Our main result is that the Steiner Point Removal (SPR) problem can always be solved with polylogarithmic distortion, which answers in the affirmative a question posed by Chan, Xia, Konjevod, and Richa (2006). Specifically, we prove that for every edge-weighted graph G = (V, E, w) and a subset of terminals T ⊆ V , there is a graph G = (T, E , w ) that is isomorphic to a minor of G, such that for every two terminals u, v ∈ T , the shortest-path distances between them in G and in G satisfyOur existence proof actually gives a randomized polynomial-time algorithm.Our proof features a new variant of metric decomposition. It is well-known that every finite metric space (X, d) admits a β-separating decomposition for β = O(log|X|), which means that for every ∆ > 0 there is a randomized partitioning of X into clusters of diameter at most ∆, satisfying the following separation property: for every x, y ∈ X, the probability they lie in different clusters of the partition is at most β d(x, y)/∆. We introduce an additional requirement in the form of a tail bound: for every shortest-path P of length d(P ) ≤ ∆/β, the number of clusters of the partition that meet the path P , denoted Z P , satisfies Pr[Z P > t] ≤ 2e −Ω(t) for all t > 0. * A preliminary version appeared in Steiner Point Removal (SPR). Let G = (V, E, w) be an edge-weighted graphand let T = {t 1 , . . . , t k } ⊆ V be a designated set of k terminals. Here and throughout, d G,w (·, ·) denotes the shortest-path metric between vertices of G according to the weights w. The Steiner Point Removal problem asks to construct on the terminals a new graph G = (T, E , w ) such that (i) distances between the terminals are distorted at most by factor α ≥ 1, formallyand (ii) the graph G is (isomorphic to) a minor of G. This formulation of the SPR problem was proposed by Chan, Xia, Konjevod, and Richa [CXKR06, Section 5] who posed the problem of bounding the distortion α (existentially and/or using an efficient algorithm). Our main result is to answer their open question. Requirement (ii) above expresses structural similarity between G and G ; for instance, if G is planar then so is G . The SPR formulation above actually came about as a generalization to a result of Gupta [Gup01], which asserts that if G is a tree, there exists a tree G , which preserves terminal distances with distortion α = 8. Later Chan et al.[CXKR06] observed that this same G is actually a minor of the original tree G, and proved the factor of 8 to be tight. The upper bound for trees was later extended by Basu and Gupta [BG08], who achieve distortion α = O(1) for the larger class of outerplanar graphs.How to construct minors. We now describe a general methodology that is natural for the SPR problem. The first step constructs a minor G with vertex set T , but without any edge weights, and is prescribed by Definition 1.2. The second step determines edge weights w such that d G ,w dominates d G,w on the terminals T , and is given in Definition 1.3. These steps are illustrated in Figure 1. Our definitions are actua...

The conjectured hardness of Boolean matrix-vector multiplication has been used with great success to prove conditional lower bounds for numerous important data structure problems, see Henzinger et al. [STOC'15]. In recent work, Larsen and Williams [SODA'17] attacked the problem from the upper bound side and gave a surprising cell probe data structure (that is, we only charge for memory accesses, while computation is free). Their cell probe data structure answers queries inÕ(n 7/4 ) time and is succinct in the sense that it stores the input matrix in read-only memory, plus an additionalÕ(n 7/4 ) bits on the side. In this paper, we essentially settle the cell probe complexity of succinct Boolean matrix-vector multiplication. We present a new cell probe data structure with query timeÕ(n 3/2 ) storing justÕ(n 3/2 ) bits on the side. We then complement our data structure with a lower bound showing that any data structure storing r bits on the side, with n < r < n 2 must have query time t satisfying tr =Ω(n 3 ). For r ≤ n, any data structure must have t =Ω(n 2 ). Since lower bounds in the cell probe model also apply to classic word-RAM data structures, the lower bounds naturally carry over. We also prove similar lower bounds for matrix-vector multiplication over

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Our main result is that the Steiner point removal (SPR) problem can always be solved with polylogarithmic distortion, which answers in the affirmative a question posed by Chan, Xia, Konjevod, and Richa in 2006. Specifically, we prove that for every edge-weighted graph G = (V, E, w) and a subset of terminals T ⊆ V , there is a graph G = (T, E , w ) that is isomorphic to a minor of G such that for every two terminals u, v ∈ T , the shortest-path distances between them in G and in (u, v). Our existence proof actually gives a randomized polynomial-time algorithm. Our proof features a new variant of metric decomposition. It is well known that every finite metric space (X, d) admits a β-separating decomposition for β = O(log|X|), which means that for every Δ > 0 there is a randomized partitioning of X into clusters of diameter at most Δ, satisfying the following separation property: for every x, y ∈ X, the probability that they lie in different clusters of the partition is at most β d(x, y)/Δ. We introduce an additional requirement in the form of a tail bound: for every shortest-path P of length d(P ) ≤ Δ/β, the number of clusters of the partition that meet the path P , denoted by Z P , satisfies Pr[Z P > t] ≤ 2e −Ω(t) for all t > 0.

Given a graph H = (U, E) and connectivity requirements r = {r(u, v) : u, v ∈ R ⊆ U }, we say that H satisfies r if it contains r(u, v) pairwise internally-disjoint uv-paths for all u, v ∈ R. We consider the Survivable Network with Minimum Number of Steiner Points (SN-MSP) problem: given a finite set V of points in a normed space (M, · ) and connectivity requirements, find a minimum size set S ⊂ M \ V of additional points, such that the unit disc graph induced by U = V ∪ S satisfies the requirements. In the (node-connectivity) Survivable Network Design Problem (SNDP) we are given a graph G = (V, E) with edge costs and connectivity requirements, and seek a min-cost subgraph H of G that satisfies the requirements. Let k = max u,v∈V r (u, v) denote the maximum connectivity requirement. We will show a natural transformation of an SN-MSP instance (V, r) into an SNDP instance (G = (V, E), c, r), such that an α-approximation algorithm for the SNDP instance implies an α · O(k 2 )-approximation algorithm for the SN-MSP instance. In particular, for the case of uniform requirement r(u, v) = k for all u, v ∈ V , we obtain for SN-MSP ratio O(k 2 ln k), which solves an open problem from [3].