An Optimal Algorithm for Minimum-Link Rectilinear

Transcript:An Optimal Algorithm for Minimum-Link Rectilinear:
An Optimal Algorithm for Minimum-Link Rectilinear Paths in Triangulated Rectilinear Domains Joseph S.B. Mitchell, Stony Brook University Valentin Polishchuk, Linköping University Mikko Sysikaski, Google Zurich Haitao Wang, Utah State University ICALP 2015 A general polygonal domain Input: a simple polygon P with some polygons (holes or obstacles) inside, and two points s and t Output: a path from s to t in P s t Minimum-link paths Making a turn in the s-t path is too “expensive” Output: a minimum-link path from s to t an s-t path of minimum number of edges (links) s t The rectilinear case of minimum-link path problem Each polygon edge is either horizontal or vertical Each edge of the sought s-t path is also either horizontal or vertical s t The rectilinear minimum-link path problem Input: a rectilinear domain P of n vertices and h holes, and s and t Output: a rectilinear minimum-link s-t path s t n = 39 h = 3 Previous work and our result The general case O(n2 α(n) log2 n) time, Mitchell, Rote, and Woeginger 92’ The rectilinear case O(n log n) time and O(n log n) space, Imai and Asano 86’ O(n log n) time and O(n) space, SSO 87’, DN 91’, MPS 14’ Ω(n + h log h) time lower bound, DN91’, MSD 00’ Our result: O(n + h log h) time and O(n) space, with a given triangulation of P triangulating P O(n + h log1+ε h) time, Bar-Yehuda and Chazelle 94’ Our result: answering single-source queries Build a data structure in O(n + h log h) time and O(n) space for s, to answer the queries: given any query point t, compute the link distance (the number of links of the rectilinear minimum-link s-t path) in O(log n) time report the actual path in additional time linear in the link distance Outline The previous work: Das and Narasimhan, 91’ Our improvement based on their work Four types of rectilinear paths Any rectilinear s-t path can be a vertical-start-vertical-end path, or v-v path a vertical-start-horizontal-end path, or v-h path a horizontal-start-vertical-end path, or h-v path a horizontal-start-horizontal-end path, or h-h path t s t s t s t s v-v path v-h path h-v path h-h path Computing a minimum-link path Compute the four paths: a minimum-link v-v path a minimum-link v-h path a minimum-link h-v path a minimum-link h-h path Return the one with