计算机算法导论_第4章.ppt

Introduction to Algorithms 计算机算法导论 20072008年第一学期 Homework Exercises 3.2-4;3.2-7 Problems 3-4 Recurrences nRecurrences nThree methods for solving recurrences nSubstitution method nRecursion-tree method nMaster method Recurrences Example Solving recurrences nThe analysis of merge sort from Lecture 1 required us to solve a recurrence. nRecurrences are like solving integrals, differential equations, etc. nLecture 2: Applications of recurrences to divide-and-conquer algorithms. Technicalities nAssumption of integer arguments of functions nBoundary conditions nFloors and ceils 4.1 The substitution method Substitution method The most general method: 1. Guess the form of the solution. 2. Use induction to find the constants and show that the solution works. Substitution method nGuess:T(n)=nlgn+n Substitution method Making a good guess T(n)=2T(n/2+17)+n nSimilar solution is reasonable nProve loose upper and lower bounds and reduce the range of uncertainty Subtleties T(n)=2T(n/2)+1 Avoiding pitfalls T(n) 2c(n/2)+n=cn+n=O(n) Wrong! Changing variables An example An example An example 4.2 The recursion-tree method An example T(n)=T(n/3)+T(2n/3)+(n) An example T(n)=T(n/3)+T(2n/3)+(n) An example T(n)=T(n/3)+T(2n/3)+(n) An example T(n)=T(n/3)+T(2n/3)+(n) 4.3 The master metho