Eyal E. Skop’s research while affiliated with Reichman University and other places

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Publications (3)


Separating a graph using a fundamental cycle separator. Left: part of a graph G is shown. Edges of T are black. Two branches of T on a fundamental cycle are solid thick. Edges of the dual tree T∗ are shown in red. Since only a part of G is shown, this illustration does not show T∗ as a tree. Center: part of the subgraph Gext of G is shown. Right: the subgraph Gint is shown. Note that the spanning trees Text and Tint are subtrees of T, and that the dual spanning trees Text∗ and Tint∗ are subtrees of T∗. Observe that, e.g., the infinite face of Gint is not triangulated, yet the maximum degree of Tint∗ remains 3
Illustration of the recursive decomposition. Edges of T are black. Edges of the cycle separator at the first level are solid thick, edges of the cycle separators at the second level are dashed. Left: part of the graph G is shown. Center: part of the subgraph G0 of G. G0 has a single hole. Right: part of the subgraph G00 of G0. G00 has two holes. The subpath of T between vertices u and v is reduced into a single edge because internal vertices on this subpath are not incident to original faces of G
The figure illustrates a part of XrQ. The dashed circle represents Sepr. Only solid arcs are part of XrQ. A shortest path from q1 ∈ Q to u ∈ V (Gr) might be enclosed in Gr (local connections C(Q,u)) or approximated through a path which intersects a separator of an ancestor of r (in the figure, it is composed of arcs from C(q1,Q̄′)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C(q_{1},{\bar Q}^{\prime })$\end{document}, Q̄′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\bar Q}^{\prime }$\end{document} and C(Q̄′,u)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C({\bar Q}^{\prime },u)$\end{document})
The figure illustrates a part of XrQ in the vertex label case. The vertices u1 and u2 are λ-labeled vertices of Gr, and are not part of XrQ. Paths from Q to λ-labeled vertices such as u1 and u2 confined to Gr are represented in XrQ by arcs between Q and λ. These arcs correspond to the local 𝜖/6 connections of λ on Q in Ĝr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${{\hat G_{r}}}$\end{document}. All solid arcs are part of XrQ. A shortest path from q1 ∈ V (Q) to λ∈ℒr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lambda \in \mathcal {L}_{r}$\end{document} is approximated by connections from q1 to a separator of an ancestor of r and from there to λ. Note that C(Q′,λ) represent distances from Q′ to λ-labeled vertices that are not necessarily in Gr
Efficient Vertex-Label Distance Oracles for Planar Graphs
  • Article
  • Publisher preview available

February 2018

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86 Reads

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6 Citations

Theory of Computing Systems

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Eyal E. Skop

We consider distance queries in vertex labeled planar graphs. For any fixed 0<ϵ1/20 < \epsilon \leq 1/2 we show how to preprocess a planar graph with vertex labels and edge lengths into a data structure that answers queries of the following form. Given a vertex u and a label λ\lambda return a (1+O(ϵ))(1+O(\epsilon))-approximation of the distance between u and its closest vertex with label λ\lambda. For an undirected n-vertex planar graph the preprocessing time is O(ϵ2nlg3n)O(\epsilon^{-2}n\lg^{3}{n}), the size is O(ϵ1nlgn)O(\epsilon^{-1}n\lg{n}), and the query time is O(lglgn+ϵ1)O(\lg\lg{n} + \epsilon^{-1}). For a directed planar graph with arc lengths bounded by N, the preprocessing time is O(ϵ2nlg3nlg(nN))O(\epsilon^{-2}n\lg^{3}{n}\lg(nN)), the data structure size is O(ϵ1nlgnlg(nN))O(\epsilon^{-1}n\lg{n}\lg(nN)), and the query time is O(lglgnlglg(nN)+ϵ1)O(\lg\lg{n}\lg\lg(nN) + \epsilon^{-1}).

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Efficient Vertex-Label Distance Oracles for Planar Graphs

September 2015

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7 Reads

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7 Citations

Lecture Notes in Computer Science

We consider distance queries in vertex labeled planar graphs. For any fixed 0<ϵ1/20 < \epsilon \le 1/2 we show how to preprocess an undirected planar graph with vertex labels and edge lengths to answer queries of the following form. Given a vertex u and a label λ\lambda return a (1+ϵ)(1+\epsilon )-approximation of the distance between u and its closest vertex with label λ\lambda . The query time of our data structure is O(lglgn+ϵ1)O(\lg \lg {n} + \epsilon ^{-1}), where n is the number of vertices. The space and preprocessing time of our data structure are nearly linear. We give a similar data structure for directed planar graphs with slightly worse performance. The best prior result for the undirected case has similar space and preprocessing bounds, but exponentially slower query time. No nontrivial results were previously considered for the directed case.


Efficient Vertex-Label Distance Oracles for Planar Graphs

April 2015

We consider distance queries in vertex-labeled planar graphs. For any fixed 0<ϵ1/20 < \epsilon \leq 1/2 we show how to preprocess a directed planar graph with vertex labels and arc lengths into a data structure that answers queries of the following form. Given a vertex u and a label λ\lambda return a (1+ϵ)(1+\epsilon)-approximation of the distance from u to its closest vertex with label λ\lambda. For a directed planar graph with n vertices, such that the ratio of the largest to smallest arc length is bounded by N, the preprocessing time is O(ϵ2nlg3nlg(nN))O(\epsilon^{-2}n\lg^{3}{n}\lg(nN)), the data structure size is O(ϵ1nlgnlg(nN))O(\epsilon^{-1}n\lg{n}\lg(nN)), and the query time is O(lglgnlglg(nN)+ϵ1)O(\lg\lg{n}\lg\lg(nN) + \epsilon^{-1}). We also point out that a vertex label distance oracle for undirected planar graphs suggested in an earlier version of this paper is incorrect.

Citations (2)


... Here Δ is the hop-diameter of the graph, which can be Θ(n). Mozes and Skop [15], building on Thorup's oracle, described a stretch-(1 + ε) distance oracle for directed planar graphs that can be stored using O(ε −1 n log n log(nN)) space, and has O(log log n log log nN + ε −1 ) query time. ...

Reference:

Efficient Dynamic Approximate Distance Oracles for Vertex-Labeled Planar Graphs
Efficient Vertex-Label Distance Oracles for Planar Graphs
  • Citing Conference Paper
  • September 2015

Lecture Notes in Computer Science

... Vertex-labeled distance oracles have received considerably more attention in the approximate setting. With (1 + ) multiplicative approximation, it is known how to getÕ(n) space andÕ(1) query time both for undirected [11] and directed planar graphs [13] and it has been shown how oracles with such guarantees can be maintained dynamically under label changes to vertices usingÕ(1) time per vertex relabel. ...

Efficient Vertex-Label Distance Oracles for Planar Graphs

Theory of Computing Systems