1D Range Searching (cont) 5.1: 1D Range Searching. Database queries

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1 kd!trees SMD156 Lecture 5 Orthogonal Range Searching Database queries Divide!and!conquer Orthogonal Range Searching Range Trees Searching for items using many ranges of criteria at a time SMD156 Computational Geometry - Håkan Jonsson 1 SMD156 Computational Geometry - Håkan Jonsson 2 5.1: 1D Range Searching 1D Range Searching (cont) We are given a set of P containing n numbers and asked to set up a data structure to answer queries in the form: " Which points lie in the interval #xmin,xmax$? Use a balanced binary search tree. The query time will be output!sensitive. " Let k be the number of numbers in the answer. SMD156 Computational Geometry - Håkan Jonsson 3 SMD156 Computational Geometry - Håkan Jonsson 4

2 1D Range Searching (cont) Takes O%n log n& time and O%n& space to build the tree %just insert the numbers one at a time&. Query time: O%k + log n& " Since the tree is balanced there are O%log n& nodes in a path from the root to a leaf. 5.2: 2D Range Searching In 2D the input is a set of n points in the plane and the query is a rectangl!. 'kd!trees' SMD156 Computational Geometry - Håkan Jonsson 5 SMD156 Computational Geometry - Håkan Jonsson 6 2D Range Searching 2D Range Searching (cont) Build a tree that corresponds to the recursive partitioning of the plane into rectangles by cutting along axis!aligned lines through input points. Divide!and!conquer and recursion: " First split in half on x!coordinate, then split %the 2 halves& by y!coordinate, then split %the 4 regions& on x!coordinate, then split %the 8 regions& on y!coordinate and so on. " A node corresponds to a dividing line. " A leaf corresponds to an input point. " The point on a line belongs to the rectangle to the left / below. SMD156 Computational Geometry - Håkan Jonsson 7 SMD156 Computational Geometry - Håkan Jonsson 8

3 2D Range Searching (cont) Performing a query SMD156 Computational Geometry - Håkan Jonsson 9 SMD156 Computational Geometry - Håkan Jonsson 10 Performing a query (cont) 2D Range Searching (cont) The median of n %orderable& items can be found in O%n& time. The algorithm that does this is fast but complicated. " Simpler: Sort all points into two arrays by x! and y!coordinate. Access a median by computing its index. No need to pre!compute all regions; compute them during the traversal of the tree. Lemma 5.3: A kd!tree for n points in the plane can be built in O%n log n& time and O%n& space. SMD156 Computational Geometry - Håkan Jonsson 11 SMD156 Computational Geometry - Håkan Jonsson 12

4 2D Range Searching (cont) kd Range Searching The query time is O%k + ) n &. " We only need to visit nodes corresponding to rectangles that the query rectangle intersects. Let Q%n& be the number of rectangles intersected by a vertical line. Then, Q%1& = O%1& O%n& = 2O%n/4& + 2 In d!dimensions: Use the same approach, split along the 1 st coordinate, 2 nd coordinate,, d th coordinate, then the 1 st coordinate, 2 nd coordinate,, d th coordinate, etc. Build takes: O%n log n& time using O%n& space. Query!time: O%k + n 1!1/d &. " As d increases, the query!time gets closer to linear. Solves to O%) n &. SMD156 Computational Geometry - Håkan Jonsson 13 SMD156 Computational Geometry - Håkan Jonsson : Range Trees Faster query!time but requires more space. Same main idea but use two levels of data structures: 'One in x, one in y'. Range Trees For each node in the (rst!level tree we store a pointer to a second!level tree. " The (rst!level is a 1D range tree by x!coordinate. " The second!level tree is a 1D range tree by y!coordinate. Each point ends up in O%log n& trees. SMD156 Computational Geometry - Håkan Jonsson 15 SMD156 Computational Geometry - Håkan Jonsson 16

5 Complexity Build: " O%n log n& time by pre!sorting the points also by y!coordinate. " O%n log n& space. Each point ends up in O%log n& second!level trees, and there are n points. Query: " We perform one 1D range search in each second!level tree pointed at by each of the O%log n& (rst!level nodes we encounter. " O% k + %log n&*%log n& & = O%k+log 2 n& time. 5.4 Higher-Dimensional Range Trees Just add extra levels. " A third!level range tree in each of the nodes in the second!level tree, " A fourth!level range tree in each of the nodes of a third! level range tree, and so on.. SMD156 Computational Geometry - Håkan Jonsson 17 SMD156 Computational Geometry - Håkan Jonsson 18 Range search in higher dimensions For points in d!dimensions, a range tree can be constructed in O%n log d!1 n& time and space with query!time O%k + log d n&. 5.5 General Sets of Points Points with equal x! or y!coordinate usually pose problems. Use composite numbers: %p x,p y &!> % %p x *p y &, %p y *p x & & SMD156 Computational Geometry - Håkan Jonsson 19 We could for instance think of this as the point %2.34, 1.20& becoming % , &. Possible since our numbers have (nite precision. They are not real numbers. It takes!%n log n& time to determine that each number in a set of size n is distinct. %This problem is called 'element uniqueness'.& SMD156 Computational Geometry - Håkan Jonsson 20