On Approximate Distance Labels and Routing Schemes with Affine Stretch

  • Ittai Abraham
  • Cyril Gavoille
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6950)


For every integral parameter k > 1, given an unweighted graph G, we construct in polynomial time, for each vertex u, a distance label L(u) of size \({\tilde{O}}(n^{2/(2k-1)})\). For any u,v ∈ G, given L(u),L(v) we can return in time O(k) an affine approximation \(\hat{d}(u,v)\) on the distance d(u,v) between u and v in G such that \(d(u,v) \leqslant \hat{d}(u,v) \leqslant (2k-2)d(u,v) + 1\). Hence we say that our distance label scheme has affine stretch of (2k − 2)d + 1. For k = 2 our construction is comparable to the O(n 5/3) size, 2d + 1 affine stretch of the distance oracle of Pǎtraşcu and Roditty (FOCS ’10), it incurs a o(logn) storage overhead while providing the benefits of a distance label. For any k > 1, given a restriction of o(n 1 + 1/(k − 1)) on the total size of the data structure, our construction provides distance labels with affine stretch of (2k − 2)d + 1 which is better than the stretch (2k − 1)d scheme of Thorup and Zwick (J. ACM ’05). Our second contribution is a compact routing scheme with poly-logarithmic addresses that provides affine stretch guarantees. With \({\tilde{O}}(n^{3/(3k-2)})\)-bit routing tables we obtain affine stretch of (4k − 6)d + 1, for any k > 1. Given a restriction of o(n 1/(k − 1)) on the table size, our routing scheme provides affine stretch which is better than the stretch (4k − 5)d routing scheme of Thorup and Zwick (SPAA ’01).


Short Path Hash Function Triangle Inequality Sparse Graph Unweighted Graph 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Ittai Abraham
    • 1
  • Cyril Gavoille
    • 2
  1. 1.Silicon Valley CenterMicrosoft ResearchUSA
  2. 2.Université de Bordeaux, LaBRIFrance

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