectivity, and 3-vertex connectivity, in the <italic>O</italic>(<italic>n</italic>) per change; and 4-vertex connectivity, in time <italic>O</italic>(<italic>n</italic>α(<italic>n</italic>)) per change. Further results speed up the insertion times to match the bounds of known partially dynamic algorithms. All our algorithms are based on a new technique that transforms an algorithm for sparse graphs into one that will work on any graph, which we call <italic>sparsification.</italic>

Dynamic graph algorithms

In dynamic graph algorithms the following provide-or-boundproblem has to be solved quickly: Given a set S containing a subset R and a way of generating random elements fromS testing for membership inR, either (i) provide an element of R or (ii) give a (small) upper bound on the size of R that holds with high probability. We give an optimal algorithm for this problem. This algorithm improves the time per operation for various dyamic graph algorithms by a factor ofO(logn). For example, it improves the time per update for fully dynamic connectivity fromO(log n) toO(log n).

Time-space-optimal string matching

Abstract Any string-matching algorithm requires at least linear time and a constant number of local storage locations. We design and analyze an algorithm which realizes both asymptotic bounds simultaneously. This can be viewed as completely eliminating the need for the tabulated “failure function” in the linear-time algorithm of Knuth, Morris, and Pratt. It makes possible a completely general implementation as a Fortran subroutine or even as a six-head finite automaton.

On the Exponent of the All Pairs Shortest Path Problem

The upper bound on the exponent,?, of matrix multiplication over a ring that was three in 1968 has decreased several times and since 1986 it has been 2.376. On the other hand, the exponent of the algorithms known for the all pairs shortest path problem has stayed at three all these years even for the very special case of directed graphs with uniform edge lengths. In this paper we give an algorithm of timeO(n?log3n),?=(3+?)/2, for the case of edge lengths in {?1, 0, 1}. Thus, for the current known bound on?, we get a bound on the exponent,?<2.688. In case of integer edge lengths with absolute value bounded above byM, the time bound isO((Mn)?log3n) and the exponent is less than 3 forM=O(n?), for?<0.116 and the current bound on?.

An improved algorithm for approximate string matching

Given a text string, a pattern string, and an integer k, a new algorithm for finding all occurrences of the pattern string in the text string with at most k differences is presented. Both its theoretical and practical variants improve upon the known algorithms.

Data structures and algorithms for approximate string matching

Abstract This paper surveys techniques for designing efficient sequential and parallel approximate string matching algorithms. Special attention is given to the methods for the construction of data structures that efficiently support primitive operations needed in approximate string matching.

Data structures and algorithms for disjoint set union problems

This paper surveys algorithmic techniques and data structures that have been proposed tosolve thesetunion problem and its variants, Thediscovery of these data structures required anew set ofalgorithmic tools that have proved useful in other areas. Special attention is devoted to recent extensions of the original set union problem, and an attempt is made to provide a unifying theoretical framework for this growing body of algorithms.