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Selected Algorithms and Data Structures (Augmenting Data Structures) Prof. Th. Ottmann Lecture 9. Page 1 Augmentation Process Augmentation is a process of extending a data structure in order to support additional functionality. It consists of four steps: 1. Choose an underlying data structure. 2. Determine the additional information to be maintained in the underlying data structure. 3. Verify that the additional information can be maintained for the basic modifying operations on the underlying data structure. 4. Develop new operations. Lecture 9. Page 2 Examples for Augmenting DS • Dynamic order statistics: Augmenting binary search trees by size information • D-dimensional range trees: Recursive construction of (static) d-dim range trees • Min-augmented dynamic range trees: Augmenting 1-dim range trees by mininformation • Interval trees • Priority search trees Lecture 9. Page 3 Examples for Augmenting DS • Dynamic order statistics: Augmenting binary search trees by size information • D-dimensional range trees: Recursive construction of (static) d-dim range trees • Min-augmented dynamic range trees: Augmenting 1-dim range trees by mininformation • Interval trees • Priority search trees Lecture 9. Page 4 Dynamic Order Statistics • Problem: Given a set S of numbers that changes under insertions and deletions, construct a data structure to store S that can be updated in O(log n) time and that can report the k-th order statistic for any k in O(log n) time. S 5 51 13 48 22 85 34 7 14 Lecture 9. Page 5 Binary Search Trees and Order Statistics 17 7 5 1 19 13 18 37 25 49 33 Lecture 9. Page 6 Binary Search Trees and Order Statistics 17 7 5 1 • • Retrieving an element with a given rank: For a given i, find the i-th smallest key in the set. • • Determining the rank of an element: For a given (pointer to a) key k, determine the rank of k in the set of keys. 19 13 18 37 25 49 33 Lecture 9. Page 7 Augmenting the Data Structure • 33 11 Every node v stores two pieces of information: 17 4 Its key The number of its descendants (The size of the subtree with root v) 9 2 4 1 51 6 21 1 48 1 92 4 73 2 124 1 81 1 Lecture 9. Page 8 How To Determine The Rank of an Element • Find the rank of key x in the tree with root node v: • Rank(v, x) • 1 if x = key(v) • 2 then return 1 + size(left(v)) • 3 if x < key(v) • 4 then return Rank(left(v), x) • 5 else return 1 + size(left(v)) + Rank(right(v), x) 33 11 51 6 17 4 9 2 4 1 21 1 48 1 92 4 73 2 124 1 81 1 Lecture 9. Page 9 How to Find the k-th Order Statistic • Find (a pointer to) the node containing the • k-th smallest key in the subtree rooted • at node v. • 33 11 51 6 17 4 9 2 Select(v, k) 1 if k = size(left(v)) + 1 4 1 2 then return v 3 if k ≤ size(left(v)) 4 then return Select(left(v), k) 5 else return Select(right(v), k – 1 – size(left(v))) 21 1 48 1 92 4 73 2 124 1 81 1 Lecture 9. Page 10 Maintaining Subtree Sizes Under Insertions 33 11 17 4 9 2 4 1 51 6 21 1 48 1 92 4 73 2 • Insert operation • Insert node as into a standard binary search tree. • Add 1 to the subtree size of every ancestor of the new node. 124 1 81 1 Lecture 9. Page 11 Maintaining Subtree Sizes Under Insertions 33 11 17 4 9 2 51 6 21 1 48 1 4 1 92 4 73 2 64 1 • Insert operation • Insert node as into a standard binary search tree • Add 1 to the subtree size of every ancestor of the new node 124 1 81 1 Lecture 9. Page 12 Maintaining Subtree Sizes Under Insertions 33 12 17 4 9 2 51 7 21 1 48 1 4 1 92 5 73 3 64 1 • Insert operation • Insert node as into a standard binary search tree • Add 1 to the subtree size of every ancestor of the new node 124 1 81 1 Lecture 9. Page 13 Maintaining Subtree Sizes Under Deletions • Delete operation • Delete node as from a standard binary search tree • Subtract 1 from the subtree size of every ancestor of the deleted node Lecture 9. Page 14 Maintaining Subtree Sizes Under Rotations s1 s1 s3 s2 s4 s5 s5 + s3 + 1 s4 s5 s3 Lecture 9. Page 15 Dynamic Order Statistics—Summary • Theorem: There exists a data structure to represent a dynamically changing set S of numbers with the following properties: • The data