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arXiv · 2007.06105

Efficient Labeling for Reachability in Digraphs

Maciej Dulęba·Paweł Gawrychowski·Wojciech Janczewski
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Abstract

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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Maciej Dulęba, Paweł Gawrychowski, Wojciech Janczewski. 2020-07-12. Efficient Labeling for Reachability in Digraphs. https://arxiv.org/abs/2007.06105

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