Almost Shortest Paths with Near-Additive Error in Weighted Graphs

Let G = (V,E,w) be a weighted undirected graph with n vertices and m edges, and fix a set of s sources S ⊆ V . We study the problem of computing almost shortest paths (ASP) for all pairs in S × V in both classical centralized and parallel (PRAM) models of computation. Consider the regime of multiplicative approximation of 1 + ϵ, for an arbitrarily small constant ϵ > 0 (henceforth (1 + ϵ)-ASP for S × V ). In this regime existing centralized algorithms require ω(min{|E|s, nω}) time, where ω < 2.372 is the matrix multiplication exponent. Existing PRAM algorithms with polylogarithmic depth (aka time) require work ω(min{|E|s, nω}). In a bold attempt to achieve centralized time close to the lower bound of m + ns, Cohen [10] devised an algorithm which, in addition to the multiplicative stretch of 1 + ϵ, allows also additive error of β Wmax, where Wmax is the maximum edge weight in G (assuming that the minimum edge weight is 1), and β = (log n)O( log 1/ρ ρ ) is polylogarithmic in n. It also depends on the (possibly) arbitrarily small parameter ρ > 0 that determines the running time O((m+ ns)nρ) of the algorithm. The tradeoff of [10] was improved in [15], whose algorithm has similar approximation guarantee and running time, but its β is (1/ρ)O( log 1/ρ ρ ). However, the latter algorithm produces distance estimates rather than actual approximate shortest paths. Also, the additive terms in [10, 15] depend linearly on a possibly quite large global maximum edge weight Wmax. In the current paper we significantly improve this state of affairs. Our centralized algorithm has running time O((m + ns)nρ), and its PRAM counterpart has polylogarithmic depth and work O((m+ ns)nρ), for an arbitrarily small constant ρ > 0. For a pair (s, v) ∈ S × V , it provides a path of length d(s, v) that satisfies d(s, v) ≤ (1 + ϵ)dG(s, v) + β W(s, v), where W(s, v) is the weight of the heaviest edge on some shortest s - v path. Hence our additive term depends linearly on a local maximum edge weight, as opposed to the global maximum edge weight in [10, 15]. Finally, our β = (1/ρ)O(1/ρ), i.e., it is significantly smaller than in [10, 15]. We also extend a centralized algorithm of Dor et al. [14]. For a parameter κ = 1, 2, . . ., this algorithm provides for unweighted graphs a purely additive approximation of 2(κ - 1) for all pairs shortest paths (APASP) in time O(n2+1/κ). Within the same running time, our algorithm for weighted graphs provides a purely additive error of 2(κ-1)W(u, v), for every vertex pair (u, v) ∈ (V 2 ) , with W(u, v) defined as above. On the way to these results we devise a suite of novel constructions of spanners, emulators and hopsets.