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Let f be a drawing in the Euclidean plane of a graph G, which is understood to be a 1-dimensional simplicial complex. We assume that every edge of G is drawn by f as a curve of constant algebraic complexity, and the ratio of the length of the longest simple path to the the length of the shortest edge is poly(n). In the drawing f, a path P of G, or its image in the drawing π = f(P), is β-stretch if π is a simple (non-self-intersecting) curve, and for every pair of distinct points p ∈ P and q ∈ P , the length of the sub-curve of π connecting f(p) with f(q) is at most β∥f(p) − f(q)∥, where ∥.∥ denotes the Euclidean distance. We introduce and study the β-stretch Path Problem (βSP for short), in which we are given a pair of vertices s and t of G, and we are to decide whether in the given drawing of G there exists a β-stretch path P connecting s and t. We also output P if it exists. The βSP quantifies a notion of “near straightness” for paths in a graph G, motivated by gerrymandering regions in a map, where edges of G represent natural geographical/political boundaries that may be chosen to bound election districts. The notion of a β-stretch path naturally extends to cycles, and the extension gives a measure of how gerrymandered a district is. Furthermore, we show that the extension is closely related to several studied measures of local fatness of geometric shapes. We prove that βSP is strongly NP-complete. We complement this result by giving a quasi-polynomial time algorithm, that for a given ε > 0, β ∈ O(poly(log |V (G)|)), and s, t ∈ V (G), outputs a β-stretch path between s and t, if a (1 − ε)β-stretch path between s and t exists in the drawing.
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Map-matching using shortest paths
We consider several variants of the map-matching problem, which seeks to find a path Q in graph G that has the smallest distance to a given trajectory P (which is likely not to be exactly on the graph). In a typical application setting, P models a noisy GPS trajectory from a person traveling on a road network, and the desired path Q should ideally correspond to the actual path in G that the person has traveled. Existing map-matching algorithms in the literature consider all possible paths in G as potential candidates for Q. We find solutions to the map-matching problem under different settings. In particular, we restrict the set of paths to shortest paths, or concatenations of shortest paths, in G. As a distance measure, we use the Fréchet distance, which is a suitable distance measure for curves since it takes the continuity of the curves into account.
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- Award ID(s):
- 1618605
- PAR ID:
- 10182027
- Date Published:
- Journal Name:
- IWISC
- Page Range / eLocation ID:
- 44 to 51
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1145/3191801.3191812
