On Compact Representations of All-Pairs-Shortest-Path-Distance Matrices

Abstract

Let G be an unweighted and undirected graph of n nodes, and let D be the n ×n matrix storing the All-Pairs-Shortest-Path distances in G. Since D contains integers in [n] ∪ + ∞, its plain storage takes n ²log(n + 1) bits. However, a simple counting argument shows that (n ² − n)/2 bits are necessary to store D. In this paper we investigate the question of finding a succinct representation of D that requires O(n ²) bits of storage and still supports constant-time access to each of its entries. This is asymptotically optimal in the worst case, and far from the information-theoretic lower-bound by a multiplicative factor log₂ 3 ≃ 1.585. As a result O(1) bits per pairs of nodes in G are enough to retain constant-time access to their shortest-path distance. We achieve this result by reducing the storage of D to the succinct storage of labeled trees and ternary sequences, for which we properly adapt and orchestrate the use of known compressed data structures.

This work has been partially supported by the Italian MIUR grants PRIN MainStream and Italy-Israel FIRB “Pattern Discovery Algorithms in Discrete Structures, with Applications to Bioinformatics”, and by the Yahoo! Research grant on “Data compression and indexing in hierarchical memories”.