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DOI: 10.4230/LIPIcs.SEA.2021.16

Multi-Level Weighted Additive Spanners

Authors: Reyan Ahmed, Greg Bodwin, Faryad Darabi Sahneh, Keaton Hamm, Stephen Kobourov, and Richard Spence

Published in: LIPIcs, Volume 190, 19th International Symposium on Experimental Algorithms (SEA 2021)

Abstract

Given a graph G = (V,E), a subgraph H is an additive +β spanner if dist_H(u,v) ≤ dist_G(u,v) + β for all u, v ∈ V. A pairwise spanner is a spanner for which the above inequality is only required to hold for specific pairs P ⊆ V × V given on input; when the pairs have the structure P = S × S for some S ⊆ V, it is called a subsetwise spanner. Additive spanners in unweighted graphs have been studied extensively in the literature, but have only recently been generalized to weighted graphs. In this paper, we consider a multi-level version of the subsetwise additive spanner in weighted graphs motivated by multi-level network design and visualization, where the vertices in S possess varying level, priority, or quality of service (QoS) requirements. The goal is to compute a nested sequence of spanners with the minimum total number of edges. We first generalize the +2 subsetwise spanner of [Pettie 2008, Cygan et al., 2013] to the weighted setting. We experimentally measure the performance of this and several existing algorithms by [Ahmed et al., 2020] for weighted additive spanners, both in terms of runtime and sparsity of the output spanner, when applied as a subroutine to multi-level problem. We provide an experimental evaluation on graphs using several different random graph generators and show that these spanner algorithms typically achieve much better guarantees in terms of sparsity and additive error compared with the theoretical maximum. By analyzing our experimental results, we additionally developed a new technique of changing a certain initialization parameter which provides better spanners in practice at the expense of a small increase in running time.

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Reyan Ahmed, Greg Bodwin, Faryad Darabi Sahneh, Keaton Hamm, Stephen Kobourov, and Richard Spence. Multi-Level Weighted Additive Spanners. In 19th International Symposium on Experimental Algorithms (SEA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 190, pp. 16:1-16:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{ahmed_et_al:LIPIcs.SEA.2021.16,
  author =	{Ahmed, Reyan and Bodwin, Greg and Sahneh, Faryad Darabi and Hamm, Keaton and Kobourov, Stephen and Spence, Richard},
  title =	{{Multi-Level Weighted Additive Spanners}},
  booktitle =	{19th International Symposium on Experimental Algorithms (SEA 2021)},
  pages =	{16:1--16:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-185-6},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{190},
  editor =	{Coudert, David and Natale, Emanuele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2021.16},
  URN =		{urn:nbn:de:0030-drops-137885},
  doi =		{10.4230/LIPIcs.SEA.2021.16},
  annote =	{Keywords: multi-level, graph spanner, approximation algorithms}
}

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DOI: 10.4230/LIPIcs.ESA.2020.4

Kruskal-Based Approximation Algorithm for the Multi-Level Steiner Tree Problem

Authors: Reyan Ahmed, Faryad Darabi Sahneh, Keaton Hamm, Stephen Kobourov, and Richard Spence

Published in: LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

Abstract

We study the multi-level Steiner tree problem: a generalization of the Steiner tree problem in graphs where terminals T require varying priority, level, or quality of service. In this problem, we seek to find a minimum cost tree containing edges of varying rates such that any two terminals u, v with priorities P(u), P(v) are connected using edges of rate min{P(u),P(v)} or better. The case where edge costs are proportional to their rate is approximable to within a constant factor of the optimal solution. For the more general case of non-proportional costs, this problem is hard to approximate with ratio c log log n, where n is the number of vertices in the graph. A simple greedy algorithm by Charikar et al., however, provides a min{2(ln |T|+1), 𝓁 ρ}-approximation in this setting, where ρ is an approximation ratio for a heuristic solver for the Steiner tree problem and 𝓁 is the number of priorities or levels (Byrka et al. give a Steiner tree algorithm with ρ≈1.39, for example). In this paper, we describe a natural generalization to the multi-level case of the classical (single-level) Steiner tree approximation algorithm based on Kruskal’s minimum spanning tree algorithm. We prove that this algorithm achieves an approximation ratio at least as good as Charikar et al., and experimentally performs better with respect to the optimum solution. We develop an integer linear programming formulation to compute an exact solution for the multi-level Steiner tree problem with non-proportional edge costs and use it to evaluate the performance of our algorithm on both random graphs and multi-level instances derived from SteinLib.

Cite as

Reyan Ahmed, Faryad Darabi Sahneh, Keaton Hamm, Stephen Kobourov, and Richard Spence. Kruskal-Based Approximation Algorithm for the Multi-Level Steiner Tree Problem. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 4:1-4:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{ahmed_et_al:LIPIcs.ESA.2020.4,
  author =	{Ahmed, Reyan and Sahneh, Faryad Darabi and Hamm, Keaton and Kobourov, Stephen and Spence, Richard},
  title =	{{Kruskal-Based Approximation Algorithm for the Multi-Level Steiner Tree Problem}},
  booktitle =	{28th Annual European Symposium on Algorithms (ESA 2020)},
  pages =	{4:1--4:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-162-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{173},
  editor =	{Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.4},
  URN =		{urn:nbn:de:0030-drops-128709},
  doi =		{10.4230/LIPIcs.ESA.2020.4},
  annote =	{Keywords: multi-level, Steiner tree, approximation algorithms}
}

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Documents authored by Hamm, Keaton

Multi-Level Weighted Additive Spanners

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Kruskal-Based Approximation Algorithm for the Multi-Level Steiner Tree Problem

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