1 / 24

Quickest Visibility Queries in Polygonal Domains

Solve quickest visibility queries in polygonal domains by computing shortest paths to segments and using a decomposition approach. This method reduces query time from O((K+h).log.h.log.n) to O(h.log.h.log.n).

dalice
Download Presentation

Quickest Visibility Queries in Polygonal Domains

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Quickest Visibility Queries in Polygonal Domains Haitao Wang Utah State University SoCG 2017, Brisbane, Australia

  2. Shortest paths queries in a polygonal domain • A polygon P of h holes with a total of n vertices, a source point s • Given a query point t, find a shortest path from s to reach t t s

  3. Our problem: Quickest visibility queries • Given a query point t, find a shortest path from s to see t a t s b

  4. A subproblem: shortest-path-to-segment queries • Given a query segment t, find a shortest path from s to any point of t • q is a closest point of t to s q t s

  5. Why a subproblem? • To answer a quickest visibility query for t • Compute the visibility polygon Vis(t) of t • Find a closest pointq on the boundary of Vis(t) • Observation: q must be on a window of t a window b Vis(t) a q The quickest visibility query can be solved by computing a closest point in each window of t, using the shortest path to segment queries t s

  6. The simple polygon case is easier • Observation: Only one window needs to be considered • Difficulty for polygons with holes: have to consider many windows s t s q t

  7. A polygonal domain • A polygon P of h holes with a total of nvertices • h could be much smaller than n • Complexities are better measured by h instead of n • O(n2) vs. O(n + h2) h = 3 n = 32

  8. Previous work: Simple polygons • Shortest-path-to-segment queries Arkin et al. 2015, Chiang and Tamassia 1997 • Preprocessing time and space: O(n) • Query time: O(log n) • Quickest visibility queries Arkin et al. 2015 • Preprocessing time and space: O(n) • Query time: O(log n) • Apply the shortest-path-to-segment query on the single window

  9. Previous work and our result: Polygonal domains • Shortest-path-to-segment queries • Quickest visibility queries Assume SPM(s) has been computed in O(n log n) time K: the size of the visibility polygon Vis(t) of the query point t, and K = O(n)

  10. The Quickest Visibility Queries - O(h log h log n) time • A preliminary result: O((K+h) log h log n) query time • First, compute Vis(t) and find all O(K) windows • O(K h log n/h) time, if applied our shortest-path-to-segment queries on each window • Reduce the time to O((K+h) log h log n) • Goal: find a closest point q in all windows q t s

  11. Extended-windows • Extended-windows: Extend each window to t • q is also the closest point on extended-windows • Assumption: q is not an endpoint of these extended-windows • Otherwise, find a shortest path from s to each endpoint • Observation: if q is on an extended-window w, then π(s,q) arrives at q from the left or the right side of w • S1: The set of points on all extended-windows whose shortest paths are from the left side • q1: a closest point in S1 • q2: a closest point for the right side • q is either q1 or q2 • Assume q is q1 w q t s

  12. Computing q (= q1) • Prune some (parts of) extended-windows • For an extended-window w, if we know that a sub-segment w’ of w does not contain a closest point, then w’ can be pruned • Observation: For any point p on an extended-window, if π(s,p) intersects another extended-window, then p cannot be a closest point and can be pruned p a t s

  13. The pruning principle • Consider two extended-windows ta and tb, and three shortest paths from: π(s,a), π(s,b), π(s,t) • Case 1: no pruning can be done • Case 2: the entire tb can be pruned • Case 3: the sub-segment tc can be pruned b a b a a b c p p s s s t t t Case 1 Case 3 Case 2

  14. An example and the algorithm • Step 1: Pruning • Step 2: Compute a closest point from s to each remaining sub-segment • O((K+h) log h log n) time in total 6 5 4 2 8 7 3 1 9 s t 10

  15. Reducing the query time: O((K+h) log h log n)  O(h log h log n) • Main Idea: • Instead of Vis(t), only compute a special subset S(t) of O(h) windows • Lemma: a closest point q must be on a window of S(t) • Apply the preliminary algorithm on all windows of S(t) • How to find S(t)? • Extended corridor structure

  16. Defining the window set S(t) q s a t

  17. Defining the window set S(t) q b s a t t

  18. Thank you for your attention! 6 5 4 2 8 7 3 1 9 s t 10

  19. Shortest-path-to-segment queries • Our approach: Using a decomposition D of P • Bisectors of the shortest path map SPM(s) of s • V: the set of all intersections between bisectors and obstacles, and triple points (intersections of bisectors) • All shortest paths from s to all points of V • Remove all bisectors • The resulting decomposition is D b a s

  20. Important properties of the decomposition D • D can be computed in O(n) time from SPM(s) • Each cell is simply connected • After O(n) time preprocessing, given any segment t in a cell C, a shortest path from s to t can be computed in O(log |C|) time • For any segment t, t can intersect O(h) cells • For any segment t in P, the shortest path from s to t can be computed in O(h log n/h) time by a “pedestrian” algorithm t b a t s

  21. Important properties of the decomposition D • D can be computed in O(n) time from SPM(s) • Each cell is simply connected • Each cell C has two super-roots r1 and r2, such that for any point t in C, the shortest s-t path is the concatenation of a shortest s-r path and a shortest r-t path in C, for some super-root r • After O(n) time preprocessing, given any segment t in a cell C, a shortest path from s to t can be computed in O(log |C|) time • For any segment t, t can intersect O(h) cells • For any segment t in P, the shortest path from s to t can be computed in O(h log n/h) time by a “pedestrian” algorithm t r2 t b a t r1 s

  22. The pruning principle • Consider two extended-windows ta and tb, and three shortest paths from: π(s,a), π(s,b), π(s,t) • Case 1: no pruning can be done • Case 2: the entire tb can be pruned • Case 3: the sub-segment tc can be pruned b a b a a b c p p s s s t t t Case 1 Case 3 Case 2

  23. An example and the algorithm • Step 1: Pruning • Step 2: Compute a closest point from s to each remaining sub-segment • O((K+h) log h log n) time using the decomposition D • only need to consider O(h) cells of D 6 5 4 2 8 7 3 1 9 s t 10

  24. A summary on the preliminary result O(n log h) • Preprocessing time: O(n log h + h2log h) • Space: O(n log h + h2) • Query time: O((K+h) log h log n) • The quadratic preprocessing is only for computing the windows of t • A new problem: Given a query set of k=O(n) segments in P intersecting at a point t, find the closest point on these segments • Preprocessing time and space: O(n log h) • Query time: O((k+h) log h log n) O(n log h) If the windows are given q s t

More Related