Advanced Algorithms Computational Geometry Prof. Karen Daniels. Fall, 2012

Transcription

1 UMass Lowell Computer Science Advanced Algorithms Computational Geometry Prof. Karen Daniels Fall, 2012 O Rourke Chapter 7 Search & Intersection

2 Chapter 7 Search & Intersection Segment-Segment Intersection Segment-Triangle Intersection Point in Polygon Point in Polyhedron Intersection of Convex Polygons Intersection of Segments Intersection of Nonconvex Polygons Extreme Point of Convex Polygon Extremal Polytope Queries Planar Point Location

3 Segment-Segment Intersection Finding the actual intersection point Approach: parametric vs. slope/intercept parametric generalizes to more complex intersectionsi Parameterize each segment L cd c a b L ab d L cd c C=d-c b a A=b-a p(s)=a+sa L ab q(t)=c+tc d Intersection: values of s, t such that p(s) =q(t) : a+sa=c+tc

4 Segment-Triangle Intersection Determine if qr intersects plane π containing triangle T. useful in ray tracing Let N=(A,B,C) be normal to π r π :(x,y,z) (A,B,C)=D (i.e.π π = Ax + By + Cz = D) π c N Find normal N using cross-products involving a, b, c p T parameterize qr: p(t) = q + t(r-q) a b D ( q N) solve for t : t = q ( r q) N using t, find point of intersection p if it exists Classify relationship between p and T p is in T iff its projection p is in a projection* T of T to xy, xz or yz-plane +-- Barycentric coordinates left edge p +-+ Check results of 3 of tests for with ++respect to T : +++ means p inside T * avoid degeneracy by projecting out largest coordinate --+ T

5 Point in Polygon An application: GUI mouse click Winding number (see Guibas/Stolfi) standing at q, turn to follow border of P sum rotation to create signed angular turn + ccw - cw divide by 2π elegant but not practical: O(n) with large constant Ray crossings extend horizontal ray from q count number of intersections with P even q is outside P odd q is inside P beware degenerate intersections! Compare with using LeftOn test when polygon is convex P P O(n) q q

6 Winding number generalizes to 3D Point in Polyhedron uses solid angle (fraction of sphere surface used by cone at point) Ray crossing generalizes to 3D q inside P if odd number of crossings q outside P if even number of crossings Running Time?? (see next slide) P q r Assume each face is a triangle. Algorithm: : POINT in POLYHEDRON (P, R, q) Compute bounding radius R loop forever r 0 = random ray of length R r=q+r r 0 crossings = 0 for each triangle T of polyhedron P do SegTriInt(T, q, r) if degenerate intersection then go back to loop else increment crossings appropriately if crossings odd then q is inside P else q is outside P Exit

7 Point in Polyhedron P q r Expected time O(ρn) ρ = expected number of tries to get nondegenerate intersection Experiment: One combinatorially dense sample polyhedron has ρ ~ Claim: ρ = 1 + ε can be achieved for any ε > 0! For analysis, replace sphere with bounding cube. In worst case, L integer points on cube face disqualified by each edge of polyhedron. Probability of hitting a degeneracy = Let E = #edges (E x L) out of L 2 disqualified = E/L E is a constant, so choosing L large enough yields any desired ε = E/L. Assume each face is a triangle. Imagine integer discretization of L. Edge e kills a line of points on face of surrounding cube.

8 Intersection of Convex Polygons For nonconvex n-vertex P and m-vertex Q, the worst-case size complexity of P Q is in Ω(nm) For convex P, Q, size is only in O ( n + m) ) and can be computed in O ( n + m) time B A (example) Q P Algorithm: : INTERSECTION of CONVEX POLYGONS Choose A and B arbitrarily /* A is directed edge on P */ repeat /* B is directed edge on Q */ if A intersects B then /* A, B chase each other */ /* A, B meet at each */ Check for termination /* boundary crossing */ Update inside flag Advance either A or B depending on geometric conditions (see next slide) until both A and B cycle their polygons Handle cases: P Q P Q P Q = 0/ O(n+m)

9 Intersection of Convex Polygons If B aims towards line containing A but does not cross it then If B aims towards line containing A, but does not cross it, then advance B to close in on possible intersection with A.

