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Bypassing Erdős’ Girth Conjecture: Hybrid Stretch and Sourcewise Spanners

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Automata, Languages, and Programming (ICALP 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8573))

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Abstract

An (α,β)-spanner of an n-vertex graph G = (V,E) is a subgraph H of G satisfying that dist(u, v, H) ≤ α·dist(u, v, G) + β for every pair (u, v) ∈ V ×V, where dist(u,v,G′) denotes the distance between u and v in G′ ⊆ G. It is known that for every integer k ≥ 1, every graph G has a polynomially constructible (2k − 1,0)-spanner of size O(n 1 + 1/k). This size-stretch bound is essentially optimal by the girth conjecture. Yet, it is important to note that any argument based on the girth only applies to adjacent vertices. It is therefore intriguing to ask if one can “bypass” the conjecture by settling for a multiplicative stretch of 2k − 1 only for neighboring vertex pairs, while maintaining a strictly better multiplicative stretch for the rest of the pairs. We answer this question in the affirmative and introduce the notion of k-hybrid spanners, in which non neighboring vertex pairs enjoy a multiplicative k stretch and the neighboring vertex pairs enjoy a multiplicative (2k − 1) stretch (hence, tight by the conjecture). We show that for every unweighted n-vertex graph G, there is a (polynomially constructible) k-hybrid spanner with O(k 2 ·n 1 + 1/k) edges. This should be compared against the current best (α,β) spanner construction of [5] that obtains (k,k − 1) stretch with O(k ·n 1 + 1/k) edges. An alternative natural approach to bypass the girth conjecture is to allow ourself to take care only of a subset of pairs S ×V for a given subset of vertices S ⊆ V referred to here as sources. Spanners in which the distances in S ×V are bounded are referred to as sourcewise spanners. Several constructions for this variant are provided (e.g., multiplicative sourcewise spanners, additive sourcewise spanners and more).

Recipient of the Google European Fellowship in distributed computing; research supported in part by this Fellowship. Supported in part by the Israel Science Foundation (grant 894/09), United States-Israel Binational Science Foundation (grant 2008348), Israel Ministry of Science and Technology (infrastructures grant), and Citi Foundation.

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Authors and Affiliations

  1. The Weizmann Institute of Science, Rehovot, Israel

    Merav Parter

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  1. Merav Parter
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Editors and Affiliations

  1. Institut für Informatik, Technische Universität München, Boltzmannstrasse 3, 85748, München, Germany

    Javier Esparza

  2. LIAFA, Université Paris Diderot-Paris 7, Case 7014, 75205, Paris Cedex 13, France

    Pierre Fraigniaud

  3. IT University of Copenhagen, Rued Langgaards Vej 7, 2300, Copenhagen, Denmark

    Thore Husfeldt

  4. Department Computer Science, University of Oxford, Wolfson Building, Parks Road, OX1 3QD, Oxford, UK,

    Elias Koutsoupias

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Parter, M. (2014). Bypassing Erdős’ Girth Conjecture: Hybrid Stretch and Sourcewise Spanners. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds) Automata, Languages, and Programming. ICALP 2014. Lecture Notes in Computer Science, vol 8573. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43951-7_49

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  • DOI: https://doi.org/10.1007/978-3-662-43951-7_49

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-43950-0

  • Online ISBN: 978-3-662-43951-7

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