Problem: Given a set of n points in Rd, preprocess them such that reporting or counting the k points Y inside a d-dimensional axis-parallel box will be most efficient. " ! Desired output-sensitive query time complexity – O(k+f(n)) for reporting and O(f(n)) for counting, where f(n)=o (n), e.g. f(n)=logn." X ! Sample application: Report all cities within 100 mile radius of Boston." 15. 25. ! In a one-dimensional world, points are !Range tree solution:" real numbers and the query is two " ! Sort points." numbers (a,b)." " ! Construct a binary balanced tree, storing the points in its -4 -2 0 1 3 5 7 11 ! Simple O(logn) algorithm:" " ! Preprocessing: Sort points in O(nlogn)." a b leaves. " " ! Query: (Binary) search for a and b in list # " ! Each tree node stores the 1 in O(logn).! largest value of its left sub-tree." " " List all values inbetween." -2 5 ! Cannot be easily generalized to higher dimensions (why not ?)." 4- 0 3 7 -4 -2 0 1 3 5 7 11 35. 45. Input Range: 3.5-8.2 !Build a data structure storing a “small” number of ! Required time for finding a leaf: O(log canonical subsets, such that:" n)." 1 ! Find the two boundaries of the given " ! The c.s. may overlap." range in the leaves u and v. " " ! Every query may be answered as the union of a “small” V-split ! Report all the leaves in maximal -2 5 number of c.s." subtrees between u and v." !The geometry of the space enables this." ! Mark the vertex at which the search -4 0 3 7 paths diverge as V-split. " !Two extremes:" " ! Singletons – O(k) query time, even for counting." -4 -2 0 1 3 5 7 11 " ! Power set – O(1) query time. O(2n) storage." ! Continue to find the two boundaries, reporting values in the subtrees:" "When going left (right), report the entire right (left) subtree." ! When reaching a leaf, check it exhaustively." 55. 65. 1 ! Given a set of points in 2D." 15 V-split ! Bound the points by a rectangle." ! Split the points into two (almost) equal size groups, using a horizontal or vertical line." 7 canonical subset 24 {9,12,14,15} ! Continue recursively to partition the subsets, until they are small enough." 3 12 20 27 ! Canonical subsets are subtrees." {4,7} {17,20} 1 4 9 14 17 {22} 22 25 29 {3} 1 3 4 7 9 12 14 15 17 20 22 24 25 27 29 31 u v 2 23 canonical subsets are subtrees 75. 85. L6 L4 L1 F L1 L6 A E L4 ! Partitions 2D space into B F ! Each node in the tree defines axis-aligned rectangular H L A E G 3 regions." L2 C an axis-parallel rectangle in the B D plane, bounded by the lines ! Nodes represent partition H L3 marked by this vertex$s L G lines, and leaves represent 2 C ancestors." L7 input points." L5 D L ! Label each node with the 7 L5 L 8 1 number of points in that# L1 rectangle. " construction complexity: L2 L3 4 4 L2 L3 L4 L5 L6 L7 2 2 2 2 L4 L5 L6 L7 A B C D E F G H A B C D E F G H 95. 105. L1 L6 ! Given an axis-parallel L4 F ! k nodes are reported. How much time is spent on internal nodes? range query R, search The nodes visited are those that are stabbed by R but not R A E for this range in the tree. " B contained in R. How many such nodes are there ?" ! Traverse only subtrees L3 ! Theorem: Every side of R stabs O(√n) cells of the tree." L C G H which represent regions 9 2 LL11 ! Proof: Extend the side to a full line (WLOG - horizontal): " intersecting R." L8 D " ! In the first level it stabs two children." ! If a subtree is contained L5 I L " ! In the next level it stabs (only) two of the four grandchildren. " entirely in R:" 4 5 7 L2 L3 " ! Thus, the recursive equation is:" " ! Counting: Add its count." " ! Reporting: 2 2 2 3 LL44 LL55 L6 L7 Report entire ! Total query time: O(√n + k)." subtree." 2 A B C D E F G L8 H I 115. 125. 2 !For a d-dimensional space:" " ! Construction time: O(d nlogn)." " ! Space Complexity: O(dn)." " ! Query time complexity: O(dn1-1/d+k)." Question: Are kd trees useful for non-orthogonal range queries, e.g. disks, convex polygons ? Fact: Using interval trees, orthogonal range queries may be solved in O(logd-1n+k) time and O(nlogd-1n) space. 135. 3 .