1、Introduction to Algorithm Analysis,Algorithm : Design & Analysis 1,Introduction to Algorithm Analysis,Goal of the Course Algorithm, the concept Algorithm Analysis: the critiria Average and Worst-Case Analysis Lower Bounds and the Complexity of Problems,Goal of the Course,Learning to solve real probl
2、ems that arise frequently in computer application Learning the basic principles and techniques used for answering the question: “How good, or, how bad is the algorithm” Getting to know a group of “very difficult problems” categorized as “NP-Complete”,Probably the Oldest Algorithm,Euclid algorithm in
3、put: nonnegative integer m,n output: gcd(m,n) procedure E1. n divides m, the remainderr E2. if r =0 then return n E3. nm; rn; goto E1,The Problem: Computing the greatest common divisor of two nonnegative integers,Specification,Euclid Algorithm: Recursive Version,Euclid algorithm input: nonnegative i
4、nteger m,n output: gcd(m,n) procedure Euclid(int m,n)if n=0then return melse return Euclid(n, m mod n),Specification,Recursion,Algorithm Pseudocode,Sequential Search, another Example,Procedure: Int seqSearch(int E, int n, int K)int ans, index;ans=-1;for (index=0; indexn; index+)if (K=Eindex)ans=inde
5、x;break; Return ans;,The Problem: Searching a list for a specific key. Input: an unordered array E with n entries, a key K to be matched Output: the location of K in E (or fail),Algorithmically Solvable Problem,Informally speaking A problem for which a computer program can be written that will produ
6、ce the correct answer for any input if we let it run long enough and allow it as much storage space as it needs. Unsolvable(or un-decidable) problem Problems for which no algorithms exist the Halting Problem for Turing Machine,Computational Complexity,Formal theory of the complexity of computable fu
7、nctions The complexity of specific problems and specific algorithms,Criteria for Algorithm Analysis,Correctness Amount of work done Amount of space used Simplicity, clarity Optimality,Correctness,Describing the “correctness”: the specification of a specified problem: Preconditions vs. post-condition
8、s Establishing the method: Preconditions+Algorithm post-conditions Proving the correctness of the implementation of the algorithm,Correctness of Euclid Algorithm,Euclid algorithm input: nonnegative integer m,n output: gcd(m,n) procedure Euclid(int m,n)if n=0then return melse return Euclid(n, m mod n
9、),How to Measure?,Not too general Giving some indication to make useful comparison for algorithms Not too precise Machine independent Language independent Programming style independent Implementation independent,Focusing the View,Counting the number of the passes through a loop while ignoring the si
10、ze of the loopThe operation of interest Search or sorting an array comparison Multiply 2 matrices multiplication Find the gcd bits of the inputs Traverse a tree processing an edge Non-iterative procedure procedure invocation,Presenting the Analysis Results,Amount of work done usually depends on the
11、size of the inputsAmount of work done usually doesnt depend on the size solely,Worst-case Complexity,Worst-case complexity, of a specified algorithm A for a specified problem P of size n: Giving the maximum number of operations performed by A on any input of size n Being a function of n Denoted as W
12、(n) W(n)=maxt(I) | IDn, Dn is the set of input,Worst-Case Complexity of Euclids,For any integer k1, if mn1 and nFk+1, then the call Euclid(m,n) makes fewer than k recursive calls. (to be proved) Since Fk is approximately , the number of recursive calls in Euclid is O(lgn).,Euclid(int m,n)if n=0then
13、return melse return Euclid(n, m mod n),measured by the number of recursive calls,Euclid Algorithm and Fibonacci,If mn1 and the invocation Euclid(m,n) performs k1 recursive calls, then mFk+2 and nFk+1. Proof by induction Basis: k=1, then n1=F2. Since mn, m2=F3. For larger k, Euclid(m,n) calls Euclid(
14、n, m mod n) which makes k-1 recursive calls. So, by inductive hypothesis, nFk+1, (m mod n)Fk. Note that m n+(m-m/nn) = n+(m mod n) Fk+1+Fk = Fk+2,Average Complexity,Weighted average A(n)How to get Pr(I) Experiences Simplifying assumption On a particular application,Pr(I) is the probability of ocurre
15、nce of input I,Average Behavior Analysis of Sequential Search,Case 1: assuming that K is in E Assuming no same entries in E Look all inputs with K in the ith location as one input (so, inputs totaling n) Each input occurs with equal probability (i.e. 1/n) Asucc(n)=i=0n-1Pr(Ii|succ)t(Ii)=i=0n-1(1/n)(
16、i+1)=(n+1)/2,Average Behavior Analysis of Sequential Search,Case 2: K may be not in E Assume that q is the probability for K in E A(n) = Pr(succ)Asucc(n)+Pr(fail) Afail(n)=q(n+1)/2)+(1-q)n Issue for discussion:Reasonable Assumptions,Optimality,“The best possible” How much work is necessary and suffi
17、cient to solve the problem. Definition of the optimal algorithm For problem P, the algorithm A does at most WA(n) steps in the worst case (upper bound) For some function F, it is provable that for any algorithm in the class under consideration, there is some input of size n for which the algorithm m
18、ust perform at least F(n) steps (lower bound) If WA=F, then A is optimal.,Complexity of the Problem,F is a lower bound for a class of algorithm means that: For any algorithm in the class, and any input of size n, there is some input of size n for which the algorithm must perform at least F(n) basic
19、operations.,Eastblishing a lower bound,Procedure: int findMax(E,n) max=E(0) for (index=1;indexn;index+)if (maxE(index)max=E(index); return max Lower bound For any algorithm A that can compare and copy numbers exclusively, if A does fewer than n-1 comparisons in any case, we can always provide a righ
20、t input so that A will output a wrong result.,The problem: Input: number array E with n entries indexed as 0,n-1 Output: Return max, the largest entry in E,Home Assignment,pp.61 1.5 1.12 1.16 1.19 Additional Other than speed, what other measures of efficiency might one use in a real-world setting? C
21、ome up with a real-world problem in which only the best solution will do. Then come up with one in which a solution that is “approximately” the best is good enough.,Algorithm vs. Computer Science,Sometimes people ask: “What really is computer science? Why dont we have telephone science? Telephone, i
22、t might be argued, are as important to modern life as computer are, perhaps even more so. A slightly more focused question is whether computer science is not covered by such classical disciplines as mathematics, physics, electrical engineering, linguistics, logic and philosophy.We would do best not
23、to pretend that we can answer these questions here and now. The hope, however, is that the course will implicitly convey something of the uniqueness and universality of the study of algorithm, and hence something of the importance of computer science as an autonomous field of study. - adapted from H
24、arel: “Algorithmics, the Spirit of Computing”,References,Classics Donald E.Knuth. The Art of Computer Programming Vol.1 Fundamental Algorithms Vol.2 Semi-numerical Algorithms Vol.3 Sorting and Searching Popular textbooks Thomas H.Cormen, etc. Introduction to Algorithms Robert Sedgewick. Algorithms (
25、with different versions using different programming languages) Advanced mathematical techniques Graham, Knuth, etc. Concrete Mathematics: A Foundation for Computer Science,You Have Choices,Design Techniques Oriented Textbooks Anany Levitin. Introduction to the Design and Analysis of Algorithms M.H.Alsuwaiyel. Algorithms Design Techniques and Analysis Evergreen Textbook Aho, Hopcroft and Ullman. The Design and Analysis of Computer Algorithm,
