算法设计与分析基础第三版PPTch06.pptVIP

Design and Analysis of Algorithms - Chapter 6 Transform and Conquer This group of techniques solves a problem by a transformation to a simpler/more convenient instance of the same problem (instance simplification)  a different representation of the same instance (representation change) a different problem for which an algorithm is already available (problem reduction)  Instance simplification - Presorting Solve a problem’s instance by transforming it into another simpler/easier instance of the same problem Presorting Many problems involving lists are easier when list is sorted, e.g. searching computing the median (selection problem) checking if all elements are distinct (element uniqueness) Also: Topological sorting helps solving some problems for dags. Presorting is used in many geometric algorithms. How fast can we sort ? Efficiency of algorithms involving sorting depends on efficiency of sorting. Theorem (see Sec. 11.2): ?log2 n!? ? n log2 n comparisons are necessary in the worst case to sort a list of size n by any comparison-based algorithm. Note: About nlog2 n comparisons are also sufficient to sort array of size n (by mergesort). Searching with presorting Problem: Search for a given K in A[0..n-1] Presorting-based algorithm: Stage 1 Sort the array by an efficient sorting algorithm Stage 2 Apply binary search Efficiency: Θ(nlog n) + O(log n) = Θ(nlog n) Good or bad? Why do we have our dictionaries, telephone directories, etc. sorted? Element Uniqueness with presorting Presorting-based algorithm Stage 1: sort by efficient sorting algorithm (e.g. mergesort) Stage 2: scan array to check pairs of adjacent elements Efficiency: Θ(nlog n) + O(n) = Θ(nlog n)  Brute force algorithm  Compare all pairs of elements Efficiency: O(n2) Another algorithm? Hashing Instance simplification – Gaussian Elimination Given: A system of n linear equations in n unknowns with an arbitrary coefficient matrix.Transform to: An equivalent system