Improved Bounds on Distance Preservers
Yanyu Chen, Yu Chen, Zihan Tan, Mingyang Yang
Given an undirected unweighted graph $G$ on $n$ vertices and a set $P$ of $p$ pairs of its vertices, a distance preserver is a sparse subgraph of $G$ that exactly preserves the shortest-path distances between all pairs in $P$. In this paper, we prove a new upper bound of $O(n^{8/9} p^{4/9})$ for distance preservers when $p\in [n^{1/2}, n]$. Our result improves the previous bound of $O\big(n^{2/3}p^{2/3}+np^{1/3}\big)$ by Bodwin and Vassilevska Williams and $O(\min\{n+n^{1/2}p,\,np^{1/2}\})$ by Coppersmith and Elkin, particularly in the regime $p\in [n^{7/10}, n]$, and narrows the gap toward the conjectured $O(n^{2/3}p^{2/3}+n)$ bound. Using the techniques developed for distance preservers, we are also able to obtain pairwise additive $+2$ spanners with $O(n+n^{9/10}p^{2/5})$ edges, pairwise additive $+4$ spanners with $O(n+n^{24/25}p^{8/25})$ edges, and new bounds for sourcewise spanners, improving upon previous results in certain parameter regimes.
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