structure can be updated in O(log n) time after every insertion or deletion into or from S. • The data structure allows us to determine the rank of an element or to find the element with a given rank in O(log n) time. • The data structure occupies O(n) space. Lecture 9. Page 16 Examples for Augmenting DS • Dynamic order statistics: Augmenting binary search trees by size information • D-dimensional range trees: Recursive construction of (static) d-dim range trees • Min-augmented dynamic range trees: Augmenting 1-dim range trees by mininformation • Interval trees • Priority search trees Lecture 9. Page 17 4-Sided Range Queries Lecture 9. Page 18 4-Sided Range Queries Goal: Build a static data structure of size O(n log n) that can answer 4-sided range queries in O(log2 n + k) time. Lecture 9. Page 19 Orthogonal d-dimensional Range Search Build a static data structure for a set P of n points in d-space that supports d-dim range queries: d-dim range query: Let R be a d-dim orthogonal hyperrectangle, given by d ranges [x1, x1‘], …, [xd, xd‘]: Find all points p = (p1, …, pd) P such that x1 ≤ p1≤ x1‘,…,xd ≤ pd ≤ xd. Special cases: 1-dim range query: 2-dim range query: x2‘ x1 x1‘ x2 x1 x1‘ Lecture 9. Page 20 1-dim Range Search Standard binary search trees support also 1-dim range queries: 68 37 99 81 55 18 23 12 42 21 61 74 90 49 30 80 Lecture 9. Page 21 1-dim Range Search Leaf-search-tree: 68 37 99 55 18 23 12 12 18 42 37 61 21 21 90 74 61 42 ∞ 81 68 74 81 90 99 49 23 49 30 55 80 Lecture 9. Page 22 1-dim Range Tree A 1-dim range tree is a leaf-search tree for the x-values (points on the line). Internal nodes have routers guiding the search to the leaves: We choose the maximal x-value in left subtree as router. Range search: In order to find all points in a given range [l, r] search for the boundary values l and r. This is a forked path; report all leaves of subtrees rooted at nodes v in between the two search paths whose parents are on the search path. Lecture 9. Page 23 The selected subtrees Split node l r Lecture 9. Page 24 Canonical Subsets The canonical subset of node v, P(v), is the subset of points of P stored at the leaves of the subtree rooted at v. If v is a leaf, P(v) is the point stored at this leaf. If v is the root, P(v) = P. Observations: For each query range [l, r] the set of points with x-coordinates falling into this range is the disjoint union of O(log n) canonical subsets of P. A node v is called an umbrella node for the range [l, r], if the x-coordinates of all points in its canonical subset P(v) fall into the range, but this does not hold for the predecessor of v. All k points stored at the leaves of a tree rooted at node v, i.e. the k points in a canonical subset P(v), can be reported in time O(k). Lecture 9. Page 25 1-dim Range Tree: Summary Let P be a set of n points in 1-dim space. P can be stored in a balanced binary leaf-search tree such that the following holds: Construction time: O(n log n) Space requirement: O(n) Insertion of a point: O(log n) time Deletion of a point: O(log n) time 1-dim-range-query: Reporting all k points falling into a given query range can be carried out in time O(log n + k). The performance of 1-dim range trees does not depend on the chosen balancing scheme! Lecture 9. Page 26 2-dim Range tree: The Primary Structure • Static binary leaf-search tree over x-coordinates of points. Lecture 9. Page 27 The Primary Structure • Static binary leaf-search tree over x-coordinates of points. Lecture 9. Page 28 The Primary Structure • Static binary leaf-search tree over x-coordinates of points. Lecture 9. Page 29 The Primary Structure • Static binary leaf-search tree over x-coordinates of points. • Every leaf represents a vertical slab of the plane. Lecture 9. Page 30 The Primary Structure • Static binary leaf-search tree over x-coordinates of points. • Every leaf represents a vertical slab of the plane. • Every internal node represents a slab that is the union of the slabs of its children. Lecture 9. Page 31 The Primary Structure • Static binary leaf-search tree over x-coordinates of points. • Every leaf represents a vertical slab of the plane. • Every