高等計算機演算法 Advanced Computer Algorithms 伍麗樵(Lih-Chyau Wuu) 雲林科技大學資工所

課程內容: Design Methods Analysis Methods The theory of NP-completeness

Design Methods The Greedy Method The Divide-and-Conquer strategy Prune-and-Search Tree Searching strategy Dynamic Programming Approximation Algorithms Randomized Algorithms On-Line Algorithms

Analysis Methods Complexity of algorithms Best, Average and Worst Case Analyses The Lower Bounds of Problems Amortized Analysis

教科書 Introduction to the Design and Analysis of Algorithms R.C.T Lee R.C. Chang S.S. Tseng Y.T. Tsai Second Edition 旗標代理

分數 期中考 35% 期末考 35% 報告 30%

Algorithm A number of rules, which are to be followed in a prescribed order, for solving a specific type of problems A method that can be used by a computer to solve a problem.

Computer Algorithm – Five Criteria Input Output Finiteness Definiteness Effectiveness

Motivations of studying Algorithms Algorithm is everywhere Use computers efficiently Optimal solution NP-complete problems(Decision problems) NP-hard problems(Optimization problems) next

Use computers efficiently Sorting algorithms 11, 7, 14, 1, 5, 9, 10 ↓sort 1, 5, 7, 9, 10, 11, 14   Insertion sort Quick sort

Comparison with Two Algorithms Implemented on Two Computers

The study of Algorithms: How to design algorithms How to validate algorithms How to analyze algorithms How to test a program

How to design algorithm The study of algorithm design is almost a study of strategies

How to analyze algorithm Measure the goodness of algorithms efficiency?(time/space) asymptotic notations: O( ) worst case average case amortized Measure the difficulty of problems NP-complete undecidable lower bound Is the algorithm optimal?

時間複雜度(time complexity)-worst case executing time O(1):常數時間(constant time) O(n):線性時間(linear time) O(log2n):次線性時間(sub-linear time) O(n2):平方時間(quadratic time) O(n3):立方時間(cubic time) O(2n):指數時間(exponential time)   O(1)<O(log2n)<O(n)<O(nlog2n)<O(n2)<O(n3)<O(2n)

Algorithm vs. Program Algorithm的表示方法 *虛擬碼(Pseudo Code) SPARKS, PASCAL-like, Nature Language *流程圖表示法(Flow-chart representation) A program is the expression of an algorithm in a programming language

Background for Learning Algorithms Clear brain Data structure Discrete mathematics

NP-complete problems:No efficient algorithms Easy problem: polynomial-time algorithm Difficult problem: exponential-time algorithm Garey & Johnson “Computers & Intracability”

N1: Satisfiability Problem Given a Boolean formula S, is there an assignment to make S true?  &: and !: or ~: not S=A& (~B !C) &~C   S=A& ~A Worst case: n variables --> 2n combinations

N2: Bin Packing Problem C1 C2 C3 C4 C5 C6 C7 C8 7 1 3 4 1 3 3 8 B=15 7 1 3 4 1 3 3 8   B=15 How many bags are necessary?

N3: 0/1 Knapsack Problem M(weight limit)=14 Value 10 5 1 9 3 4 11 17 Weight 7 22 15 M(weight limit)=14 best solution: P1, P2, P3, P5(optimal) This problem is NP-complete.

N4: Traveling Salesperson Problem Given a graph (vertex-city, edge-flying cost between two cities), the salesperson travels to every city once and only once, and the cost should also be minimal. Given a set of n planar points Find: A closed tour which includes all points exactly once such that its total length is minimized. This problem is NP-complete.

N5: Partition Problem Given: A set of positive integers S Find: S1 and S2 such that S1S2=, S1S2=S, iS1i=iS2 i (partition into S1 and S2 such that the sum of S1 is equal to S2) e.g. S={1, 7, 10, 9, 5, 8, 3, 13} S1={1, 10, 9, 8} S2={7, 5, 3, 13} This problem is NP-complete.

N6: Art Gallery Problem Given an art gallery in the form of a polygon, determine the minimum number of guards and their placements such that the entire art gallery can be monitored by these guards.

P1: Minimal Spanning Tree Problem Given a graph, find a spanning tree (a graph without cycle) with the minimum total edges length.

P1: Minimal Spanning Tree Problem graph: greedy method geometry(on a plane): divide-and-conquer # of possible spanning trees for n points: nn-2 n=10→108, n=100→10196

P1: Minimal Spanning Tree Problem graph: greedy method geometry(on a plane): divide-and-conquer # of possible spanning trees for n points: nn-2 (Cayley’s theorem) n=10→108, n=100→10196

P2: Convex Hull Problem Given a set of 2-dimensional points, find a smallest convex polygon to contain all of these points.   Convex Polygon: any line connecting any two points inside the polygon must itself lie inside the polygon.

P2: Convex Hull Problem It is not obvious to find a convex hull by examining all possible solutions divide-and-conquer

P3: One-Center Problem Given a set of points, find a smallest circle covering all of these points. prune-and-search !!