10 Intersection of Segments Goal: Output-size sensitive polygon intersection algorithm Core Problem: Output-size sensitive line segment intersection algorithm Bentley-Ottmann plane sweep: O((n+k)logn) time k = number of intersection points in output Intuition: sweep line (discrete event ent simulation) First, review algorithm that returns TRUE iff there exists an intersection

11 Intersection of >2 Line Segments Sweep-Line Algorithmic Paradigm: 33.4 source: textbook Cormen et al.

12 Intersection of >2 Line Segments Sweep-Line Algorithmic Paradigm: source: textbook Cormen et al.

13 Intersection of >2 Line Segments Time to decide if any 2 segments intersect: segments intersect:o(n lg n) Balanced BST stores segments in order of intersection with sweep line. Associated operations take O(lgn) time. (See next slide.) Note that it exits as soon as one intersection is detected source: source: textbook Cormen et al.

14 Intersection of >2 Line Segments Balanced BST stores segments in order of intersection with sweep line. Associated operations take O(lgn) time. Each segment has a leaf node. At each internal node, store segment from rightmost leaf in its left subtree. What does this allow us to do? source: deberg et al.

15 Intersection of Segments Goal: Output-size sensitive line segment intersection algorithm that actually computes all intersection points Bentley-Ottmann plane sweep: O((n+k)log(n+k))= O((n+k)logn) time k = number of intersection points in output Intuition: sweep vertical line rightwards just before intersection, 2 segments are adjacent in sweep-line intersection structure check for intersection only adjacent segments insert intersection event into sweep-line structure event types: (example on previous slide) left endpoint of a segment right endpoint of a segment intersection between 2 segments swap order Improved to O(nlogn+k) [Chazelle/Edelsbrunner] l, Computational Geometry in C

16 Intersection of Nonconvex Polygons Variation on Bentley- Ottmann sweep Maintain status for each piece of sweep line: 0: exterior to P, Q P: inside P, outside Q Q: inside Q, outside P PQ: inside P, inside Q Useful for CAD/CAM P Q For n-vertex P,, m-vertex Q,, O((n+m n+m)log( )log(n+m)+k) time to compute: P Q P Q P \ Q

17 Extreme Point of Convex Polygon Algorithm: HIGHEST POINT of CONVEX POLYGON Initialize a and b /* highest h point is in [a,b] */ repeat forever c index midway from a to b if P[c] is locally highest then return c if A points up and C points down then [a,b] [a,c] /* highest point is in [a,c] */ else if A points down and C points up then [a,b] [c,b] /* highest point is in [c,b] */ else if A points up and C points up if P[a] is above P[c] then [a,b] [a,c] else [a,b] [c,b] /* highest point is in [c,b] */ else if A points down and C points down if P[a] is below P[c] then [a,b] [a,c] /* highest point is in [a,c] */ else [a,b] [c,b] /* highest point is in [c,b] */ I iti li d b A Geometric Binary Search C c b A Unimodality allows B Binary Search O(lg n) a

18 Stabbing a Convex Polygon Extreme-Finding algorithm can stab a convex polygon extreme in + u direction a +u y L x b extreme in - u direction If a and b straddle L, binary search on [a,b] yields x binary search on [b,a] yields y O(lg n)

19 Extremal Polytope Queries: Main Idea Form sequence of O(log n) simpler nested polytopes in O(n) time (see next slide) To answer a (w.l.o.g.. vertical) query in O(logn logn) time: (see slide after next) Find extreme with respect to inner polytope, then work outwards Need only check small number of candidate vertices in next polytope Key idea: independent sets in planar graphs are large ; vertices of low degree to construct next (inner) polytope, remove independent set of vertices deleting constant fraction of vertices at each step produces O(logn logn) polytopes lt (derivation derivation) O(log n) An independent set of a polytope graph of n vertices produced by INDEPENDENT SET has size at least n/18 (to be shown later).

20 Extremal Polytope Queries: Main Idea (continued)

21 Extremal Polytope Queries: Main Idea (continued) For maximal z query: Moving from P i+1 to P i is like raising a plane orthogonal to z axis from a i+1 1( (highest point on P i+1 ) to a i a 0

22 Extremal Polytope Queries Details Maximum Independent Set is NP-complete, but greedy heuristic performs well. Choose vertex of lowest degree, not adjacent to vertices already chosen. Algorithm 7.4: : INDEPENDENT SET Input: : graph G Output: : independent set I I 0 Mark all nodes of G of degree >= 9 while some nodes remain unmarked do Choose an unmarked node v Mark v and all neighbors of v I I U {v} spurious edge? Goal: tetrahedron triangulate polytope: lt use convex hll hull (d) octahedron (a) Icosahedron Schlegel diagram: 5 triangles meet at each vertex. There exist at least n/2 vertices of degree at most 8 (derivation derivation). An independent set of a polytope graph of n vertices produced by INDEPENDENT SET has size at least n/18 (derivation).

