2015
DOI: 10.1137/140951382
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Cutting Corners Cheaply, or How to Remove Steiner Points

Abstract: 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 actu… Show more

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Cited by 11 publications
(11 citation statements)
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“…Basu and Gupta [BG08] provided an O(1) upper bound for the family of outerplanar graphs. 5 For general nvertex graphs with k terminals the author [Fil18,Fil19c] recently proved an O(log k) upper bound for the SPR problem using the Relaxed-Voronoi framework, improving upon previous works by Kamma, Krauthgamer, and Nguyen [KKN15] (O(log 5 k)), and Cheung [Che18] (O(log 2 k)) (which were based on the Ball-Growing algorithm). Interestingly, there are no results on any other restricted graph family, although several attempts have been made (see [EGK + 14, KNZ14, CGH16]).…”
Section: Previous Resultsmentioning
confidence: 88%
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“…Basu and Gupta [BG08] provided an O(1) upper bound for the family of outerplanar graphs. 5 For general nvertex graphs with k terminals the author [Fil18,Fil19c] recently proved an O(log k) upper bound for the SPR problem using the Relaxed-Voronoi framework, improving upon previous works by Kamma, Krauthgamer, and Nguyen [KKN15] (O(log 5 k)), and Cheung [Che18] (O(log 2 k)) (which were based on the Ball-Growing algorithm). Interestingly, there are no results on any other restricted graph family, although several attempts have been made (see [EGK + 14, KNZ14, CGH16]).…”
Section: Previous Resultsmentioning
confidence: 88%
“…Scattering Partitions As we are the first to define scattering partitions there is not much previous work. Nonetheless, Kamma et al [KKN15] implicitly proved that general n-vertex graphs are (O(log n), O(log n))-scatterable. 7 Sparse Covers and Partitions Awerbuch and Peleg [AP90] introduced the notion of sparse covers and constructed (O(log n), O(log n))-strong sparse cover scheme for n-vertex weighted graphs.…”
Section: Previous Resultsmentioning
confidence: 99%
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“…For SPR in general graphs there is currently a huge gap. The best lower bound known is just 8, known for trees, and recently Filtser [Fil18a] showed an O(log k) upper bound (improving over [KKN15,Che18]). No better upper bound is known even for seemingly much simpler cases such as planar graphs, and the only other bound known is α = O(1) for outerplanar graphs [BG08].…”
Section: Steiner Point Removal (Spr)mentioning
confidence: 99%
“…This version of the Relaxed-Voronoi algorithm was first proposed by Filtser [Fil18b] for the SPR problem in general graphs. It is simpler to describe and to analyze than the Ball-Growing algorithm of previous work [KKN15,Che18,Fil18a]. 1 Filtser also showed that the Relaxed-Voronoi algorithm can be implemented in time O(|E| log |V |).…”
Section: Algorithmic Frameworkmentioning
confidence: 99%