Randomized Graph Data-Structures for Approximate Shortest Paths:Further Reading and Bibliography.

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Further Reading and Bibliography

Zwick [10] presents a very recent and comprehensive survey on the existing algorithms for all- pairs approximate/exact shortest paths. Based on the fastest known matrix multiplication

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algorithms given by Coppersmith and Winograd [3], the best bound for computing all-pairs shortest paths is O(n2.575) [11].

Approximate distance oracles are designed by Thorup and Zwick [9]. Based on a 1963 girth conjecture of Erdo˝s [6], they also show that Ω(n1+1/k ) space is needed in the worst case for any oracle that achieves stretch strictly smaller than (2k + 1). The space requirement of their approximate distance oracle is, therefore, essentially optimal. Also note that the preprocessing time of (2k − 1)-approximate distance oracle is O(mn1/k ), which is sub-cubic.

However, for further improvement in the computation time for approximate distance oracles, Thorup and Zwick pose the following question : Can (2k −1)-approximate distance oracle be computed in O˜(n2) time? Recently Baswana and Sen [2] answer their question in affirmative for unweighted graphs. However, the question for weighted graphs is still open.

For maintaining fully dynamic all-pairs shortest paths in graphs, the best known algorithm is due to Demetrescu and Italiano [5]. They show that it takes O(n2) amortized time to maintain all-pairs exact shortest paths after each update in the graph. Baswana et al. [1] present a hierarchical data-structure based on random sampling that provides efficient decremental algorithm for maintaining APASP in undirected unweighted graphs. In addition to achieving o(nd) update time for the problem APASP-d (as described in this chapter), they also employ the same hierarchical scheme for designing efficient data-structures for maintaining approximate distance information for all-pairs of vertices separated by distance in an interval [a, b], 1 ≤ a< b ≤ n.

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