internal node represents a slab that is the union of the slabs of its children. Lecture 9. Page 32 The Primary Structure • Static binary leaf-search tree over x-coordinates of points. • Every leaf represents a vertical slab of the plane. • Every internal node represents a slab that is the union of the slabs of its children. Lecture 9. Page 33 The Primary Structure • Static binary leaf-search tree over x-coordinates of points. • Every leaf represents a vertical slab of the plane. • Every internal node represents a slab that is the union of the slabs of its children. Lecture 9. Page 34 Answering 2-dim Range Queries • Normalize queries to end on slab boundaries. • Query decomposes into O(log n) subqueries. • Every subquery is a 1-dimensional range query on ycoordinates of all points in the slab of the corresponding node. (x-coordinates do not matter!) Lecture 9. Page 35 The selected subtrees Split node l r Lecture 9. Page 36 Answering Queries • Normalize queries to end on slab boundaries. • Query decomposes into O(log n) subqueries. • Every subquery is a 1-dimensional range query on ycoordinates of all points in the slab of the corresponding node. (x-coordinates do not matter!) Lecture 9. Page 37 Answering Queries • Normalize queries to end on slab boundaries. • Query decomposes into O(lg n) subqueries. • Every subquery is a 1-dimensional range query on ycoordinates of all points in the slab of the corresponding node. (x-coordinates do not matter!) Lecture 9. Page 38 Answering Queries • Normalize queries to end on slab boundaries. • Query decomposes into O(log n) subqueries. • Every subquery is a 1-dimensional range query on ycoordinates of all points in the slab of the corresponding node. (x-coordinates do not matter!) Lecture 9. Page 39 2-dim Range Tree y Ty(v) x Ix(v) v Tx Lecture 9. Page 40 2-dim Range Tree A 2-dimensional range tree for storing a set P of n points in the x-y-plane is: • A 1-dim-range tree Tx for the x-coordinates of points. • Each node v of Tx has a pointer to a 1-dim-range-tree Ty(v) storing all points which fall into the interval Ix(v). That is: Ty(v) is a 1-dim-range-tree based on the ycoordinates of all points p P with p Ix(v). Leaf-search-tree on y-coordinates of poins v Leaf-search-tree on x-coordinates of points Lecture 9. Page 41 2-dim Range Tree A 2-dim range tree on a set of n points in the plane requires O(n log n) space. A point p is stored in all associated range trees Ty(v) for all nodes v on the search path to px in Tx. p Hence, for each depth d, each point p occurs in only one associated search structure Ty(v) for a node v of depth d in Tx. The 2-dim range tree can be constructed in time O(n log n). (Presort the points on y-coordinates!) p p p Lecture 9. Page 42 The 2-Dimensional Range Tree • Primary structure: Leaf-search tree on x-coordinates of points • Every node stores a secondary structure: Balanced binary search tree on ycoordinates of points in the node’s slab. Every point is stored in secondary structures of O(log n) nodes. Space: O(n log n) Lecture 9. Page 43 Answering Queries • Every 2-dimensional range query decomposes into O(log n) 1dimensional range queries • Each such query takes O(log n + k′) time • Total query complexity: • O(log2 n + k) Lecture 9. Page 44 2-dim Range Query Let P be a set of points in the plane stored in a 2-dim range tree and let a 2-dim range R defined by the two intervals [x, x‘], [y, y‘] be given. The all k points of P falling into the range R can be reported as follows: 1. Determine the O(log n) umbrella nodes for the range [x, x‘], i.e. determine the canonical subsets of P that together contain exactly the points with x-coordinates in the range [x, x‘]. (This is a 1-dim range query on the x-coordinates.) 