Mike Paterson

The complexity of mean payoff games on graphs

Abstract We study the complexity of finding the values and optimal strategies of mean payoff games on graphs, a family of perfect information games introduced by Ehrenfeucht and Mycielski and considered by Gurvich, Karzanov and Khachiyan. We describe a pseudo-polynomial-time algorithm for the solution of such games, the decision problem for which is in NP ∩ coNP . Finally, we describe a polynomial reduction from mean payoff games to the simple stochastic games studied by Condon. These games are also known to be in NP ∩ coNP , but no polynomial or pseudo-polynomial-time algorithm is known for them.

Selection and sorting with limited storage

When selecting from, or sorting, a file stored on a read-only tape and the internal storage is rather limited, several passes of the input tape may be required. We study the relation between the amount of internal storage available and the number of passes required to select the Kth highest of N inputs. We show, for example, that to find the median in two passes requires at least ω(N12) and at most O(N12log N) internal storage. For probabilistic methods, θ(N12) internal storage is necessary and sufficient for a single pass method which finds the median with arbitrarily high probability.

On Nearest-Neighbor Graphs

The “nearest-neighbor” relation, or more generally the “k-nearest-neighbors” relation, defined for a set of points in a metric space, has found many uses in computational geometry and clustering analysis, yet surprisingly little is known about some of its basic properties. In this paper we consider some natural questions that are motivated by geometric embedding problems. We derive bounds on the relationship between size and depth for the components of a nearest-neighbor graph and prove some probabilistic properties of the k-nearest-neighbors graph for a random set of points.

On the Approximability of Numerical Taxonomy (Fitting Distances by Tree Metrics)

We consider the problem of fitting an n × n distance matrix D by a tree metric T. Let

Bounds on minimax edge length for complete binary trees

\varepsilon

A Deterministic Subexponential Algorithm for Solving Parity Games

be the distance to the closest tree metric under the

Improved sorting networks withO(logN) depth

L_{\infty}

Progress in Selection

norm; that is,

Upper bounds for the expected length of a longest common subsequence of two binary sequences

\varepsilon=\min_T\{\parallel T-D\parallel{\infty}\}

A family of NFAs which need 2 n - α deterministic states

. First we present an O(n2) algorithm for finding a tree metric T such that