Mihai Pǎtraşcu: obituary and open problems
ACM SIGACT News
Brief announcement: a simple stretch 2 distance oracle
Proceedings of the 2013 ACM symposium on Principles of distributed computing
Shortest-path queries in static networks
ACM Computing Surveys (CSUR)
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Given a weighted undirected graph, our basic goal is to represent all pair wise distances using much less than quadratic space, such that we can estimate the distance between query vertices in constant time. We will study the inherent trade-off between space of the representation and the stretch (multiplicative approximation disallowing underestimates) of the estimates when the input graph is sparse with $m=\wt O(n)$ edges. In this paper, for any fixed positive integers $k$ and $\ell$, we obtain stretches $\alpha=2k+1\pm\frac{2}{\ell}= 2k+1-\frac{2}{\ell}, 2k+1+\frac{2}{\ell}$, using space $S(\alpha, m) = \wt O(m^{1+2/(\alpha+1)})$. The query time is $O(k+\ell)=O(1)$. For integer stretches, this coincides with the previous bounds (odd stretches with $\ell=1$ and even stretches with $\ell=2$). The infinity of fractional stretches between consecutive integers are all new (even though $\ell$ is fixed as a constant independent of the input, the number of integers $\ell$ is still countably infinite). % We will argue that the new fractional points are not just arbitrary, but that they, at least for fixed stretches below $3$, provide a complete picture of the inherent trade-off between stretch and space in $m$. Consider any fixed stretch $\alpha