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Sorting

Sorting. 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

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Sorting

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  1. Sorting

  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 • Bubble Sort • Merge Sort 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(1)>A(2) 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 A(1) is now Nth 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 4th largest entry A(N-2) is still 3rd largest entry A(N-1) is still 2nd largest entry A(N) is still largest entry 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 A(N-2) is now 3rd largest entry A(N-1) is still 2nd largest entry A(N) is still largest enry A(N-1) is now 2nd largest entry A(N) is still largest enry A(N) is now 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(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 1 step 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 N-3 steps N-2 steps N-1 steps

  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(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(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 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 end 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 end 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 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 1st half Sort 2nd half Merge 2 sorted arrays

  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 R(n/2) to sort array of length n/2 n steps to merge two sorted arrays of total length n 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=2m, it is true that R(n) ≤ n log2(n) Case (m=0): true, since log2(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.

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