1 / 11

CS 330: Algorithms Quick Select

CS 330: Algorithms Quick Select. Gene Itkis. QuickSelect: random divide & concur. Q Sort (A[], F, L): if F>L then return; k  Partition (A [],F,L); Q Sort (A[], F, k-1); Q Sort (A[], k+1, L); ----------------------- Worst case: T(n)= O(n) + T(n-1)  T(n)=O(n 2 ) Best Case:

ismail
Download Presentation

CS 330: Algorithms Quick Select

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. CS 330: AlgorithmsQuick Select Gene Itkis

  2. QuickSelect: random divide & concur QSort(A[], F, L): • if F>L then return; • k  Partition(A[],F,L); • QSort(A[], F, k-1); • QSort(A[], k+1, L); ----------------------- • Worst case: • T(n)=O(n)+ T(n-1)  T(n)=O(n2) • Best Case: • T(n)=O(n)+ 2T(n/2)  T(n)=O(n lg n) • Average: ??? QSel(A[], F, L, r): • if F>L then return; • k  Partition(A[],F,L); • if k=r thenreturn A[k]; • if k>r then returnQSel(A[],F,k-1,r); • if k<r then returnQSel(A[],k+1,L,r); ----------------------- • Worst case: • T(n)=O(n)+ T(n-1)  T(n)=O(n2) • Best Case: • T(n)=O(n)+ T(n/2)  (or even on 1st call) T(n)=O(n) • Average: ??? Gene Itkis

  3. QSort Analysis • Average performance = Expected Number of Compares • Notation: • zi = i-th smallest element of A[1..n] • Ci,j=Cj,j = the cost of comparing zi and zj • Pi,j = the probability of QSort comparing zi and zj • E[#compares]= all i,j>i Ci,j Pi,j =all i,j>i Pi,j • zi and zj are not compared if for some k: i<k<j, zk is chosen as a pivot before either zi or zj •  Pi,j=2/(j-i+1) Gene Itkis

  4. QSel Analysis(same as QSort!) • Average performance = Expected # of Compares • Notation: • zi = i-th smallest element of A[1..n] • Ci,j=Cj,j = the cost of comparing zi and zj • Pi,j = the probability of QSel comparing zi and zj • E[#compares]= all i,j>i Ci,j Pi,j =all i,j>i Pi,j • zi and zj are not compared if for some k: i<k<j, zk is chosen as a pivot before either zi or zj • NOT as in QSel: zi and zj are alsonot compared if for some k: r<k<j, zk is chosen as a pivot before either zr or zj • Recall: zr is the element QSel is looking for Gene Itkis

  5. QSel Analysis ? zi<zj • Thus, Pi,j = 2/(max{|i-j|, |i-r|, |j-r|} +1) • Let • j= r+D (-r<D<n-r) • i= r+D’ (|D’|<|D|) • Then Pi,j < 2/D zj zi zk zr D D’ For QSort: Pi,j < 2/d; d=|j-i| Gene Itkis

  6. QSort Analysis • Thus • E[#compares]=all i,j>iPi,j= all i,j>i 2/(j-i+1)< all i,j>i 2/(j-i) • Let j=i+d, where 0<d  n-i. Then • E[#compares]< all i,j>i 2/(j-i)= all i,d2/d= = i=1..n d=1..n-i 2/d < i=1..nd=1..n 2/d  2i=1..nln n  1.4 n lg n Gene Itkis

  7. QSel Analysis E[#compares] = all i,j>iPi,j =all i,d2/d < i=1..nd=1..n2/d  2i=1..nln n  1.4n lg n • Thus • E[#compares]=all i,j>i Pi,j < all D,D’ (|D’|<|D|) 2/D≤ -r<D<n-r D’= 1-|D|…|D|-12/D< -r<D<n-r2D2/D= -r<D<n-r 4= 4 n 2ndmight be “over-counting”: e.g. r=5, D=10, then D’=-4…9, not -9…9 Gene Itkis

  8. Conclusion • Can find r-th smallest element in linear time: O(n) • E.g. median (useful for hw and tests ;-) • No need to sort – works faster without sorting • Murphy is not always right • Sometimes best case is more common than worst: • QSort • Worst case: O(n2); Best and Average: O(n lg n) • QSel • Worst case: O(n2); Best and Average: O(n) • Randomized algorithms are COOL! Gene Itkis

  9. Hiring Problem • Another example of the above principles • Problem • N candidates come for a job (at large intervals) • You are the big boss • You want the best, but do not want to wait • Your hiring strategy – greedy: • Get the best you’ve seen so far (i.e. better than the current choice) • Assume: easy to compare candidates/workers • Each “fire current & hire new” procedure costs $1K • How much you will spend eventually? Gene Itkis

  10. Hiring Problem • Best case: • Best candidate comes first • Cost: $1k • Probability: • 1/N • Worst case: • Best candidate comes last • Not enough – e.g. if 2nd best comes first, cost=$2k • Worst case: every candidate is hired => • Candidates come in increasing quality order • Cost: $Nk • Probability: • 1/(N!) • Average: ??? Gene Itkis

  11. Expected Number of Hires • QSort/Qsel – expected # of compareshere: expected # of hires E(#hires) • Let Pi= probability that i-th candidate is hired • E(#hires) = all i $1k×Pi = $1k×all i Pi= • Pi=Prob[i-th hired] = Prob[i-th is best of 1..i] • Pi = 1/i • E(#hires) = $1k×all iPi = = $1k×i=1..N1/i   $1k×ln n  $0.7k lg n Gene Itkis

More Related