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A graph G is given and two players, a cop and a robber, play the folioking game: the cop chooses a vertex, then the robber chooses a vertex, then the players move alternately beginning with the cop. A move consists of staying at one's present vertex or moving to an adjacent vertex; each move is seen by both players. The cop wins if he manages to occupy the… (More)

We show that the problem of counting the number of linear extensions of a given partially ordered set is #P-complete. This settles a long-standing open question and contrssts with recent results giving randomized polynomial-time algo-rit hms for estimating the number of linear extensions. One consequence is that computing the volume of a rational… (More)

The following problem arose in the planning of optical communications networks which use bidirectional SONET rings. Traac demands d i;j are given for each pair of nodes in an n-node ring; each demand must be routed one of the two possible ways around the ring. The object is to minimize the maximum load on the cycle, where the load of an edge is the sum of… (More)

Let G be the digraph consisting of two oppositely-directed rings on the same set of n nodes. We provide a polynomial-time algorithm which, given a list of demands-each requiring a path from a specified source node to a specified target node-routes the demands so as to minimize the largest number of paths through any of the 2n directed links of G. The… (More)

17 There is one further consideration, which leads perhaps to the most intriguing conjecture of all. Let us put two tokens on a graph and let them take random walks, as before; but now suppose the schedule demon is clairvoyant|that is, he can see where each token will go, innnitely far into the future. The question is, with this advantage, can he now keep… (More)

An n-bit binary Gray code is an enumeration of all n-bit binary strings so that successive elements differ in exactly one bit position; equivalently, a hamilton path in the Hasse diagram of B n (the partially ordered set of subsets of an n-element set, ordered by inclusion.) We construct, for each n, a hamilton path in B n with the following additional… (More)

- David Aldous, Ll Aszll O Lovv Asz, Peter Winkler
- 1996

Consider the class of discrete time, general state space Markov chains which satisfy a \uniform ergodicity under sampling" condition. There are many ways to quantify the notion of \mixing time", that is time to approach stationarity from a worst initial state. We prove results asserting equivalence (up to universal constants) of diierent quantiications of… (More)

Let lf be the transition matrix, and a the initial state distribution. for a discrete-time finite-state irreducible Markov chain. A stopping rule for M is an algorithm which observes the progress of the chain and then stops it at some random time r; the distribution of the final state is denoted by ar. We give a useful characterization for stopping rules… (More)

We analyze and solve a game in which a player chooses which of several Markov chains to advance, with the object of minimizing the expected time (or cost) for one of the chains to reach a target state. The solution entails computing (in polynomial time) a function γ—a variety of " Gittins index " —on the states of the individual chains, the minimization of… (More)