On Approximate Distance Labels and Routing Schemes with Affine Stretch

Abraham, Ittai; Gavoille, Cyril

doi:10.1007/978-3-642-24100-0_39

Cited by 31 publications

(98 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This includes both the odd stretches from [TZ05] and the even stretches from [PR10], [AG11]. For arbitrary (noninteger stretches) α, the best known space was S( α , m).…”

Section: B Our Resultsmentioning

confidence: 99%

“…This leaves open a polynomial gap for stretch 2. Abraham and Gavoille [AG11] implicitly (we shall shortly discuss what they did explicitly) generalized the stretch 2 oracle to all even stretches. With stretch 2k, the space becomes O(m 1+1/(k+1/2) ).…”

Section: A State-of-the-artmentioning

confidence: 97%

“…In fact, they start with this simpler multiplicative-additive version for unweighted graphs before they get the clean stretch 2 for weighted graphs. Abraham and Gavoille [AG11] continued with the multiplicative-additive version, showing how to generalize it to estimatesδ(u, v) ≤ 2kδ(u, v) + 1, using space O(n 1+1/(k+1/2) ), and using it for labeling and routing schemes. They say they will handle the weighted case in the full version.…”

Section: A State-of-the-artmentioning

confidence: 98%

See 2 more Smart Citations

A New Infinity of Distance Oracles for Sparse Graphs

Pătraşcu

Roditty

Thorup

2012

2012 IEEE 53rd Annual Symposium on Foundations of Computer Science

View full text Add to dashboard Cite

Given a weighted undirected graph, our basic goal is to represent all pairwise distances using much less than quadratic space, such that we can estimate the distance between query vertices in constant time. We will study the inherent trade-off between space of the representation and the stretch (multiplicative approximation disallowing underestimates) of the estimates when the input graph is sparse with m = O(n) edges.In this paper, for any fixed positive integers k and , we obtain stretches α = 2kFor integer stretches, this coincides with the previous bounds (odd stretches with = 1 and even stretches with = 2). The infinity of fractional stretches between consecutive integers are all new (even though is fixed as a constant independent of the input, the number of integers is still countably infinite). We will argue that the new fractional points are not just arbitrary, but that they, at least for fixed stretches below 3, provide a complete picture of the inherent trade-off between stretch and space in m. Consider any fixed stretch α < 3. Based on the hardness of set intersection, we argue that if is the largest integer such that 3 − 2/ ≤ α, then Ω(S(3 − 2 , m)) space is needed for stretch α. In particular, for fixed stretch below 2 2 3 , we improve Pǎtraşcu and Roditty's lower bound from Ω(m 3/2 ) to Ω(m 5/3 ), thus matching their upper bound for stretch 2. For space in terms of m, this is the first hardness matching the space of a non-trivial/sub-quadratic distance oracle.

show abstract

“…This includes both the odd stretches from [TZ05] and the even stretches from [PR10], [AG11]. For arbitrary (noninteger stretches) α, the best known space was S( α , m).…”

Section: B Our Resultsmentioning

confidence: 99%

Section: A State-of-the-artmentioning

confidence: 97%

Section: A State-of-the-artmentioning

confidence: 98%

See 1 more Smart Citation

A New Infinity of Distance Oracles for Sparse Graphs

Pătraşcu

Roditty

Thorup

2012

2012 IEEE 53rd Annual Symposium on Foundations of Computer Science

View full text Add to dashboard Cite

show abstract

“…1 The stretch α of a distance oracle D(G) is the smallest (in fact, infimum) value such that for every u,…”

Section: Distance Oracles For General Graphsmentioning

confidence: 99%

“…(The algorithm actually explores only original vertices. For the sake of this argument we imagine that when the algorithm reaches a vertex y it reaches its first copy y (1) . Right after that it reaches the next copy y (2) , etc., and then reaches y (d(y)) .…”

Section: An Extension To General Graphsmentioning

confidence: 99%

A Linear-Size Logarithmic Stretch Path-Reporting Distance Oracle for General Graphs

Elkin

Pettie

2016

ACM Trans. Algorithms

View full text Add to dashboard Cite

In [33] Thorup and Zwick came up with a landmark distance oracle. Given an n-vertex undirected graph G = (V, E) and a parameter k = 1, 2, . . ., their oracle has size O(kn 1+1/k ), and upon a query (u, v) it constructs a path Π between u and v of length δ(u, v) such that u, v). The query time of the oracle from [33] is O(k) (in addition to the length of the returned path), and it was subsequently improved to O(1) [36,13]. A major drawback of the oracle of [33] is that its space is Ω(n · log n). Mendel and Naor [23] devised an oracle with space O(n 1+1/k ) and stretch O(k), but their oracle can only report distance estimates and not actual paths. In this paper we devise a path-reporting distance oracle with size O(n 1+1/k ), stretch O(k) and query time O(n ǫ ), for an arbitrarily small ǫ > 0. In particular, for k = log n our oracle provides logarithmic stretch using linear size. Another variant of our oracle has size O(n log log n), polylogarithmic stretch, and query time O(log log n).For unweighted graphs we devise a distance oracle with multiplicative stretch O(1), additive stretch O(β(k)), for a function β(·), space O(n 1+1/k · β), and query time O(n ǫ ), for an arbitrarily small constant ǫ > 0. The tradeoff between multiplicative stretch and size in these oracles is far below Erdős's girth conjecture threshold (which is stretch 2k − 1 and size O(n 1+1/k )). Breaking the girth conjecture tradeoff is achieved by exhibiting a tradeoff of different nature between additive stretch β(k) and size O(n 1+1/k ). A similar type of tradeoff was exhibited by a construction of (1 + ǫ, β)-spanners due to Elkin and Peleg [18]. However, so far (1 + ǫ, β)-spanners had no counterpart in the distance oracles' world.An important novel tool that we develop on the way to these results is a distancepreserving path-reporting oracle. We believe that this oracle is of independent interest. * A preliminary version of this paper was published in SODA '15 [19].

show abstract

Approximate Distance Oracles with Improved Stretch for Sparse Graphs

Roditty

Tov

2021

Lecture Notes in Computer Science

View full text Add to dashboard Cite

On Approximate Distance Labels and Routing Schemes with Affine Stretch

Cited by 31 publications

References 9 publications

A New Infinity of Distance Oracles for Sparse Graphs

A New Infinity of Distance Oracles for Sparse Graphs

A Linear-Size Logarithmic Stretch Path-Reporting Distance Oracle for General Graphs

Approximate Distance Oracles with Improved Stretch for Sparse Graphs

Contact Info

Product

Resources

About