2. For each umbrella node v obtained in 1, use the associated 1-dim range tree Ty(v) in order to select the subset P(v) of points with y-coordinates in the range [y, y‘]. (This is a 1-dim range query for each of the O(log n) canonical subsets obtained in 1.) Time to report all k points in the 2-dim range R: O(log2 n + k). Query time can be reduced to O(log n +k) by a technique known as fractional cascading. Lecture 9. Page 45 The 3-Dimensional Range Tree • Primary structure: Search tree on x-coordinates of points • Every node stores a secondary structure: 2-dimensional range tree on points in the node’s slab. Every point is stored in secondary structures of O(log n) nodes. Space: O(n log2 n) Lecture 9. Page 46 Answering Queries • Every 3-dimensional range query decomposes into O(log n) 2dimensional range queries • Each such query takes O(log2 n + k′) time • Total query complexity: • O(log3 n + k) Lecture 9. Page 47 d-Dimensional Range Queries • • • • Primary structure: Search tree on x-coordinates Secondary structures: (d – 1)-dimensional range trees Space requirement: O(n logd – 1 n) Query time: O(n logd – 1 n) Lecture 9. Page 48 Updates are difficult! Insertion or deletion of a point p in a 2-dim range tree requires: 1. Insertion or deletion of p into the primary range tree Tx according to the xcoordinate of p 2. For each node v on the search path to the leaf storing p in Tx, insertion or deletion of p in the associated secondary range tree Ty(v). Maintaining the primary range tree balanced is difficult, except for the case d = 1! Rotations in the primary tree may require to completely rebuild the associated range trees along the search path! Lecture 9. Page 49 Range Trees–Summary • Theorem: There exists a data structure to represent a static set S of n points in d dimensions with the following properties: The data structure allows us to answer range queries in O(logd n + k) time. The data structure occupies O(n logd – 1 n) space. • Note: The query complexity can be reduced to O(logd – 1 n + k), for d ≥ 2, using a very beautiful technique called fractional cascading. Lecture 9. Page 50 Examples for Augmenting DS • Dynamic order statistics: Augmenting binary search trees by size information • D-dimensional range trees: Recursive construction of (static) d-dim range trees • Min-augmented dynamic range trees: Augmenting 1-dim range trees by mininformation • Interval trees • Priority search trees Lecture 9. Page 51 minXinRectangle Queries Problem: Given a set P of points that changes under insertions and deletions, construct a data structure to store P that can be updated in O(log n) time and that can find the point with minimal x-coordinate in a given range below a given threshold in O(log n) time. y0 minXinRectangle(l, r, y0) l r Assumption: All points have pairwise different x-coordinates Lecture 9. Page 52 minXinRectangle Queries minXinRectangle(l, r, y0) y0 l r Assumption: All points have pairwise different x-coordinates Lecture 9. Page 53 Min-augmented Range Tree Two data structures in one: 11 2 (2, 12) 3 14 3 2 2 4 4 3 (3, 4) (4, 11) 5 Leaf-search tree on x-coordinates of points Min-tournament tree on y-coordinates of points 17 (14, 7) 2 8 15 21 3 2 8 (11, 21) (15, 2) (17, 30) (21, 8) 3 (5, 3) (8, 5) Lecture 9. Page 54 minXinRectangle(l, r, y0) Search for the boundary values l, r. Find the leftmost umbrella node with a min-field ≤ y0. l Split node r Lecture 9. Page 55 minXinRectangle(l, r, y0) Search for the boundary values l, r. Find the leftmost umbrella node with a min-field ≤ y0. Split node Proceed to the left son of the current node, if its min-field is ≤ y0, and to the right son, otherwise. Return the point at the leaf. l minXinRectangle(l, r, y0) can be found in time O(height of tree). r Lecture 9. Page 56 Updates Insert operation Insert node as into a standard binary leaf search tree. Adjust min-fields of every ancestor of the new node by playing a min tournament for each node and its sibling along the search path. Delete operation: Similar Lecture 9. Page 57 Maintaining min-fields under Rotations s1 s1 s3 s2 s4 s5 min{s5, s3} s4 s5 s3 Lecture 9. Page 58 Min-augmented Range Trees–Summary • Theorem: There exists a data structure to represent a dynamic set S of n points in the plane with the following properties: The data structure allows updates and to answer minXinRectangle(l, r, y0) queries in O(log n) time. The data structure occupies O(n) space. • Note: The data structure can be based on an arbitrary scheme of balanced binary leaf search trees. Lecture 9. Page 59