Dynamic Order Statistics CS 332: Algorithms Dynamic Order Statistics David Luebke 1 8/26/2018

Review: Choosing A Hash Function Choosing the hash function well is crucial Bad hash function puts all elements in same slot A good hash function: Should distribute keys uniformly into slots Should not depend on patterns in the data We discussed three methods: Division method Multiplication method Universal hashing David Luebke 2 8/26/2018

Review: Universal Hashing When attempting to foil an malicious adversary, randomize the algorithm Universal hashing: pick a hash function randomly when the algorithm begins (not upon every insert!) Guarantees good performance on average, no matter what keys adversary chooses Need a family of hash functions to choose from David Luebke 3 8/26/2018

Review: Universal Hashing A family of hash functions  is said to be universal if: With a random hash function from , the chance of a collision between x and y is exactly 1/m (x  y) We can use this to get good expected performance: Choose h from a universal family of hash functions Hash n keys into a table of m slots, n  m Then the expected number of collisions involving a particular key x is less than 1 David Luebke 4 8/26/2018

Review: A Universal Hash Function Choose table size m to be prime Decompose key x into r+1 bytes, so that x = {x0, x1, …, xr} Only requirement is that max value of byte < m Let a = {a0, a1, …, ar} denote a sequence of r+1 elements chosen randomly from {0, 1, …, m - 1} Define corresponding hash function ha  : With this definition,  has mr+1 members David Luebke 5 8/26/2018

Review: A Universal Hash Function  is a universal collection of hash functions (Theorem 12.4) How to use: Pick r based on m and the range of keys in U Pick a hash function by (randomly) picking the a’s Use that hash function on all keys David Luebke 6 8/26/2018

Review: Order Statistic Trees OS Trees augment red-black trees: Associate a size field with each node in the tree x->size records the size of subtree rooted at x, including x itself: M 8 C 5 P 2 Q 1 A 1 F 3 D 1 H 1 David Luebke 7 8/26/2018

How can we use this property to select the ith element of the set? Selection On OS Trees M 8 C 5 P 2 Q 1 A 1 F 3 D 1 H 1 How can we use this property to select the ith element of the set? David Luebke 8 8/26/2018

OS-Select OS-Select(x, i) { r = x->left->size + 1; if (i == r) return x; else if (i < r) return OS-Select(x->left, i); else return OS-Select(x->right, i-r); } David Luebke 9 8/26/2018

OS-Select Example Example: show OS-Select(root, 5): OS-Select(x, i) { r = x->left->size + 1; if (i == r) return x; else if (i < r) return OS-Select(x->left, i); else return OS-Select(x->right, i-r); } M 8 C 5 P 2 Q 1 A 1 F 3 D 1 H 1 David Luebke 10 8/26/2018

OS-Select Example Example: show OS-Select(root, 5): OS-Select(x, i) { r = x->left->size + 1; if (i == r) return x; else if (i < r) return OS-Select(x->left, i); else return OS-Select(x->right, i-r); } M 8 i = 5 r = 6 C 5 P 2 A 1 F 3 Q 1 D 1 H 1 David Luebke 11 8/26/2018

OS-Select Example Example: show OS-Select(root, 5): OS-Select(x, i) { r = x->left->size + 1; if (i == r) return x; else if (i < r) return OS-Select(x->left, i); else return OS-Select(x->right, i-r); } M 8 i = 5 r = 6 C 5 i = 5 r = 2 P 2 A 1 F 3 Q 1 D 1 H 1 David Luebke 12 8/26/2018

OS-Select Example Example: show OS-Select(root, 5): OS-Select(x, i) { r = x->left->size + 1; if (i == r) return x; else if (i < r) return OS-Select(x->left, i); else return OS-Select(x->right, i-r); } M 8 i = 5 r = 6 C 5 i = 5 r = 2 P 2 A 1 F 3 i = 3 r = 2 Q 1 D 1 H 1 David Luebke 13 8/26/2018

OS-Select Example Example: show OS-Select(root, 5): OS-Select(x, i) { r = x->left->size + 1; if (i == r) return x; else if (i < r) return OS-Select(x->left, i); else return OS-Select(x->right, i-r); } M 8 C 5 P 2 Q 1 A 1 F 3 D 1 H 1 i = 5 r = 6 i = 5 r = 2 i = 3 r = 2 i = 1 r = 1 David Luebke 14 8/26/2018

OS-Select: A Subtlety What happens at the leaves? OS-Select(x, i) { r = x->left->size + 1; if (i == r) return x; else if (i < r) return OS-Select(x->left, i); else return OS-Select(x->right, i-r); } What happens at the leaves? How can we deal elegantly with this? Oops… David Luebke 15 8/26/2018

OS-Select What will be the running time? OS-Select(x, i) { r = x->left->size + 1; if (i == r) return x; else if (i < r) return OS-Select(x->left, i); else return OS-Select(x->right, i-r); } What will be the running time? David Luebke 16 8/26/2018

Determining The Rank Of An Element C 5 P 2 Q 1 A 1 F 3 D 1 H 1 What is the rank of this element? David Luebke 17 8/26/2018

Determining The Rank Of An Element C 5 P 2 Q 1 A 1 F 3 D 1 H 1 Of this one? Why? David Luebke 18 8/26/2018

Determining The Rank Of An Element C 5 P 2 Q 1 A 1 F 3 D 1 H 1 Of the root? What’s the pattern here? David Luebke 19 8/26/2018

Determining The Rank Of An Element C 5 P 2 Q 1 A 1 F 3 D 1 H 1 What about the rank of this element? David Luebke 20 8/26/2018

Determining The Rank Of An Element C 5 P 2 Q 1 A 1 F 3 D 1 H 1 This one? What’s the pattern here? David Luebke 21 8/26/2018

OS-Rank What will be the running time? OS-Rank(T, x) { r = x->left->size + 1; y = x; while (y != T->root) if (y == y->p->right) r = r + y->p->left->size + 1; y = y->p; return r; } What will be the running time? David Luebke 22 8/26/2018

OS-Trees: Maintaining Sizes So we’ve shown that with subtree sizes, order statistic operations can be done in O(lg n) time Next step: maintain sizes during Insert() and Delete() operations How would we adjust the size fields during insertion on a plain binary search tree? David Luebke 23 8/26/2018

OS-Trees: Maintaining Sizes So we’ve shown that with subtree sizes, order statistic operations can be done in O(lg n) time Next step: maintain sizes during Insert() and Delete() operations How would we adjust the size fields during insertion on a plain binary search tree? A: increment sizes of nodes traversed during search David Luebke 24 8/26/2018

OS-Trees: Maintaining Sizes So we’ve shown that with subtree sizes, order statistic operations can be done in O(lg n) time Next step: maintain sizes during Insert() and Delete() operations How would we adjust the size fields during insertion on a plain binary search tree? A: increment sizes of nodes traversed during search Why won’t this work on red-black trees? David Luebke 25 8/26/2018

Maintaining Size Through Rotation y 19 x 19 rightRotate(y) x 11 y 12 7 6 leftRotate(x) 6 4 4 7 Salient point: rotation invalidates only x and y Can recalculate their sizes in constant time Why? David Luebke 26 8/26/2018

Augmenting Data Structures: Methodology Choose underlying data structure E.g., red-black trees Determine additional information to maintain E.g., subtree sizes Verify that information can be maintained for operations that modify the structure E.g., Insert(), Delete() (don’t forget rotations!) Develop new operations E.g., OS-Rank(), OS-Select() David Luebke 27 8/26/2018