Analysis of Algorithms--preface

Analysis of Algorithms:

• First part of the course is focused on analysis.
• Second part of the course is focused on design.

The analysis of algorithm is the theoretical study.(算法分析是理论研究)

The theoretical study of computer-program performance and resource usage.（理论研究是关于计算机性能和资源利用的研究）

In programming, what is more important than performance?

• correctness
• simplicity
• maintainability
• cost
• stability
• functionablity
• fearures
• modularity
• security
• scalability
• user-friendly

Why do we bother and why study algorithms and performance?

• Algorithms is the feasible versus infeasible.
• Algorithms give you a lauguage of talking about program behavior.
• We study algorithms performance is it's tons of fun.

The problem of sorting(排序问题)

• input:sequence<a₁,a₂,a₃...a_n>
• output:permutation<A₁,A₂...A_n>

Such that:A₁<A₂<...<A_n

Insertion-Sort:

for (int i = 1; i < a.length; i++) {
            key=a[i];
            int j=i-1;
            while (j>=0&&a[j]>key) {
                a[j+1]=a[j];
                a[j]=key;
                j--;
            }
            
            for (int k = 0; k < a.length; k++) {
                System.out.print(a[k]+" ");
            }
            System.out.println();
            
        }
}

running time:

　　one thing it depends on is the input itself.

• Depends on input self(eg:already sorted)
• Depends on input size(eg:6 elements vs 6*10⁹)

　　--parameterize things in the input size.

• want upper bounds.guarantee to the user

Kinds of analysis:

• worst-case analysis(usually):T(n)=max time on any input of size n.
• Average case analysis(sometimes):T(n)=expected time over all inputs of size n.
• best-case analysis(bogus:假象)No good.

What is insertion sorts worst-case time?

　　Depends n computer.

• relative speed (on same machine) 
• absolute speed (on defferent machine)

BIG Idea of algorithms:

on same machine analysis algorithms performance use asymptotic analysis(渐进分析)

asymptotic analysis:

• ignore machine-dependent constants
• look at the growth of the running time,look at growth of T(n) as n->∞

asymptotic notation(渐进符号Θ)

Θ-notation:

• drop low order terms(弃去它的低阶级)
• ignore leading constants(忽略前面的常量因子)
• EX:3n³+90n²-5n+6046=Θ(n³)

Insertion-Sort worst-case sorted:T(n)=∑Θ(j)=Θ(n²)

Is insertion sort fast?

It's turns out for small n it is moderately fast;but it is not at all for large n.