Geometric search: Overview Types of data. Points, lines, planes, polygons, circles, ... Geometric Algorithms This lecture. Sets of N objects. Geometric problems extend to higher dimensions. ! Good algorithms also extend to higher dimensions. ! Curse of dimensionality. Basic problems. ! Range searching. ! Nearest neighbor. ! Finding intersections of geometric objects. Reference: Chapters 26-27, Algorithms in C, 2nd Edition, Robert Sedgewick. Robert Sedgewick and Kevin Wayne • Copyright © 2005 • http://www.Princeton.EDU/~cos226 2 1D Range Search 7.3 Range Searching Extension to symbol-table ADT with comparable keys. ! Insert key-value pair. ! Search for key k. ! How many records have keys between k1 and k2? ! Iterate over all records with keys between k1 and k2. Application: database queries. insert B B insert D B D insert A A B D insert I A B D I Geometric intuition. insert H A B D H I ! Keys are point on a line. insert F A B D F H I ! How many points in a given interval? insert P A B D F H I P count G to K 2 search G to K H I Robert Sedgewick and Kevin Wayne • Copyright © 2005 • http://www.Princeton.EDU/~cos226 4 1D Range Search Implementations 2D Orthogonal Range Search Range search. How many records have keys between k1 and k2? Extension to symbol-table ADT with 2D keys. ! Insert a 2D key. Ordered array. Slow insert, binary search for k1 and k2 to find range. ! Search for a 2D key. Hash table. No reasonable algorithm (key order lost in hash). ! Range search: find all keys that lie in a 2D range? ! Range count: how many keys lie in a 2D range? BST. In each node x, maintain number of nodes in tree rooted at x. Search for smallest element ! k1 and largest element " k2. Applications: networking, circuit design, databases. Geometric interpretation. insert count range ! Keys are point in the plane. ordered array N log N R + log N ! Find all points in a given h-v rectangle? hash table 1 N N BST log N log N R + log N nodes examined N = # records within interval R = # records that match not touched 5 6 2D Orhtogonal Range Search: Grid Implementation 2D Orthogonal Range Search: Grid Implementation Costs Grid implementation. [Sedgewick 3.18] Space-time tradeoff. 2 ! Divide space into M-by-M grid of squares. ! Space: M + N. 2 ! Create linked list for each square. ! Time: 1 + N / M per grid cell examined on average. ! Use 2D array to directly access relevant square. ! Insert: insert (x, y) into corresponding grid square. Choose grid square size to tune performance. ! Range search: examine only those grid squares that could have ! Too small: wastes space. points in the rectangle. ! Too large: too many points per grid square. ! Rule of thumb: ! N by ! N grid. Time costs. [if points are evenly distributed] RT RT ! Initialize: O(N). ! Insert: O(1). ! Range: O(1) per point in range. LB LB 7 8 Clustering Space Partitioning Trees Grid implementation. Fast, simple solution for well-distributed points. Space partitioning tree. Use a tree to represent the recursive Problem. Clustering is a well-known phenomenon in geometric data. hierarchical subdivision of d-dimensional space. BSP tree. Recursively divide space into two regions. Quadtree. Recursively divide plane into four quadrants. Octree. Recursively divide 3D space into eight octants. kD tree. Recursively divide k-dimensional space into two half-spaces. Applications. Ex: USA map data. ! Ray tracing. ! Flight simulators. ! 80,000 points, 20,000 grid squares. ! N-body simulation. ! Half the grid squares are empty. Grid kD tree ! Collision detection. ! Half the points have ! 10 others in same grid square. ! Astronomical databases. ! Ten percent have ! 100 others in same grid square. ! Adaptive mesh generation. ! Accelerate rendering in Doom. Need data structure that gracefully adapts to data. ! Hidden surface removal and shadow casting. Quadtree BSP tree 9 10 Quad Trees Curse of Dimensionality Quad tree. Recursively partition plane into 4 quadrants. Range search / nearest neighbor in k dimensions? Main application. Multi-dimensional databases. Implementation: 4-way tree. public class QuadTree { Quad quad; 3D space. Octrees: recursively divide 3D space into 8 octants. Value value; QuadTree NW, NE, SW, SE; 100D space. Centrees: recursively divide into 2100 centrants??? } b c a d a h d e f g e f b c g h Good clustering performance is a primary reason to choose quad trees over grid methods. Raytracing with octrees http://graphics.cs.ucdavis.edu/~gregorsk/graphics/275.html 11 12 2D Trees kD Trees 2D tree. Recursively partition plane into 2 halfplanes. kD tree. Recursively partition k-dimensional space into 2 halfspaces. Implementation: BST, but alternate using x and y coordinates as key. Implementation: BST, but cycle through dimensions ala 2D trees. ! Search gives rectangle containing point. ! Insert further subdivides the plane. p level ! i (mod k) p points points whose ith whose ith coordinate coordinate is less than p’s is greater than p’s points points left of p right of p even levels Efficient, simple data structure for processing k-dimensional data. q ! Adapts well to clustered data. ! Adapts well to high dimensional data. points points ! Discovered by an undergrad in an algorithms class! below q above q odd levels 13 14 Summary Basis of many geometric algorithms: search in a planar subdivision. 