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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)
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)
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)
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)
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)