23 Extremal Polytope Queries Details (continued) (creation) i i +1 1 (creation)

24 Extremal Polytope Queries Details (continued) To use nested polytope hierarchy to answer an extreme point query: Find extreme with respect to inner polytope P k (brute-force search) Move from polytope P i+1 to P i Let a i and a i+1 be uniquely highest h vertices of P i and P i+1. Then (Lemma ) either: a i = a i+1 or a i+1 is the highest among the vertices adjacent to a i (See proof sketch in next slide.) (Assume w.l.o.g.. vertical query direction) π (Plane effect )

25 Extremal Polytope Queries Dtil Details (continued) Lemma : (repeated) Let a i and a i+1 be uniquely highest vertices of P i and P i+1. Then either: a i = a i+1 or a i+1 is the highest among the vertices adjacent to a i Proof Sketch: (by cases) a i is vertex of both P i and P i+1 a i = a i+1 else a i deleted in construction of P i+1 Let b i+1 = highest vertex of P i+1 among those adjacent to a i in P i Claim: b i+1 is highest vertex of P i+1 Proof uses polygonal-faced cone See book for further details But we still need bound on # vertices adjacent to a i+1

26 Extremal Polytope Queries Dtil Details (continued) Lemma7103:Leta : Let a i and a i+1 be uniquely highest vertices of P i and P i+1. Then either (to provide constant-time search): a i = a i+1 or a i is the highest among the parents of the extreme edges L i+1 and R i+1 Proof Sketch: π Project P i+1 onto plane orthogonal to π (e.g. xz plane) Let L i+1, R i+1 = 2 edges of P i+1 projecting to 2 edges of P i+1 incident to a i+1 Parents of edge are vertices of P i from L π i+1 = R i+1 which it derives (during P i+1 creation) tetrahedron New extreme edges are close to the old. projected See p for remaining details. onto xz plane P i+1

27 Extremal Polytope Queries Summary Algorithm: EXTREME POINT of a POLYTOPE Input: polytope P and direction vector u Output: vertex a of P extreme in u direction Construct nested polytope p hierarchy P = P 0 0,, P 1 1,,...,, P k a k vertex of P k extreme in u direction Compute L k and R k for i = k - 1, k - 2,...,1, 0 do a i extreme vertex among a i+1 and parents of L i+1 and R i+1 if a i = a i+1 then i i 1 for all edges incident to a i do π u P k P 0 save extreme edges L i and R i else (a i = a i+1 ) compute L i from L i+1 etc... After O(n) time and space preprocessing, a polytope extreme-point query can be answered in O(log n) time

28 Planar Point Location Goal: Given a planar subdivision of n vertices, preprocess it so that point location query can be quickly answered. A polygonal planar subdivision can be preprocessed in O(n) time and space for O(log n) query. 2D version/variant of independent set/nested approach ( may need to triangulate polygonal faces) Monotone subdivision approach* Randomized dtrapezoidal decomposition* i * See next slides.

29 Planar Point Location (continued) Monotone subdivision approach: -Partition subdivision (e.g. Voronoi diagram) into horizontal strips. -Double binary search: -Vertical search on strips to locate query point between 2 separators -Horizontal search to locate it within 1 strip -O(log 2 n) query time

30 Planar Point Location (continued) And now for something completely different As foreshadowed in Ch. 2 (polygon partitioning/triangulation) Seidel s randomized trapezoidal decomposition for non-crossing segments: * * O(nlogn) construction Assume no 2 points are at same height. time: extend horizontal line through each endpoint. O(logn) query time. * can build binary tree

31 Planar Point Location (continued) Lemma allows us to incrementally build binary search tree. 3 types of nodes: 1. internal X nodes, which branch left or right of a segment s i 2. internal Y nodes, which branch above or below a segment endpoint 3. Leaf trapezoid nodes. Identify each trapezoid cut by s 1. B C

32 Planar Point Location (continued)

33 Planar Point Location (continued)

34 Planar Point Location (continued) query point q search path for query point q