第05章_基本技术Fundamental_Techniques.pptVIP

Chapter 5: Techniques Merge Sort Fundamental Techniques Outline and Reading The Greedy Method Technique (§5.1) Fractional Knapsack Problem (§5.1.1) Task Scheduling (§5.1.2) Divide-and-conquer paradigm (§5.2) Recurrence Equations (§5.2.1) Integer Multiplication (§5.2.2) Optional: Matrix Multiplication (§5.2.3) Dynamic Programming (§5.3) Matrix Chain-Product (§5.3.1) The General Technique (§5.3.2) 0-1 Knapsack Problem (§5.3.3) The Greedy Method Technique The greedy method is a general algorithm design paradigm, built on the following elements: configurations: different choices, collections, or values to find objective function: a score assigned to configurations, which we want to either maximize or minimize It works best when applied to problems with the greedy-choice property: a globally-optimal solution can always be found by a series of local improvements from a starting configuration. Making Change Problem: A dollar amount to reach and a collection of coin amounts to use to get there. Configuration: A dollar amount yet to return to a customer plus the coins already returned Objective function: Minimize number of coins returned. Greedy solution: Always return the largest coin you can Example 1: Coins are valued $.32, $.08, $.01 Has the greedy-choice property, since no amount over $.32 can be made with a minimum number of coins by omitting a $.32 coin (similarly for amounts over $.08, but under $.32). Example 2: Coins are valued $.30, $.20, $.05, $.01 Does not have greedy-choice property, since $.40 is best made with two $.20’s, but the greedy solution will pick three coins (which ones?) The Fractional Knapsack Problem Given: A set S of n items, with each item i having bi - a positive benefit wi - a positive weight Goal: Choose items with maximum total benefit but with weight at most W. If we are allowed to take fractional amounts, then this is the fractional knapsack problem. In this case, we let xi denote the amount we take of item i Objective: maximize C