Sorting What makes it hard? Chapter 7 in DS&AA Chapter 8 in DS&PS

Insertion Sort Algorithm – Conceptually, incremental add element to sorted array or list, starting with an empty array (list). –Incremental or batch algorithm. Analysis –In best case, input is sorted: time is O(N) –In worst case, input is reverse sorted: time is O(N 2 ). –Average case is (loose argument) is O(N 2 ) Inversion: elements out of order –critical variable for determining algorithm time-cost –each swap removes exactly 1 inversion

Inversions What is average number of inversions, over all inputs? Let A be any array of integers Let revA be the reverse of A Note: if (i,j) are in order in A they are out of order in revA. And vice versa. Total number of pairs (i,j) is N*(N-1)/2 so average number of inversions is N*(N-1)/4 which is O(N 2 ) Corollary: any algorithm that only removes a single inversion at a time will take time at least O(N 2 )! To do better, we need to remove more than one inversion at a time.

BubbleSort Most frequently used sorting algorithm Algorithm: for j=n-1 to 1 …. O(n) for i=0 to j ….. O(j) if A[i] and A[i+1] are out of order, swap them (that’s the bubble) …. O(1) Analysis –Bubblesort is O(n^2) Appropriate for small arrays Appropriate for nearly sorted arrays Comparision versus swaps ?

Shell Sort: 1959 by Shell Motivated by inversion result - need to move far elements Still quadratic Only in text books Historical interest and theoretical interest - not fully understood. Algorithm: (with schedule 1, 3, 5) –bubble sort things spaced 5 apart –bubble sort things 3 apart –bubble sort things 1 apart Faster than insertion sort, but still O(N^2) No one knows the best schedule

Divide and Conquer: Merge Sort Let A be array of integers of length n define Sort (A) recursively via auxSort(A,0,N) where Define array[] Sort(A,low, high) –if (low == high) return –Else mid = (low+high)/2 temp1 = sort(A,low,mid) temp2 = sort(A,mid,high) temp3 = merge(temp1,temp2)

Merge Int[] Merge(int[] temp1, int[] temp2) –int[] temp = new int[ temp1.length+temp2.length] –int i,j,k –repeat if (temp1[i]<temp2[j]) temp[k++]=temp1[i++] else temp[k++] = temp2[j++] –for all appropriate i, j. Analysis of Merge: – time: O( temp1.length+temp2.length) – memory: O(temp1.length+temp2.length)

Analysis of Merge Sort Time –Let N be number of elements –Number of levels is O(logN) –At each level, O(N) work –Total is O(N*logN) –This is best possible for sorting. Space –At each level, O(N) temporary space –Space can be freed, but calls to new costly –Needs O(N) space –Bad - better to have an in place sort –Quick Sort (chapter 8) is the sort of choice.

Quicksort: Algorithm QuickSort - fastest algorithm QuickSort(S) –1. If size of S is 0 or 1, return S –2. Pick element v in S (pivot) –3. Construct L = all elements less than v and R = all elements greater than v. –4. Return QuickSort(L), then v, then QuickSort(R) Algorithm can be done in situ (in place). On average runs in O(NlogN), but can take O(N 2 ) time –depends on choice of pivot.

Quicksort: Analysis Worst Case: –T(N) = worst case sorting time –T(1) = 1 –if bad pivot, T(N) = T(N-1)+N –Via Telescope argument (expand and add) –T(N) = O(N^2) Average Case (text argument) –Assume equally likely subproblem sizes Note: chance of picking ith is 1/N –T(N) average cost to sort

Analysis continued –T(left branch) = T(right branch) (average) so –T(N) = 2* ( T(0)+T(1)….T(N-1) )/N + N, where N is cost of partitioning –Multiply by N: NT(N) = 2(T(0)+…+T(N-1)) +N^2 (*) –Subtract N-1 case of (*) NT(N) - (N-1)T(N-1) = 2T(N-1) +2N-1 –Rearrange and drop -1 NT(N) = (N+1)T(N-1) + 2N -1 –Divide by N(N+1) T(N)/(N+1) = T(N-1)/N + 2/(N+1)

Last Step Substitute N-1, N-2,... 3 for N –T(N-1)/N = T(N-2)/(N-1) + 2/N –… – T(2)/3 = T(1)/2 +2/3 Add –T(N)/(N+1) = T(1)/2+ 2(1/3+1/ /(N+1) – = 2( 1+1/2 +…) -5/2 since T(1) = 0 – = O(logN) Hence T(N) = N logN In literature, more accurate proof. For best results, choose pivot as median of 3 random values from array.

Lower Bound for Sorting Theorem: if you sort by comparisons, then must use at least log(N!) comparisons. Hence Ω(N logN) algorithm. Proof: –N items can be rearranged in N! ways. –Consider a decision tree where each internal node is a comparison. –Each possible array goes down one path –Number of leaves N! –minimum depth of a decision tree is log(N!) –log(N!) = log1+log2+…+log(N) is O(N logN) –Proof: use partition trick sum log(N/2) + log(N/2+1)….log(N) >N/2*log(N/2)

Summary For online sorting, use heapsort. –Online : get elements one at at time –Offline or Batch: have all elements available For small collections, bubble sort is fine For large collections, use quicksort You may hybridize the algorithms, e.g –use quicksort until the size is below some k –then use bubble sort Sorting is important and well-studied and often inefficiently done. Libraries often contain sorting routines, but beware: the quicksort routine in Visual C++ seems to run in quadratic time. Java sorts in Collections are fine.