Alan M. Frieze

Min-Wise Independent Permutations

We define and study the notion of min-wise independent families of permutations. We say that F?Sn (the symmetric group) is min-wise independent if for any set X?n and any x?X, when ? is chosen at random in F we havePr(min{?(X)}=?(x))=1|X| . In other words we require that all the elements of any fixed set X have an equal chance to become the minimum element of the image of X under ?. Our research was motivated by the fact that such a family (under some relaxations) is essential to the algorithm used in practice by the AltaVista web index software to detect and filter near-duplicate documents. However, in the course of our investigation we have discovered interesting and challenging theoretical questions related to this concept?we present the solutions to some of them and we list the rest as open problems.

A random polynomial-time algorithm for approximating the volume of convex bodies

A randomized polynomial-time algorithm for approximating the volume of a convex body <italic>K</italic> in <italic>n</italic>-dimensional Euclidean space is presented. The proof of correctness of the algorithm relies on recent theory of rapidly mixing Markov chains and isoperimetric inequalities to show that a certain random walk can be used to sample nearly uniformly from within <italic>K</italic>.

Clustering Large Graphs via the Singular Value Decomposition

We consider the problem of partitioning a set of m points in the n-dimensional Euclidean space into k clusters (usually m and n are variable, while k is fixed), so as to minimize the sum of squared distances between each point and its cluster center. This formulation is usually the objective of the k-means clustering algorithm (Kanungo et al. (2000)). We prove that this problem in NP-hard even for k = 2, and we consider a continuous relaxation of this discrete problem: find the k-dimensional subspace V that minimizes the sum of squared distances to V of the m points. This relaxation can be solved by computing the Singular Value Decomposition (SVD) of the m × n matrix A that represents the m points; this solution can be used to get a 2-approximation algorithm for the original problem. We then argue that in fact the relaxation provides a generalized clustering which is useful in its own right.Finally, we show that the SVD of a random submatrix—chosen according to a suitable probability distribution—of a given matrix provides an approximation to the SVD of the whole matrix, thus yielding a very fast randomized algorithm. We expect this algorithm to be the main contribution of this paper, since it can be applied to problems of very large size which typically arise in modern applications.

Improved approximation algorithms for MAX k-CUT and MAX BISECTION

Polynomial-time approximation algorithms with nontrivial performance guarantees are presented for the problems of (a) partitioning the vertices of a weighted graph intok blocks so as to maximize the weight of crossing edges, and (b) partitioning the vertices of a weighted graph into two blocks of equal cardinality, again so as to maximize the weight of crossing edges. The approach, pioneered by Goemans and Williamson, is via a semidefinite programming relaxation.

A general model of web graphs

We describe a very general model of a random graph process whose proportional degree sequence obeys a power law. Such laws have recently been observed in graphs associated with the world wide web.

Min-wise independent permutations (extended abstract)

We define and study the notion of min-wise independent families of permutations. We say that F ⊆ Sn is min-wise independent if for any set X ⊆ [n] and any x ∈ X , when π is chosen at random in F we have Pr ( min{π(X)} = π(x) ) = 1 |X | . In other words we require that all the elements of any fixed set X have an equal chance to become the minimum element of the image of X under π. Our research was motivated by the fact that such a family (under some relaxations) is essential to the algorithm used in practice by the AltaVista web index software to detect and filter near-duplicate documents. However, in the course of ∗Digital SRC, 130 Lytton Avenue, Palo Alto, CA 94301, USA. E-mail: broder@pa.dec.com. †Computer Science Department, Stanford University, CA 94305, USA. E-mail: moses@cs.stanford.edu. Part of this work was done while this author was a summer intern at Digital SRC. Supported by the Pierre and Christine Lamond Fellowship and in part by an ARO MURI Grant DAAH04-96-1-0007 and NSF Award CCR-9357849, with matching funds from IBM, Schlumberger Foundation, Shell Foundation, and Xerox Corporation. ‡Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA. Part of this work was done while this author was visiting Digital SRC. Supported in part by NSF grant CCR9530974. E-mail: af1p@andrew.cmu.edu §Digital SRC, 130 Lytton Avenue, Palo Alto, CA 94301, USA. E-mail: michaelm@pa.dec.com. our investigation we have discovered interesting and challenging theoretical questions related to this concept – we present the solution to some of them and we list the rest as open problems.

The shortest-path problem for graphs with random arc-lengths

We consider the problem of finding the shortest distance between all pairs of vertices in a complete digraph on n vertices, whose arc-lengths are non-negative random variables. We describe an algorithm which solves this problem in O(n(m+nlogn)) expected time, where m is the expected number of arcs with finite lenght. If m is small enough, this represents a small improvement over the bound in Bloniarz [3]. We consider also the case when the arc-lengths are random variables which are independently distributed with distribution function F, where F(0)=0 and F is differentiable at 0; for this case, we describe an algorithm which runs in O(n2logn) expected time. In our treatment of the shortest-path problem we consider the following problem in combinatorial probability theory. A town contains n people, one of whom knows a rumour. At the first stage he tells someone chosen randomly from the town; at each stage, each person who knows the rumour tells someone else, chosen randomly from the town and indeependently of all other choices. Let Sn be the number of stages before the whole town rnows the rumour. We show that Sn/log2n → 1 + loge2 in probability as n → ∞, and estimate the probabilities of large deviations in Sn.

On the complexity of computing the volume of a polyhedron

We show that computing the volume of a polyhedron given either as a list of facets or as a list of vertices is as hard as computing the permanent of a matrix.

On the worst-case performance of some algorithms for the asymmetric traveling salesman problem

We consider the asymmetric traveling salesman problem for which the triangular inequality is satisfied. For various heuristics we construct examples to show that the worst-case ratio of length of tour found to minimum length tour is (n) for n city problems. We also provide a new O([log2n]) heuristic.

The cover times of random walks on hypergraphs

Random walks in graphs have been applied to various network exploration and network maintenance problems. In some applications, however, it may be more natural, and more accurate, to model the underlying network not as a graph but as a hypergraph, and solutions based on random walks require a notion of random walks in hypergraphs. At each step, a random walk on a hypergraph moves from its current position v to a random vertex in a randomly selected hyperedge containing v. We consider two definitions of cover time of a hypergraph H. If the walk sees only the vertices it moves between, then the usual definition of cover time, C(H), applies. If the walk sees the complete edge during the transition, then an alternative definition of cover time, the inform time I(H) is used. The notion of inform time models passive listening which fits the following types of situations. The particle is a rumor passing between friends, which is overheard by other friends present in the group at the same time. The particle is a message transmitted randomly from location to location by a directional transmission in an ad-hoc network, but all receivers within the transmission range can hear. In this paper we give an expression for C(H) which is tractable for many classes of hypergraphs, and calculate C(H) and I(H) exactly for random r-regular, s-uniform hypergraphs. We find that for such hypergraph whp C(H)/I(H) = Θ(s) for large s.