Computer Science CS 330: Algorithms Quick Sort Gene Itkis

Computer Science CS-330: Algorithms, Fall 2008Gene Itkis2 QuickSort: randomly divide & concur QSort(A[], F, L):  if F>L then return;  k  Partition(A[], F, L);  QSort  QSort(A[], F, k-1);  QSort  QSort(A[], k+1, L);  Worst case:  T(n)=O(n)+ T(n-1)  T(n)=O(n 2 )  Best Case:  T(n)=O(n)+ 2T(n/2)  T(n)=O(n lg n)  Average: ??? O(L- F)

Computer Science CS-330: Algorithms, Fall 2008Gene Itkis3 QSort Analysis  Average Performance = expected #compares  Notation:  z i = i-th smallest element of A[1..n]  C i,j =C j,i = the cost of comparing z i and z j  P i,j = the probability of QSort comparing z i and z j  E[#compares]=  all i,j>i C i,j  P i,j =  all i,j>i P i,j  z i and z j are not compared if for some k: i<k<j, z k is chosen as a pivot before either z i or z j  (only pivot is compared with other elements)   P i,j =2/(j-i+1)  ( 2 elements – z i, z j – out of j-i+1 result in the comparison )

Computer Science CS-330: Algorithms, Fall 2008Gene Itkis4 QSort Analysis  Thus +1  E[#compares] =  all i,j>i P i,j =  all i,j>i 2/(j-i+1) i 2/(j-i)  Let j=i+d, where 0<d  n-i. Then n  E[#compares] i 2/(j-i)=  all i,d 2/d = =  all i=1..n  d=1..n-i 2/d <  all i=1..n  d=1..n 2/d   2  all i=1..n ln n  2 n ln n  1.4 n lg n

Computer Science CS-330: Algorithms, Fall 2008Gene Itkis5 Randomized Algorithms vs. Average performance  Average over  Inputs distribution  Always same performance for the same input O(n 2 )  E.g. deterministic QSort (say pivot  1 st element): ordered arrays always O(n 2 )  Internal coin-flips distribution  On any input most of the times get better performance although sometimes can slow down to the worst case O(n lg n) O(n 2 )  Example: QSort (w/randomized partition) most of the times runs in O(n lg n), but occasionally can be as bad as O(n 2 )  Not good for real-time and/or critical systems