In this paper we introduce a new tool for controlling the knowledge transfer process in cryptographic protocol design. It is applied to solve a general class of problems which include most of the two-party cryptographic problems in the literature. Specifically, we show how two parties A and B can interactively generate a random integer N = p?q such that its secret, i.e., the prime factors (p, q), is hidden from either party individually but is recoverable jointly if desired. This can be utilized to give a protocol for two parties with private values i and j to compute any polynomially computable functions f(i,j) and g(i,j) with minimal knowledge transfer and a strong fairness property. As a special case, A and B can exchange a pair of secrets sA, sB, e.g. the factorization of an integer and a Hamiltonian circuit in a graph, in such a way that sA becomes computable by B when and only when sB becomes computable by A. All these results are proved assuming only that the problem of factoring large intergers is computationally intractable.

How to generate and exchange secrets

The book is outstanding and admirable in many respects. ... is necessary reading for all kinds of readers from undergraduate students to top authorities in the field. Journal of Symbolic Logic Written by two experts in the field, this is the only comprehensive and unified treatment of the central ideas and their applications of Kolmogorov complexity. The book presents a thorough treatment of the subject with a wide range of illustrative applications. Such applications include the randomness of finite objects or infinite sequences, Martin-Loef tests for randomness, information theory, computational learning theory, the complexity of algorithms, and the thermodynamics of computing. It will be ideal for advanced undergraduate students, graduate students, and researchers in computer science, mathematics, cognitive sciences, philosophy, artificial intelligence, statistics, and physics. The book is self-contained in that it contains the basic requirements from mathematics and computer science. Included are also numerous problem sets, comments, source references, and hints to solutions of problems. New topics in this edition include Omega numbers, KolmogorovLoveland randomness, universal learning, communication complexity, Kolmogorov's random graphs, time-limited universal distribution, Shannon information and others.

An Introduction to Kolmogorov Complexity and Its Applications

We survey the current techniques to cope with the problem of string matching that allows errors. This is becoming a more and more relevant issue for many fast growing areas such as information retrieval and computational biology. We focus on online searching and mostly on edit distance, explaining the problem and its relevance, its statistical behavior, its history and current developments, and the central ideas of the algorithms and their complexities. We present a number of experiments to compare the performance of the different algorithms and show which are the best choices. We conclude with some directions for future work and open problems.

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A guided tour to approximate string matching

In this paper we develop a new data structure for implementing heaps (priority queues). Our structure, Fibonacci heaps (abbreviated F-heaps), extends the binomial queues proposed by Vuillemin and studied further by Brown. F-heaps support arbitrary deletion from an n-item heap in O(log n) amortized time and all other standard heap operations in O(1) amortized time. Using F-heaps we are able to obtain improved running times for several network optimization algorithms. In particular, we obtain the following worst-case bounds, where n is the number of vertices and m the number of edges in the problem graph: O(n log n + m) for the single-source shortest path problem with nonnegative edge lengths, improved from O(mlog(m/n+2)n);O(n2log n + nm) for the all-pairs shortest path problem, improved from O(nm log(m/n+2)n);O(n2log n + nm) for the assignment problem (weighted bipartite matching), improved from O(nmlog(m/n+2)n);O(mβ(m, n)) for the minimum spanning tree problem, improved from O(mlog log(m/n+2)n); where β(m, n) = min {i | log(i)n ≤ m/n}. Note that β(m, n) ≤ log*n if m ≥ n.Of these results, the improved bound for minimum spanning trees is the most striking, although all the results give asymptotic improvements for graphs of appropriate densities.

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Fibonacci heaps and their uses in improved network optimization algorithms

1 Defining Network Flow A flow network is a directed graph G = (V,E) in which each edge (u, v) ∈ E has non-negative capacity c(u, v) ≥ 0. We require that if (u, v) ∈ E, then (v, u) / ∈ E. That is, if an edge exists, then the edge between the same vertices going the reverse direction does not exist. Every flow network has a source s and a sink t, and we assume that for every v ∈ V , there is some path s→ · · · → v → · · · → t. Note that this implies that flow networks are connected. Informally, the intuition behind network flow is to think of the edges as pipes and the weights on the edges as the capacity its corresponding pipe per unit of time. The question we are trying to ask is: how many units of water can we send from the source to the sink per unit of time? Formally, a flow in G is a function f : V × V → R that satisfies the following: • Capacity constraint. For all u, v ∈ V , we require 0 ≤ f(u, v) ≤ c(u, v). Our pipe cannot hold more than is allowed as dictated by its capacity. • Flow conservation. For u ∈ V − {s, t}, we require ∑

Network Flows

1. Off-Line Serial Exact String Searching 2. Off-Line Parallel Exact String Searching 3. On-Line String Searching 4. Serial Computations of Levenshtein Distances 5. Parallel Computations of Levenshtein Distances 6. Approximate String Searching 7. Dynamic Programming: Special Cases 8. Shortest Common Superstrings 9. Two Dimensional Matching 10. Suffix Tree Data Structures for Matrices 11. Tree Pattern Matching

Pattern matching 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>

Sparsification-A Technique for Speeding up Dynamic Graph Algorithms (Extended Abstract)

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?.

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Zvi Galil

Papers

Pattern matching algorithms

Sparsification-A Technique for Speeding up Dynamic Graph Algorithms (Extended Abstract)

On the exponent of the all pairs shortest path problem

NP completeness of finding the chromatic index of regular graphs

Proceedings of the 30th IEEE symposium on Foundations of computer science