A new algorithm for Euclidean shortest paths in the plane

Wang, Haitao

doi:10.1145/3406325.3451037

Cited by 3 publications

(2 citation statements)

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“…Because we do not consider ∂H to be part of B u , this is a possibly disconnected curve that consists of edges from the shortest path map of u and ∆. The shortest path map of a point p in a polygonal domain P partitions the free space into maximal regions, such that for any two points in the same region the shortest paths from u to both points use the same vertices of P [37]. We call the curves of the shortest path map for which there are two topologically distinct paths from p to any point on the curve walls.…”

Section: Definition 19mentioning

confidence: 99%

“…In the augmented shortest path map, we include the shortest path tree of u, essentially triangulating each region in the shortest path map using the vertices of P . This can be constructed in O(m log m) time [37] (or even O(m + h log h) time, where h is the number of holes in P ). We then find the region of SPM u that contains p ∈ S for all p in O(n log m) time.…”

Section: Figure 13mentioning

confidence: 99%

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The Complexity of Geodesic Spanners

Berg¹,

Kreveld²,

Staals³

2023

Preprint

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A geometric t-spanner for a set S of n point sites is an edge-weighted graph for which the (weighted) distance between any two sites p, q ∈ S is at most t times the original distance between p and q. We study geometric t-spanners for point sets in a constrained two-dimensional environment P . In such cases, the edges of the spanner may have non-constant complexity. Hence, we introduce a novel spanner property: the spanner complexity, that is, the total complexity of all edges in the spanner. Let S be a set of n point sites in a simple polygon P with m vertices. We present an algorithm to construct, for any constant ε > 0 and fixed integer k ≥ 1, a (2k + ε)-spanner with complexity O(mn 1/k + n log 2 n) in O(n log 2 n + m log n + K) time, where K denotes the output complexity. When we consider sites in a polygonal domain P with holes, we can construct such a (2k + ε)-spanner of similar complexity in O(n 2 log m + nm log m + K) time. Additionally, for any constant ε ∈ (0, 1) and integer constant t ≥ 2, we show a lower bound for the complexity of any (t − ε)-spanner of Ω(mn 1/(t−1) + n).

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Section: Definition 19mentioning

confidence: 99%

Section: Figure 13mentioning

confidence: 99%