{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article13555","name":"Computing \u03b2-Stretch Paths in Drawings of Graphs","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 \u03c0=f(P), is \u03b2-stretch if \u03c0 is a simple (non-self-intersecting) curve, and for every pair of distinct points p\u2208P and q\u2208P, the length of the sub-curve of \u03c0 connecting f(p) with f(q) is at most \u03b2||f(p)-f(q)\u2016, where \u2016.\u2016 denotes the Euclidean distance. We introduce and study the \u03b2-stretch Path Problem (\u03b2SP 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 \u03b2-stretch path P connecting s and t. The \u03b2SP also asks that we output P if it exists. \r\nThe \u03b2SP 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 \u03b2-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. \r\nWe prove that \u03b2SP is strongly NP-complete. We complement this result by giving a quasi-polynomial time algorithm, that for a given \u03b5>0, \u03b2\u2208O(poly(log |V(G)|)), and s,t\u2208V(G), outputs a \u03b2-stretch path between s and t, if a (1-\u03b5)\u03b2-stretch path between s and t exists in the drawing.","keywords":["stretch factor","dilation","geometric spanners"],"author":[{"@type":"Person","name":"Arkin, Esther M.","givenName":"Esther M.","familyName":"Arkin","affiliation":"Stony Brook University, NY, USA"},{"@type":"Person","name":"Sahneh, Faryad Darabi","givenName":"Faryad Darabi","familyName":"Sahneh","affiliation":"University of Arizona, Tucson, AZ, USA"},{"@type":"Person","name":"Efrat, Alon","givenName":"Alon","familyName":"Efrat","affiliation":"University of Arizona, Tucson, AZ, USA"},{"@type":"Person","name":"Frank, Fabian","givenName":"Fabian","familyName":"Frank","affiliation":"University of Arizona, Tucson, AZ, USA"},{"@type":"Person","name":"Fulek, Radoslav","givenName":"Radoslav","familyName":"Fulek","affiliation":"University of Arizona, Tucson, AZ, USA"},{"@type":"Person","name":"Kobourov, Stephen","givenName":"Stephen","familyName":"Kobourov","affiliation":"University of Arizona, Tucson, AZ, USA"},{"@type":"Person","name":"Mitchell, Joseph S. B.","givenName":"Joseph S. B.","familyName":"Mitchell","affiliation":"Stony Brook University, NY, USA"}],"position":7,"pageStart":"7:1","pageEnd":"7:20","dateCreated":"2020-06-12","datePublished":"2020-06-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Arkin, Esther M.","givenName":"Esther M.","familyName":"Arkin","affiliation":"Stony Brook University, NY, USA"},{"@type":"Person","name":"Sahneh, Faryad Darabi","givenName":"Faryad Darabi","familyName":"Sahneh","affiliation":"University of Arizona, Tucson, AZ, USA"},{"@type":"Person","name":"Efrat, Alon","givenName":"Alon","familyName":"Efrat","affiliation":"University of Arizona, Tucson, AZ, USA"},{"@type":"Person","name":"Frank, Fabian","givenName":"Fabian","familyName":"Frank","affiliation":"University of Arizona, Tucson, AZ, USA"},{"@type":"Person","name":"Fulek, Radoslav","givenName":"Radoslav","familyName":"Fulek","affiliation":"University of Arizona, Tucson, AZ, USA"},{"@type":"Person","name":"Kobourov, Stephen","givenName":"Stephen","familyName":"Kobourov","affiliation":"University of Arizona, Tucson, AZ, USA"},{"@type":"Person","name":"Mitchell, Joseph S. B.","givenName":"Joseph S. B.","familyName":"Mitchell","affiliation":"Stony Brook University, NY, USA"}],"copyrightYear":"2020","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.SWAT.2020.7","funding":"This work is supported in part by NSF grants CCF-1526406, CCF-1740858, CCF-1712119, and DMS-1839274.","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume6365","volumeNumber":162,"name":"17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)","dateCreated":"2020-06-12","datePublished":"2020-06-12","editor":{"@type":"Person","name":"Albers, Susanne","givenName":"Susanne","familyName":"Albers","email":"mailto:albers@in.tum.de","affiliation":"Department of Computer Science, Technical University of Munich, 85748 Garching, Germany"},"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article13555","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6365"}}}