7.4 Geometric Intersection Voronoi intersecting grid 2D tree diagram lines basis #N h-v lines N points N points #N lines 2D array N-node representation N-node BST ~N-node BST of N lists multilist cells ~N squares N rectangles N polygons ~N triangles search cost 1 log N log N log N cells too use (k-1)D extend to kD? too many cells easy complicated hyperplane 15 Robert Sedgewick and Kevin Wayne • Copyright © 2005 • http://www.Princeton.EDU/~cos226 Geometric Intersection Orthogonal Segment Intersection: Sweep Line Algorithm Problem. Find all intersecting pairs among set of N geometric objects. Sweep vertical line from left to right. Applications. CAD, games, movies, virtual reality. ! Event times: x-coordinates of h-v line segments. ! Left endpoint of h-segment: insert y coordinate into ST. ! Right endpoint of h-segment: remove y coordinate from ST. Simple version: 2D, all objects are horizontal or vertical line segments. ! v-segment: range search for interval of y endpoints. Brute force. Test all $(N2) pairs of line segments for intersection. Sweep line. Efficient solution extends to 3D and general objects. insert y range search delete y 17 18 Orthogonal Segment Intersection: Sweep Line Algorithm Immutable H-V Segment ADT Sweep line: reduces 2D orthogonal segment intersection problem to 1D range searching! public final class SegmentHV implements Comparable<SegmentHV> { Running time of sweep line algorithm. public final int x1, y1; public final int x2, y2; ! Put x-coordinates on a PQ (or sort). O(N log N) N = # line segments ! Insert y-coordinate into ST. O(N log N) R = # intersections public SegmentHV(int x1, int y1, int x2, int y2) public boolean isHorizontal() { … } ! Delete y-coordinate from ST. O(N log N) public boolean isVertical() { … } ! Range search. O(R + N log N) public int compareTo(SegmentHV b) { … } public String toString() { … } } compare by y-coordinate; break ties by x-coordinate Efficiency relies on judicious use of data structures. (x1, y2) (x1, y1) (x2, y1) (x1, y1) horizontal segment vertical segment 19 20 Sweep Line Event Sweep Line Algorithm: Initialize Events // initialize events public class Event implements Comparable<Event> { MinPQ<Event> pq = new MinPQ<Event>(); int time; for (int i = 0; i < N; i++) { SegmentHV segment; if (segments[i].isVertical()) { Event e = new Event(segments[i].x1, segments[i]); public Event(int time, SegmentHV segment) { pq.insert(e); this.time = time; } this.segment = segment; else if (segments[i].isHorizontal()) { } Event e1 = new Event(segments[i].x1, segments[i]); Event e2 = new Event(segments[i].x2, segments[i]); public int compareTo(Event b) { pq.insert(e1); return a.time - b.time; pq.insert(e2); } } } } 21 22 Sweep Line Algorithm: Simulate the Sweep Line General Line Segment Intersection // simulate the sweep line Use horizontal sweep line moving from left to right. int INF = Integer.MAX_VALUE; ! Maintain order of segments that intersect sweep line by y-coordinate. RangeSearch<SegmentHV> st = new RangeSearch<SegmentHV>(); ! Intersections can only occur between adjacent segments. while (!pq.isEmpty()) { ! Add/delete line segment % one new pair of adjacent segments. Event e = pq.delMin(); int sweep = e.time; ! Intersection % two new pairs of adjacent segments. SegmentHV segment = e.segment; if (segment.isVertical()) { SegmentHV seg1, seg2; seg1 = new SegmentHV(-INF, segment.y1, -INF, segment.y1); seg2 = new SegmentHV(+INF, segment.y2, +INF, segment.y2); A Iterable<SegmentHV> list = st.range(seg1, seg2); B for (SegmentHV seg : list) System.out.println(segment + " intersects " + seg); } C else if (sweep == segment.x1) st.add(segment); insert segment else if (sweep == segment.x2) st.remove(segment); D delete segment } A AB ABC ACB ACBD ACD CAD CA A intersection order of segments 23 24 Line Segment Intersection: Implementation Efficient implementation of sweep line algorithm. VLSI Rules Checking ! Maintain PQ of important x-coordinates: endpoints and intersections. ! Maintain ST of segments intersecting sweep line, sorted by y. ! O(R log N + N log N). Implementation issues. ! Degeneracy. ! Floating point precision. ! Use PQ since intersection events aren't known ahead of time. 25 Robert Sedgewick and Kevin Wayne • Copyright © 2005 • http://www.Princeton.EDU/~cos226 Algorithms and Moore's Law Algorithms and Moore's Law Rectangle intersection. Find all intersections among h-v rectangles. "Moore’s Law." Processing power doubles every 18 months. Application. VLSI rules checking in microprocessor design. ! 197x: need to check N rectangles. ! 197(x+1.5): need to check 2N rectangles on a 2x-faster computer.