1. Sorting UC Berkeley Fall 2004, E77 http://jagger.me.berkeley.edu/~pack/e77 Copyright 2005, Andy Packard. This work is licensed under the Creative Commons Attribution-ShareAlike License. To view a copy of this license, visit http://creativecommons.org/licenses/by-sa/2.0/ or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA. http://jagger.me.berkeley.edu/~pack/e77http://creativecommons.org/licenses/by-sa/2.0/

2. Sorting Keeping data in “order” allows it to be searched more efficiently Example: Phone Book –Sorted by Last Name (“lots” of work to do this) Easy to look someone up if you know their last name Tedious (but straightforward) to find by First name or Address Important if data will be searched many times Two algorithms for sorting today –BubbleSort –MergeSort Searching: next lecture

3. Bubble Sort (“Sink” sort here) If A(1)>A(2) switch If A(2)>A(3) switch If A(3)>A(4) switch If A(4)>A(5) switch … If A(N-3)>A(N-2) switch If A(N-2)>A(N-1) switch If A(N-1)>A(N) switch If A(1)>A(2) switch If A(2)>A(3) switch If A(3)>A(4) switch If A(4)>A(5) switch If A(N-3)>A(N-2) switch If A(N-2)>A(N-1) switch If A(1)>A(2) switch If A(2)>A(3) switch If A(3)>A(4) switch If A(4)>A(5) switch If A(N-3)>A(N-2) switch If A(1)>A(2) switch A(N) is now largest entry A(N-1) is now 2 nd largest entry A(N) is still largest enry A(N-2) is now 3 rd largest entry A(N-1) is still 2 nd largest entry A(N) is still largest enry A(1) is now N th largest entry. A(2) is still (N-1) th largest entry. A(3) is still (N-2) th largest entry. A(N-3) is still 4 th largest entry A(N-2) is still 3 rd largest entry A(N-1) is still 2 nd largest entry A(N) is still largest entry

4. Bubble Sort (“Sink” sort here) If A(1)>A(2) switch If A(2)>A(3) switch If A(3)>A(4) switch If A(4)>A(5) switch … If A(N-3)>A(N-2) switch If A(N-2)>A(N-1) switch If A(N-1)>A(N) switch If A(1)>A(2) switch If A(2)>A(3) switch If A(3)>A(4) switch If A(4)>A(5) switch If A(N-3)>A(N-2) switch If A(N-2)>A(N-1) switch If A(1)>A(2) switch If A(2)>A(3) switch If A(3)>A(4) switch If A(4)>A(5) switch If A(N-3)>A(N-2) switch If A(1)>A(2) switch N-1 steps N-2 steps N-3 steps 1 step

5. Bubble Sort (“Sink” sort here) If A(1)>A(2) switch If A(2)>A(3) switch If A(3)>A(4) switch If A(4)>A(5) switch … If A(N-3)>A(N-2) switch If A(N-2)>A(N-1) switch If A(N-1)>A(N) switch If A(1)>A(2) switch If A(2)>A(3) switch If A(3)>A(4) switch If A(4)>A(5) switch If A(N-3)>A(N-2) switch If A(N-2)>A(N-1) switch If A(1)>A(2) switch If A(2)>A(3) switch If A(3)>A(4) switch If A(4)>A(5) switch If A(N-3)>A(N-2) switch If A(1)>A(2) switch for lastcompare=N-1:-1:1 for i=1:lastcompare if A(i)>A(i+1)

6. Matlab code for Bubble Sort function S = bubblesort(A) % Assume A row/column; Copy A to S S = A; N = length(S); for lastcompare=N-1:-1:1 for i=1:lastcompare if S(i)>S(i+1) tmp = S(i); S(i) = S(i+1); S(i+1) = tmp; end What about returning an Index vector Idx, with the property that S = A(Idx) ?

7. Matlab code for Bubble Sort function [S,Idx] = bubblesort(A) % Assume A row/column; Copy A to S N = length(A); S = A; Idx = 1:N; % A(Idx) equals S for lastcompare=N-1:-1:1 for i=1:lastcompare if S(i)>S(i+1) tmp = S(i); tmpi = Idx(i); S(i) = S(i+1); Idx(i) = Idx(i+1); S(i+1) = tmp; Idx(i+1) = tmpi; end If we switch two entries of S, then exchange the same two entries of Idx. This keeps A(Idx) equaling S

8. Merging two already sorted arrays Suppose A and B are two sorted arrays (different lengths) How do you “merge” these into a sorted array C? Chalkboard…

9. Pseudo-code: Merging two already sorted arrays function C = merge(A,B) nA = length(A); nB = length(B); iA = 1; iB = 1; %smallest unused element C = zeros(1,nA+nB); for iC=1:nA+nB if A(iA)<B(iB) %compare smallest unused C(iC) = A(iA); iA = iA+1; %use A else C(iC) = B(iB); iB = iB+1; %use B end

10. MergeSort function S = mergeSort(A) n = length(A); if n==1 S = A; else hn = floor(n/2); S1 = mergeSort(A(1:hn)); S2 = mergeSort(A(hn+1:end)); S = merge(S1,S2); end Base Case Split in half Sort 2 nd half Merge 2 sorted arrays Sort 1 st half

11. Rough Operation Count for MergeSort Let R(n) denote the number of operations necessary to sort (using mergeSort) an array of length n. function S = mergeSort(A) n = length(A); if n==1 S = A; else hn = floor(n/2); S1 = mergeSort(A(1:hn)); S2 = mergeSort(A(hn+1:end)); S = merge(S1,S2); end R(1) = 0 R(n/2) to sort array of length n/2 n steps to merge two sorted arrays of total length n R(n/2) to sort array of length n/2 Recursive relation: R(1)=0, R(n) = 2*R(n/2) + n

12. Rough Operation Count for MergeSort The recursive relation for R R(1)=0, R(n) = 2*R(n/2) + n Claim: For n=2 m, it is true that R(n) ≤ n log 2 (n) Case (m=0): true, since log 2 (1)=0 Case (m=k+1 from m=k) Recursive relation Induction hypothesis

13. Matlab command: sort Syntax is [S] = sort(A) If A is a vector, then S is a vector in ascending order The indices which rearrange A into S are also available. [S,Idx] = sort(A) S is the sorted values of A, and A(Idx) equals S.