Geometric Data Structures Reading: Chapter 10 of the Textbook Driving Applications Windowing Queries Related Application Query Image Database UNC Chapel Hill M. C. Lin

Windowing Queries Given a rectangular region (2D or 3D), or a window, the algorithm must determine the part of the database that lie in the window and report them. Data are normally not points, but line segments, polygons, etc. 2D Windowing: navigation using a map or an electronic GIS; circuit-board layout, etc. 3D Windowing: view frustrum culling; viewing a complex CAD/CAM model by zooming in to a small portion, etc. UNC Chapel Hill M. C. Lin

Orthogonal Window Queries Let S be a set of n axis-parallel line segments. A query asks for the segments intersecting a 2D query window, W := [x:x’] x [y:y’] Most cases, the segment has at least one point inside of W. We can find such segments by performing a range query with W in the set of 2n endpoints of the segments in S, by using a 2D range tree T. 2D range tree can answer a range query in O(log2n + k) time; query time can be improved to O(logn + k) by fractional cascading. Find segments that intersect W twice either at the left and right edges or top and bottom edges. This reduces to finding horiz. (v) segments intersecting a vert. (h) line. UNC Chapel Hill M. C. Lin

Classification of Segments w.r.t. xmid An interval [x:x’] contains l := (x=qx) iff x  qx  x’ We can classify a set of segments w.r.t to xmid : Imid: those containing xmid Ileft: those complete lie to the left & cannot intersect l Iright: those complete lie to the right & cannot intersect l Check Imid to find intersecting segments with l Store the intervals in 2 sorted lists: increasing left endpoints and decreasing right endpoints qx can be contained in an interval if it’s also contained in all its predecessors in the sorted list We can simply walk along sorted lists reporting intervals & stop when we encounter I s.t. qx  I UNC Chapel Hill M. C. Lin

Interval Trees If I = 0, then the interval tree is a leaf Otherwise, let xmid be the median of the endpoints of the intervals, let Ileft := { [xj:xj’]  I : xj’ < xmid }, Imid := { [xj:xj’]  I : xj  xmid  xj’ }, Iright := { [xj:xj’]  I : xmid < xj }. The interval trees consists of a root node v storing xmid. The set Imid is stored twice: once in a list Lleft(v) that is stored on the left endpoints of the intervals, once in a list Lright(v) that is stored on the right endpoints of intervals The left subtree of v is an interval tree for Ileft The right subtree of v is an interval tree for Iright UNC Chapel Hill M. C. Lin

ConstructIntervalTree(I) Input: A set I of intervals on the real line. Output: The root of an interval tree for I . 1. if I = 0 2. then return an empty leave 3. else Create a node v. Compute xmid, the median of the set of interval endpoints, and store xmid with v. 4. Compute Imid and construct two sorted lists for Imid: a list Lleft(v) sorted on left endpoint & a list Lright (v) sorted on right endpoint. Store these two lists at v. 5. lc(v)  ConstructIntervalTree(Ileft) 6. rc(v)  ConstructIntervalTree(Iright) 7. return v UNC Chapel Hill M. C. Lin

QueryIntervalTree(v, qx) Input: The root v of an interval tree and a query point qx Output: All intervals that contain qx 1. if v is not a leaf 2. then if qx < xmid(v) 3. then Walk along the list Lleft(v), starting at the interval with the leftmost endpoint, reporting all the intervals that contain qx. Stop as soon as an interval does not contain qx. 4. QueryIntervalTree(lc(v), qx) 5. else Walk along the list Lright(v), starting at the interval with the rightmost endpoint, reporting 6. QueryIntervalTree(rc(v), qx) UNC Chapel Hill M. C. Lin

Algorithm Analysis An interval tree for a set I of n intervals uses O(n) storage, has depth O(logn) and can be built in O(n log n) time. Using the interval tree we can report all intervals containing a query point in O(k + log n) time, where k is no. of reported intervals. Let S be a set of n horizontal segments in the plane. The segments intersecting a vertical query segment can be reported in O(log2n + k) time with a data structure of O(n log n) storage, where k is the number of reported segments. The structure can be built in O(n log n) time. Let S be a set of n axis-parallel segments in the plane. The segments intersecting an axis-parallel rectangular query window can be reported in O(log2n + k) time with a data structure of O(n log n) storage, where k is number of reported segments. The structure can be built in O(n log n) time. UNC Chapel Hill M. C. Lin

Motivation for Priority Search Trees Replacing RangeTrees: A 2d rectangle query on P asks for points in P lying inside a query window (-:qx] x [qy:qy’]. Use ideas from 1d range query. Use a heap to answer 1d query if px (-:qx] and partition using the y-coordinates. Root of the tree stores the point with minimum x-value and the remainder is partitioned into 2 equal sets. The 2 sets are partitioned into sets above & below Construct the tree recursively UNC Chapel Hill M. C. Lin

