Divide and Conquer.(Merge Sort) by sixleaves

algo-C1-Introduction

Merge Sort:一.Motivation and Examplewhy study merge sort?The Sorting ProblemExample:二.Pseudocode:Pseudocode for Merge:Merge Sort Running Time:Running Time of Merge:Running Time of Merge SortRecall:三.AnalysisClaim:Question1:Question2Proof of claim

Merge Sort:

一.Motivation and Example

why study merge sort?

• Good introduction to divide & conquer
• Calibrate your preparation
• Motivates guiding principles for algorithm analysis(worst-case and asymptotic analysis)
• Analysis generalizes to “Master Method"

The Sorting Problem

Input: array of n numbers, unsorted. (5 4 1 8 7 2 6 3)

Output: same numbers, sorted in increasing order (1 … 8)

Example:

二.Pseudocode:

• recursively sort 1st half of input array
• recursively sort 2nd half of input array
• merge two sorted sublists into one[ ignores base cases]

Pseudocode for Merge:

Merge Sort Running Time:

Key Question: running time of MergeSort on array of n numbers?

[Running time ~ #of mines of code executed ]

Running Time of Merge:

running time of merge on array of m numbers is <= 4m + 2 <= 6m(since m >= 1)

Running Time of Merge Sort

Claim: MergeSort requires <=

6nlog2n+6n

operations for sort n numbers

Recall:

log2n

of times you divide by 2 until you get a number that drops below one

So if you plug in 32 you got to divide five time by two get down to one.

三.Analysis

Claim:

For every Input array of n numbers, Merge Sort produces a sorted output array and uses at most 6nlog_2n + 6n operations.

Proof of calim (assuming n = power of 2):

Question1:

how many levels does this recursion tree have.===>

beacause: {level0, level1……. level(log2n)} === > (log_2n - 0) + 1 ===>

log2n+1

Question2

at each level j = 0,1 ,2 , log_2n, there are 2^j subproblems, each of size n / 2^j.

第二个的原因很简单, 这颗递归树的第j层的子问题有 2^j 个, 例如第0层是1个, 第1层是两个, 由此类推, 所以每个的大小都是n / 2^j.

Proof of claim

第j层总的合并程序中所含有的指令次数是 <= 2 ^j * 6(n / 2^j) = 6n.也就是所每一层的合并程序内部所做的指令次数都是固定的, 为6n次, 所以对于这课共有log2n + 1个层的数, 总的操作次数为6n( log2n + 1) = 6nlog_2n + 6n.