Space bounds for a game on graphs
STOC '76 Proceedings of the eighth annual ACM symposium on Theory of computing
Universal circuits (Preliminary Report)
STOC '76 Proceedings of the eighth annual ACM symposium on Theory of computing
Upper and lower bounds on time-space tradeoffs
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Expressing graph algorithms using generalized active messages
Proceedings of the 27th international ACM conference on International conference on supercomputing
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The following problem was raised by H.-J. Stoss in connection with certain questions related to the complexity of Boolean functions. An acyclic directed graph G is said to have property P(m,n) if for any set X of m vertices of G, there is a directed path of length n in G which does not intersect X. Let f(m,n) denote the minimum number of edges a graph with porperty P(m,n) can have. The problem is to estimate f(m,n). For the remainder of the paper, we shall restrict ourselves to the case m = n. We shall prove (1) $c_1$n log n/log log n f(n,n) $c_2$n log n (where $c_1$,$c_2$,..., will hereafter denote suitable positive constaints). In fact, the graph we construct in order to establish the upper bound on f(n,n) in (1) will have just $c_3$n vertices. In this case the upper bound in (1) is essentially best possible since it will also be shown that for $c_4$ sufficiently large, every graph on $c_4$n vertices having property P(n,n) must have at least $c_5$n log n edges.