Efficient Labeling for Reachability in Digraphs
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We consider labeling nodes of a directed graph for reachability queries. A reachability labeling scheme for such a graph assigns a binary string, called a label, to each node. Then, given the labels of nodes u and v and no other information about the underlying graph, it should be possible to determine whether there exists a directed path from u to v. By a simple information theoretical argument and invoking the bound on the number of partial orders, in any scheme some labels need to consist of at least n/4 bits, where n is the number of nodes. On the other hand, it is not hard to design a scheme with labels consisting of n/2+O(log n) bits. In the classical centralised setting, Munro and Nicholson designed a data structure for reachability queries consisting of n^2/4+o(n^2) bits (which is optimal, up to the lower order term). We extend their approach to obtain a scheme with labels consisting of n/3+o(n) bits.
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