算法导论--初级1

算法分析是关于计算机程序性能（performance）与资源利用（communication, memory）的理论研究。Correctness, simplicity, maintainability, cost（time, robustness）feature, security, user-friendliness et al. may important than performance. performance会决定算法的feasible or infeasible，是算法实现的基础。performance应用于尚未成熟领域。

sorting problem：input sequence<a1,a2,……an>, output is permutation(rearrangement) of those numbers<b1,b2,……bn>, where b1<=b2<=……<=bn。

插入排序Insertion sort: sorting A[1..n]

for j <- 2 to n

    do    key <- A[j]

            i <- j-1

    while i > 0 and A[i] > key

        do   A[i+1] <- A[i]

               i <- i-1

    A[i+1]<-key

running time：input data itself(sorted), input data size(parametrize in input size), upper bounds(guarantee to user)

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(make an assumption of the statical distribution of the inputs -> expected time, uniform distribution)

Best-case(bogus假象): 

for insertion sort:Asymptotic analysis渐近分析(ignore machine-dependant constants, look the growth of actual running time)

Worst-case: （in reverse order）T(n)=n^2，适合小数据，不适合大数据

并归排序Merge sort: sorting A[1..n]

1. if n = 1, done

2. recursively sort A[1..n/2] and A[n/2+1..n]

3. Merge 2 sorted lists

Key subroutine: Merge, Time = n。

T(n) = 2T(n/2) + n , if n>1 

~Recursion tree. -----T(n) = nlog n