Priority Search Trees If P = 0, then the priority search tree is an empty leaf Otherwise, let pmin be the point in P with the smallest x-coordinate and ymid be the y-median of the rest in P. Let Pbelow := { p  P\{pmin} : py < ymid }, Pabove := { p  P\{pmin} : py > ymid }. The priority search trees consists of a root node v where the point p(v) := pmin and the value y(v) := ymin are stored. The left subtree of v is a priority search tree for Pbelow The right subtree of v is a priority search tree for Pabove UNC Chapel Hill M. C. Lin

ReportInSubTree(v, qx) Input:The root v of a subtree of a priority search tree and a value qx Output: All points in the subtree with x-coordinate at most qx 1. if v is not a leaf and (p(v))x  qx 2. then Report p(v) 3. ReportInSubTree(lc(v), qx) 4. ReportInSubTree(rc(v), qx) UNC Chapel Hill M. C. Lin

QueryPrioSearchTree(T,(-:qx]x[qy:qy’]) Input: A priority search tree and a range unbounded to the left Output: All points lying in the range 1. Search with qy and qy’ in T. Let vsplit be node where 2 search paths split 2. for each node v on the search path of qy or qy’ 3. do if p(v)  (-:qx]x[qy:qy’] then report p(v) 4. for each node v on the search path of qy in the left subtree of vsplit 5. do if the search path goes left at v 6. then ReportInSubtree(rc(v), qx) 7. for each node v on the search path of qy’ in the right subtree of vsplit 8. do if the search path goes right at v 9. then ReportInSubtree(lc(v), qx) UNC Chapel Hill M. C. Lin

Algorithm Analysis A priority search tree for a set P of n points in the plane uses O(n) storage & can be built in O(n log n) time. Using the priority search tree we can report all points in a query range of the form (-:qx] x [qy:qy’] in O(k + log n) time, where k is the number of reported points. UNC Chapel Hill M. C. Lin

Segment Tree Data Structures The skeleton of the segment tree is a balanced binary tree T. The leaves of T correspond to the elementary intervals induced by the endpoints of the intervals in I in an ordered way: the leftmost leaf corresponds to the leftmost elementary interval, and so on. The elementary interval corresponding to leaf u is denoted Int(u). The internal nodes of correspond to intervals that are the union of elementary intervals: the interval Int(v) corresponding to node v is the union of the elementary intervals Int(u) of the leaves in the subtree rooted at v. (implying Int(v) is the union of its two children) Each node or leaf v in T stores the interval Int(v) and a set I(v) I of intervals (e.g. in a linked list). This canonical subset of node v contains the intervals [x:x’]I s.t. Int(v)[x:x’] & Int(parent(v))[x:x’] (See the diagrams in class) UNC Chapel Hill M. C. Lin

QuerySegmentTree(v, qx) Input:The root v of a (subtree of a) segment tree and a query point qx Output: All intervals in the tree containing qx 1. Report all the intervals in I(v) 2. if v is not a leaf 3. then if qx  Int(lc(v)) 4. then QuerySegmentTree(lc(v), qx) 5. else QuerySegmentTree(rc(v), qx) UNC Chapel Hill M. C. Lin

Constructing Segment Trees Sort the endpoints on the intervals in I in O(n log n) time. This gives elementary intervals. Construct a balanced binary tree on the elementary intervals & determine for each node v of the tree the interval Int(v) it represents. Compute the canonical subsets by inserting one interval at a time into T. (See the next procedure) UNC Chapel Hill M. C. Lin

InsertSegmentTree(v, [x:x’]) Input: The root of a (subtree of a) segment tree and an interval Output: The interval will be stored in the subtree 1. if Int(v)  [x:x’] 2. then store [x:x’] at v 3. else if Int(lc(v))  [x:x’]  0 4. then InsertSegmentTree(lc(v), qx) 5. if Int(rc(v))  [x:x’]  0 6. then InsertSegmentTree(rc(v), qx) UNC Chapel Hill M. C. Lin

Algorithm Analysis A segment tree for a set I of n intervals uses O(n log n) storage and can be built in O(n log n). Using the segment tree we can report all intervals that contain a query point in O(log n + k) time, where k is the number of reported intervals. Let S be a set of n segments in the plane with disjoint interiors. The segments intersecting an axis-parallel rectangular query window can be reported in O(log2n+k) time with a data structure that uses O(n log n) storage, where k is the number of reported segments. The structure can be built in O(n log n) time. UNC Chapel Hill M. C. Lin