算法分析(三)

算法分析

前言：开始对自己的学习做一个记录，看看自己的知识图谱是什么样子。如果可以帮助到其他人是也是一件非常开心的事，更多的是对学习的东西的一个整理。

学习地址：网易公开课的算法导论。链接：
http://open.163.com/special/opencourse/algorithms.html

QuickSort(快速排序)

·Dicide-And-Conquer
·Sort “in place”
·ver practical (with turnning)

1.Divide :Partition the input array into 2 subarray around an priot $elems n lower subarray <= x <= elems in upper subarray

2.Conquer : Recursively sort two subarrys

3.Combome : Tivial

key : Linear-time θ(n) partitioning subroution

    partition(A,p,q) //A[p...q]
        x<-A[p]
        l<-p
        for j<-p+1 to q
            do if A[j] <= x
                then i <- l+1
                    exch A[j] <-> A[j]
        exch A[p] <-> A[i]
        return i

Time = θ(n) for n-elem subarray

Ex 6,10,13,5,8,3,2,11 (x<-6)

6,5,3,10,8,13,2,11
6,5,3,2,10,8,13,11
6,5,3,2,10,8,13,11
2,5,3,6,10,8,13,11

    QuickSort(A,p,q)
        if p<q
            then r<- Partition(A,p,q)
                QuickSort(A,p,r-1)
                QuickSort(A,r+1,q)

Init call QuickSort(A,1,n)

Analysis - assume all elems distinct
T(n) = warst-case time
· input sorted and reverse sorted
· one side of the partition of each partion has no elements

T(n) = T(n) + T(n-1) + θ(n)
= θ(1) + T(n-1) + θ(n)
=T(n-1) + θ(n)
=θ(n^2) ( arithmetic , like insertion sort)

//average case is good

Recursion Tree : T(n) = T(n) + T(n-1) +cn

T(n) = ···cn····························= cn
·········· /········ \·························/·······\
······T(0)·······T(n-1)··············T(0)·····c(n-1)
·························································/·······\
·······················································T(0)···T(n-1)

=······························ cn·······································
·····························/··········\··································
······················T(0)········c(n-1)·····························
·····································/·········\···························
································T(0)··········c(n-1)···················
·························································· 。 ···············
································································ 。·········
····································································θ(n)·····

$c(n-1)+c(n-2)+…+ θ(n)= \sum\limits_{k=1}^nck$
T(n) = θ(n) ＋ θ(n^2) = θ(n^2)

Best-cast analysis (intution only)

Partition splits array n/2…n/2

T(n) = 2T(n/2) + θ(n)=θ(nlgn)

split is always 1/10 : 9/10

T(n) = T(n/10) + T(9n/10) + θ(n)

T(n) = ···cn················································= ················cn·····················································
·········· /········ \············································/···············································\·····························
······T(n/10)·······T(9n/10)··················n/10·········································9n/10··················
···························································/·······\···········································/···········\···················
·······················································T(n/100)···T(9n/100)·······T(9n/100)···T(81n/100)

$log_{10n/9} <= T(n)$

We alterrate spilt lucky，unlucky

L(n) = 2U(n/2) + θ(n) - - - - - - lucky
U(n) = 2L(n-1) + θ(n) - - - - - - unlucky

L(n) = 2(L(n/2 - 1) + θ(n/2)) + θ(n)
= 2L(n/2 - 1) + θ(n)
= θ(n)

Randomized quickSort(随机函数)

· running time is independent of the input ording
· no assumptions about the input distribution
· no specific input that can elict worst-case behavior
· worse-case is determined only by a random-number generator pivot on a random element

Analysis
T(N) = random version for running time assumiry rand #’s indep

For k = 0,1,2,3,..,n
$\begin{eqnarray}f(x)= \begin{cases} 1, & {if parttition generates a k=n-k-1} \cr 0, & otherwise \cr \end{cases} \end{eqnarray}$

E[Xk] = 0·Pr{Xk = 0} +1·Pr{Xk = 1}
=Pr{Xk = 1}
=1/n

$\begin{equation} T(n)=\left\{ \begin{array}{rcl} T(0)+T(n-1)+ θ(n) & & {if 0:n-1 spilt }\\ T(1)+T(n)+ θ(n)& & {if 1:n-2 spilt }\\ & & { }\\ T(n-1)+T(0)+ θ(n) & & if n-1:0 spilt { } \end{array} \right. \end{equation}$

$=\sum\limits_{k=0}^{n-1}(T(k)+T(n-k-1)+θ(n))$

$E(T(n))=\sum\limits_{k=0}^{n-1} E [Xk(T(k)+T(n-k-1)+θ(n))]$

$=\sum\limits_{k=0}^{n-1} E [Xk(T(k)]E[T(n-k-1)+θ(n))]$

$=\sum\limits_{k=0}^{n-1} E [Xk(T(k)]E[T(n-k-1)+θ(n))]$

$=1/n\sum\limits_{k=0}^{n-1} E [Xk(T(k)]E[T(n-k-1)+θ(n))]$