Goal: Maintain a dynamic subset S of size n of the universe U = {0, 1, …, u – 1} of size u subject to these operations:

Computational geometry Algorithms for solving “geometric problems” in 2D and higher. Fundamental objects: point line segment line Basic structures: point set polygon

Balanced search tree: A search-tree data structure for which a height of O(lg n) is guaranteed when implementing a dynamic set of n items

The depth of a node = the number of comparisons made during TREE-INSERT. Assuming all input permutations are equally likely, we have

Algorithms Day 12 L8.2 A weakness of hashing Problem: For any hash function h, a set of keys exists that can cause the average access time of a hash table to skyrocket. IDEA: Choose the hash function at random, independently of the keys

Symbol-table problem Symbol table T holding n records: key key[x] [x]

Order statistics Select the ith smallest of n elements (the element with rank i)

All the sorting algorithms we have seen so far are comparison sorts: only use comparisons to determine the relative order of elements

Proposed by C.A.R. Hoare in 1962. Divide-and-conquer algorithm. Sorts “in place” (like insertion sort, but notlike merge sort). Very practical (with tuning)

1. Divide the problem (instance) into subproblems. 2. Conquer the subproblems by solving them recursively. 3. Combine subproblem solutions