A Linear-Size Logarithmic Stretch Path-Reporting Distance Oracle for General Graphs

In 2001 Thorup and Zwick devised a distance oracle, which given an $n$-vertex undirected graph and a parameter $k$, has size $O(k n^{1+1/k})$. Upon a query $(u,v)$ their oracle constructs a $(2k-1)$-approximate path $Π$ between $u$ and $v$. The query time of the Thorup-Zwick's oracle is $O(k)$, and it was subsequently improved to $O(1)$ by Chechik. A major drawback of the oracle of Thorup and Zwick is that its space is $Ω(n \cdot \log n)$. Mendel and Naor devised an oracle with space $O(n^{1+1/k})$ and stretch $O(k)$, but their oracle can only report distance estimates and not actual paths. In this paper we devise a path-reporting distance oracle with size $O(n^{1+1/k})$, stretch $O(k)$ and query time $O(n^ε)$, for an arbitrarily small $ε> 0$. In particular, our oracle can provide logarithmic stretch using linear size. Another variant of our oracle has size $O(n \log\log n)$, polylogarithmic stretch, and query time $O(\log\log n)$. For unweighted graphs we devise a distance oracle with multiplicative stretch $O(1)$, additive stretch $O(β(k))$, for a function $β(\cdot)$, space $O(n^{1+1/k} \cdot β)$, and query time $O(n^ε)$, for an arbitrarily small constant $ε>0$. The tradeoff between multiplicative stretch and size in these oracles is far below girth conjecture threshold (which is stretch $2k-1$ and size $O(n^{1+1/k})$). Breaking the girth conjecture tradeoff is achieved by exhibiting a tradeoff of different nature between additive stretch $β(k)$ and size $O(n^{1+1/k})$. A similar type of tradeoff was exhibited by a construction of $(1+ε,β)$-spanners due to Elkin and Peleg. However, so far $(1+ε,β)$-spanners had no counterpart in the distance oracles' world. An important novel tool that we develop on the way to these results is a {distance-preserving path-reporting oracle}.

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