Polylog-time and near-linear work approximation scheme for undirected shortest paths

Shortest paths computations constitute one of the most fundamental network problems. Nonetheless, known parallel shortest-paths algorithms are generally inefficient: they perform significantly more work (product of time and processors) than their sequential counterparts. This gap, known in the literature as the “transitive closure bottleneck,” poses a long-standing open problem. Our main result is an <inline-equation> <f> O<fen lp="par">mn^{<g>e</g><inf>0</inf>}+s<fen lp="par"> m+n^{1+<g>e</g><inf>0</inf>}<rp post="par"></fen><rp post="par"></fen> </f> </inline-equation> work polylog-time randomized algorithm that computes paths within (1 + <italic>O</italic>(1/polylog <italic>n</italic>) of shortest from <italic>s</italic> source nodes to all other nodes in weighted undirected networks with <italic>n</italic> nodes and <italic>m</italic> edges (for any fixed ε<subscrpt>0</subscrpt>>0). This work bound nearly matches the <inline-equation> <f> <a><ac>O</ac><ac>&d5;</ac></a><fen lp="par">sm<rp post="par"></fen> </f> </inline-equation> sequential time. In contrast, previous polylog-time algorithms required nearly <inline-equation> <f> <rf>min</rf><fen lp="cub"><a><ac>O</ac><ac>&d5;</ac></a><fen lp="par"> n³<rp post="par"></fen>,<a><ac>O</ac><ac>&d5;</ac></a> <fen lp="par">m²<rp post="par"></fen><rp post="cub"></fen> </f> </inline-equation> work (even when <italic>s</italic>=1), and previous near-linear work algorithms required near-<italic>O</italic>(<italic>n</italic>) time. We also present faster sequential algorithms that provide good approximate distances only between “distant” vertices: We obtain an <inline-equation> <f> O<fen lp="par"><fen lp="par">m+sn<rp post="par"></fen>n^{<g>e</g><inf> 0</inf>}<rp post="par"></fen></f> </inline-equation> time algorithm that computes paths of weight (1+<italic>O</italic>(1/polylog <italic>n</italic>) dist + <italic>O</italic>(<italic>w</italic><subscrpt>max</subscrpt> polylog <italic>n</italic>), where dist is the corresponding distance and <italic>w</italic><subscrpt>max</subscrpt> is the maximum edge weight. Our chief instrument, which is of independent interest, are efficient constructions of sparse <italic>hop sets</italic>. A (<italic>d</italic>,ε)-hop set of a network <italic>G</italic>=(<italic>V,E</italic>) is a set <italic>E</italic>* of new weighted edges such that mimimum-weight <italic>d</italic>-edge paths in <inline-equation> <f> <fen lp="par">V,E∪E^*<rp post="par"></fen></f> </inline-equation> have weight within (1+ε) of the respective distances in <italic>G</italic>. We construct hop sets of size <inline-equation> <f> O<fen lp="par">n^{1+<g>e</g><inf>0</inf>}<rp post="par"></fen> </f> </inline-equation> where ε=<italic>O</italic>(1/polylog <italic>n</italic>) and <italic>d</italic>=<italic>O</italic>(polylog <italic>n</italic>).