1. 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

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

3. 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

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

5. 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

6. 分數
期中考 35%
期末考 35%
報告 %

7. 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.

8. Computer Algorithm – Five Criteria
Input
Output
Finiteness
Definiteness
Effectiveness

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

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

11. Comparison with Two Algorithms Implemented on Two Computers

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

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

14. 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?

15. 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)

16. 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

17. Background for Learning Algorithms
Clear brain
Data structure
Discrete mathematics

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

19. 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

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

21. 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.

22. 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.

23. 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.

24. 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.

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

26. 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

27. 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

28. 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.

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

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