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- Average-Case Bounds on the Complexity of Path-Search, August 19, 1997 Nicholas Pippenger
Average-Case Bounds for the Complexity of Path-Search Nicholas Pippenger A channel graph is the union of all paths between a given input and a given output in an interconnection network. At any moment in time, each vertex in such a graph is either idle or busy. The search problem we consider is to find a path (from the given input to the given output) consisting entirely of idle vertices, or to find a cut (separating the given input from the given output) consisting entirely of busy vertices. We shall also allow the search to fail to find either a path or a cut with some probability bounded by a parameter called the failure probability. This is to be accomplished by sequentially probing the idle-or-busy status of vertices, where the vertex chosen for each probe may depend on the outcome of previous probes. Thus a search algorithm may be modelled as a decision tree. For average-case analysis, we assume that each vertex is independently idle with some fixed probability, called the vacancy probability (and therefore busy with the complementary probability). For one commonly studied type channel graph, the parallel graph, we show that the expected number of probes is at most proportional to the length of a path, irrespective of the vacancy probability, and even if the allowed failure probability is zero. Another type of channel graph we study is the spider-web graph, which is superior to the parallel graph as regard linking probability (the probability that an idle path, rather than a busy cut, exists). For this graph we give an algorithm for which, as the vacancy probability is varied while the positive failure probability is held fixed, the expected number of probes reaches its maximum near the critical vacancy probability (where the linking probability make a rapid transition from a very small value to a substantial value). This maximum expected number of probes is about the cube-root of the diversity (the number of paths between the input and output).

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