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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,n^ω}) time, where ω < 2.372 is the matrix multiplication exponent. Existing PRAM algorithms with polylogarithmic depth (aka time) require work Ω(min{|E|s,n^ω}). In a bold attempt to achieve centralized time close to the lower bound of m + n s, Cohen [Edith Cohen, 2000] devised an algorithm which, in addition to the multiplicative stretch of 1+ε, allows also additive error of β ⋅ W_{max}, where W_{max} 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)n^ρ) of the algorithm. The tradeoff of [Edith Cohen, 2000] was improved in [M. Elkin, 2001], 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 [Edith Cohen, 2000; M. Elkin, 2001] depend linearly on a possibly quite large global maximum edge weight W_{max}. In the current paper we significantly improve this state of affairs. Our centralized algorithm has running time O((m+ ns)n^ρ), and its PRAM counterpart has polylogarithmic depth and work O((m + ns)n^ρ), for an arbitrarily small constant ρ > 0. For a pair (s,v) ∈ S× V, it provides a path of length d̂(s,v) that satisfies d̂(s,v) ≤ (1+ε)d_G(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 [Edith Cohen, 2000; M. Elkin, 2001]. Finally, our β = (1/ρ)^{O(1/ρ)}, i.e., it is significantly smaller than in [Edith Cohen, 2000; M. Elkin, 2001]. We also extend a centralized algorithm of Dor et al. [D. Dor et al., 2000]. 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 Õ(n^{2+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) ∈ binom(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.

Graph spanners and emulators are sparse structures that approximately preserve distances of the original graph. While there has been an extensive amount of work on additive spanners, so far little attention was given to weighted graphs. Only very recently [Abu Reyan Ahmed et al., 2020] extended the classical +2 (respectively, +4) spanners for unweighted graphs of size O(n^{3/2}) (resp., O(n^{7/5})) to the weighted setting, where the additive error is +2W (resp., +4W). This means that for every pair u,v, the additive stretch is at most +2W_{u,v}, where W_{u,v} is the maximal edge weight on the shortest u-v path (weights are normalized so that the minimum edge weight is 1). In addition, [Abu Reyan Ahmed et al., 2020] showed a randomized algorithm yielding a +8W_{max} spanner of size O(n^{4/3}), here W_{max} is the maximum edge weight in the entire graph. In this work we improve the latter result by devising a simple deterministic algorithm for a +(6+ε)W spanner for weighted graphs with size O(n^{4/3}) (for any constant ε > 0), thus nearly matching the classical +6 spanner of size O(n^{4/3}) for unweighted graphs. Furthermore, we show a +(2+ε)W subsetwise spanner of size O(n⋅√{|S|}), improving the +4W_{max} result of [Abu Reyan Ahmed et al., 2020] (that had the same size). We also show a simple randomized algorithm for a +4W emulator of size Õ(n^{4/3}). In addition, we show that our technique is applicable for very sparse additive spanners, that have linear size. It is known that such spanners must suffer polynomially large stretch. For weighted graphs, we use a variant of our simple deterministic algorithm that yields a linear size +Õ(√n⋅ W) spanner, and we also obtain a tradeoff between size and stretch. Finally, generalizing the technique of [D. Dor et al., 2000] for unweighted graphs, we devise an efficient randomized algorithm producing a +2W spanner for weighted graphs of size Õ(n^{3/2}) in Õ(n²) time.