算法导论 第一章 插入与归并排序

算法排序：input：sequence 《a1,a2,a3…..an》output: permutation 《a11,a22,a33,..ann》$ a110

 insertion sort

---

Insertioon-Sort(A,n) //Sorts A[1..n]    peusocode

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

---

EX:[8,2,4,9,3,6]

start j = 2;

…..

[2,3,4,6,8,9]

Running time

—depends on input (e.g. already sorted)

—depends on input size (6 elem vs 6*10^9)

—parameter in input size

—Want upper bounds

Kinds of analysis

Worst -case(usually)

T(n) = max time on any input of size n

Average-case(sometimes)

T(n)=expected time overall input of size n

(Need assumption of stat distribution——equal )

Best-case(bogus)

Worst -case

—relative speed

—absolute speed

Big Idea  Asymptotic analysis

1.Ignore machine dependent

2.Look at growth of T(n) as n->oo

Asymptotic notation

Big O (theta notation) Drop low order terms Ignore leading constants

EX:3n^3 + 90 n^2-5n + 6046=Theta(n^3)

Insertion sort analysis

Worst -case: input reverse sorted

T(n)=qiuhe (j <—2–n)O(j) = O(n^2)

Is Insertion fast?  Not for large n

Merge sort

Merge sort A[1..n]    T(n)

1.if n = 1 ,done    //Theta(1)

2.Recursive sort  //2T(n/2)

A[1..n/2] and A[n/2+1…n]

3.Merge 2 sorted list  //Theta(n)

Key subroutine  Merge

Time = Theta(n) Linear time

Recurrence

T(n) ={ theta(1) if n=1;  2T(n/2)+O(n) if n >1}

Recursion Tree T(n) = 2T(n/2)+c*n (c>0)

T(n) = cn  =                  cn =……  =        cn    cn

/    \                  /      \                      ….    cn

T(n/2) T(n/2)    cn/2    cn/2            O(1)    O(n)

/    \        /    \

T(n/4) T(n/4) T(n/4) T(n/4)

Height of the tree is lg(n)  leaves = n

Total = cn * lg(n) + O(n)= O(n*lg(n))

bounds is 30