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**11 - 12**of**12**### unknown title

"... Let G be a planar graph with n vertices and non-negative edge-lengths. Given a set of k pairs of vertices, we are interested in computing the distance in G between those k pairs of vertices. We describe how this can be achieved in O(n 2/3 k 2/3 log n + n 4/3 log 1/3 n) time, improving previous resul ..."

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Let G be a planar graph with n vertices and non-negative edge-lengths. Given a set of k pairs of vertices, we are interested in computing the distance in G between those k pairs of vertices. We describe how this can be achieved in O(n 2/3 k 2/3 log n + n 4/3 log 1/3 n) time, improving previous results for a large range of k. As possible applications, we show how this result speeds up previous algorithms for finding shortest non-contractible cycles for graphs on a bounded-genus surface or for computing the dilation of a geometric planar graph. 1

### Planar Reachability in Linear Space and Constant Time

, 2014

"... We show how to represent a planar digraph in linear space so that distance queries can be answered in constant time. The data structure can be constructed in linear time. This representation of reachability is thus optimal in both time and space, and has optimal construction time. The previous best ..."

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We show how to represent a planar digraph in linear space so that distance queries can be answered in constant time. The data structure can be constructed in linear time. This representation of reachability is thus optimal in both time and space, and has optimal construction time. The previous best solution used O(n log n) space for constant query time [Thorup FOCS’01].