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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
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We consider distance queries in vertex labeled planar graphs. For any fixed
$0 < \epsilon \leq 1/2$ 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 $\lambda$ return a
$(1+O(\epsilon))$-approximation of the distance between $...
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Citations
... 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. ...
Given an undirected n-vertex planar graph with non-negative edge weight function and given an assigned label to each vertex, a vertex-labeled distance oracle is a data structure which for any query consisting of a vertex u and a label reports the shortest path distance from u to the nearest vertex with label . We show that if there is a distance oracle for undirected n-vertex planar graphs with non-negative edge weights using s(n) space and with query time q(n), then there is a vertex-labeled distance oracle with space and query time. Using the state-of-the-art distance oracle of Long and Pettie, our construction produces a vertex-labeled distance oracle using space and query time at one extreme, space and query time at the other extreme, as well as such oracles for the full tradeoff between space and query time obtained in their paper. This is the first non-trivial exact vertex-labeled distance oracle for planar graphs and, to our knowledge, for any interesting graph class other than trees.
... We observe that by modifying the basic recursive decomposition of Thorup to use fundamental cycle separators instead of root path separators (this modification was previously used in e.g. [2,45]), we can assume that the endpoints of P in fact lie on a single face of G. This additional assumption enables us to achieve two important things. ...
In this paper we present an efficient reachability oracle under single-edge or single-vertex failures for planar directed graphs. Specifically, we show that a planar digraph G can be preprocessed in time, producing an -space data structure that can answer in time whether u can reach v in G if the vertex x (the edge~f) is removed from G, for any query vertices u,v and failed vertex x (failed edge f). To the best of our knowledge, this is the first data structure for planar directed graphs with nearly optimal preprocessing time that answers all-pairs queries under any kind of failures in polylogarithmic time. We also consider 2-reachability problems, where we are given a planar digraph G and we wish to determine if there are two vertex-disjoint (edge-disjoint) paths from u to v, for query vertices u,v. In this setting we provide a nearly optimal 2-reachability oracle, which is the existential variant of the reachability oracle under single failures, with the following bounds. We can construct in time an -space data structure that can check in time for any query vertices u,v whether v is 2-reachable from u, or otherwise find some separating vertex (edge) x lying on all paths from u to v in G. To obtain our results, we follow the general recursive approach of Thorup for reachability in planar graphs [J.~ACM~'04] and we present new data structures which generalize dominator trees and previous data structures for strong-connectivity under failures [Georgiadis et al., SODA~'17]. Our new data structures work also for general digraphs and may be of independent interest.
... We observe that by modifying the basic recursive decomposition of Thorup to use fundamental cycle separators instead of root path separators (this modification was previously used in e.g. [2,43]), we can assume that the endpoints of P in fact lie on a single face of G. This additional assumption enables us to achieve two important things. ...
... To guarantee that G r i (i ∈ {1, 2}) is smaller than G r by a constant factor [19], C r is replaced in G r i by the reduction of C r to the vertices of C r that have neighbors in V (G r i ) \ V (C r ). See [16] for a more comprehensive description of the recursive decomposition and the decomposition tree. Since we use this decomposition as a black box, we only highlight the following useful properties of T G . ...
Let G be a graph where each vertex is associated with a label. A vertex-labeled approximate distance oracle is a data structure that, given a vertex v and a label λ, returns a (1 + ε)-approximation of the distance from v to the closest vertex with label λ in G. Such an oracle is dynamic if it also supports label changes. In this paper we present three different dynamic approximate vertex-labeled distance oracles for planar graphs, all with polylogarithmic query and update times, and nearly linear space requirements. No such oracles were previously known.
... Here ∆ is the hop-diameter of the graph, which can be Θ(n). Mozes and Skop [9], 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 lg n lg(nN )) space, and has O(lg lg n lg lg nN + ε −1 ) query time. ...
... Our oracle for directed planar graphs (Section 4) is based on the static vertexlabeled distance oracle of [9], which uses connection for sets of vertices (i.e., a label) rather than connections for individual vertices. We show how to efficiently maintain the connections for a dynamically changing set of vertices using a bottom-up approach along the decomposition of the graph. ...
Let G be a graph where each vertex is associated with a label. A Vertex-Labeled Approximate Distance Oracle is a data structure that, given a vertex v and a label , returns a -approximation of the distance from v to the closest vertex with label in G. Such an oracle is dynamic if it also supports label changes. In this paper we present three different dynamic approximate vertex-labeled distance oracles for planar graphs, all with polylogarithmic query and update times, and nearly linear space requirements.
We start a systematic study of data structures for the nearest colored node problem on trees. Given a tree with colored nodes and weighted edges, we want to answer queries (v,c) asking for the nearest node to node v that has color c. This is a natural generalization of the well-known nearest marked ancestor problem. We give an O(n)-space O(log log n)-query solution and show that this is optimal. We also consider the dynamic case where updates can change a node's color and show that in O(n) space we can support both updates and queries in O(log n) time. We complement this by showing that O(polylogn) update time implies Ω(log nlog log n) query time. Finally, we consider the case where updates can change the edges of the tree (link-cut operations). There is a known (top-tree based) solution that requires update time that is roughly linear in the number of colors. We show that this solution is probably optimal by showing that a strictly sublinear update time implies a strictly subcubic time algorithm for the classical all pairs shortest paths problem on a general graph. We also consider versions where the tree is rooted, and the query asks for the nearest ancestor/descendant of node v that has color c, and present efficient data structures for both variants in the static and the dynamic setting.
