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# Computing β-Stretch Paths in Drawings of Graphs

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LIPIcs.SWAT.2020.7.pdf
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## Cite As

Esther M. Arkin, Faryad Darabi Sahneh, Alon Efrat, Fabian Frank, Radoslav Fulek, Stephen Kobourov, and Joseph S. B. Mitchell. Computing β-Stretch Paths in Drawings of Graphs. In 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 162, pp. 7:1-7:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.SWAT.2020.7

## Abstract

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. The βSP also asks that we 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.

## Subject Classification

##### ACM Subject Classification
• Mathematics of computing → Paths and connectivity problems
• Theory of computation → Computational geometry
##### Keywords
• stretch factor
• dilation
• geometric spanners

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## References

1. Soroush Alamdari, Timothy M Chan, Elyot Grant, Anna Lubiw, and Vinayak Pathak. Self-approaching graphs. In International Symposium on Graph Drawing, pages 260-271. Springer, 2012.
2. Boris Aronov, Mark De Berg, Esther Ezra, and Micha Sharir. Improved bounds for the union of locally fat objects in the plane. SIAM Journal on Computing, 43(2):543-572, 2014.
3. Jonas Azzam and Raanan Schul. How to take shortcuts in Euclidean space: making a given set into a short quasi-convex set. Proceedings of the London Mathematical Society, 105(2):367-392, 2012.
4. Prosenjit Bose and Michiel Smid. On plane geometric spanners: A survey and open problems. Computational Geometry, 46(7):818-830, 2013.
5. Ke Chen, Adrian Dumitrescu, Wolfgang Mulzer, and Csaba D Tóth. On the stretch factor of polygonal chains. arXiv preprint arXiv:1906.10217, 2019.
6. Edith Cohen. Fast algorithms for constructing t-spanners and paths with stretch t. SIAM Journal on Computing, 28(1):210-236, 1998.
7. Mark de Berg. Improved bounds on the union complexity of fat objects. Discrete & Computational Geometry, 40(1):127-140, 2008.
8. Annette Ebbers-Baumann, Rolf Klein, Elmar Langetepe, and Andrzej Lingas. A fast algorithm for approximating the detour of a polygonal chain. Computational Geometry, 27(2):123-134, 2004.
9. David Eppstein. Spanning trees and spanners. Handbook of computational geometry, pages 425-461, 1999.
10. David Eppstein, Michael T. Goodrich, Doruk Korkmaz, and Nil Mamano. Defining equitable geographic districts in road networks via stable matching. In Proceedings of the 25th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems, GIS 2017, Redondo Beach, CA, USA, November 7-10, 2017, pages 52:1-52:4, 2017.
11. Mohammad Farshi, Panos Giannopoulos, and Joachim Gudmundsson. Improving the stretch factor of a geometric network by edge augmentation. SIAM Journal on Computing, 38(1):226-240, 2008.
12. Christian Icking, Rolf Klein, and Elmar Langetepe. Self-approaching curves. Mathematical Proceedings of the Cambridge Philosophical Society, 125(3):441-453, 1999.
13. Richard M Karp. Reducibility among combinatorial problems. In Complexity of computer computations, pages 85-103. Springer, 1972.
14. Rolf Klein and Martin Kutz. Computing geometric minimum-dilation graphs is np-hard. In International Symposium on Graph Drawing, pages 196-207. Springer, 2006.
15. Giri Narasimhan and Michiel Smid. Approximating the stretch factor of euclidean graphs. SIAM Journal on Computing, 30(3):978-989, 2000.
16. Giri Narasimhan and Michiel Smid. Geometric spanner networks. Cambridge University Press, 2007.
17. David Peleg and Alejandro A Schäffer. Graph spanners. Journal of graph theory, 13(1):99-116, 1989.
18. Daniel D. Polsby and Robert D. Popper. The third criterion: Compactness as a procedural safeguard against partisan gerrymandering. Yale Law and Policy Review, 9(2):301-353, 1991.
19. Kristopher Tapp. Measuring political gerrymandering. The American Mathematical Monthly, 126(7):593-609, 2019.
20. A Frank van der Stappen, Dan Halperin, and Mark H Overmars. The complexity of the free space for a robot moving amidst fat obstacles. Computational Geometry, 3(6):353-373, 1993.
21. A Frank van der Stappen and Mark H Overmars. Motion planning amidst fat obstacles. In Proceedings of the tenth annual symposium on Computational geometry, pages 31-40. ACM, 1994.
22. Vijay V Vazirani. Approximation algorithms. Springer Science & Business Media, 2013.