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Mergesort without Animation
 

Mergesort without Animation

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Understand mergesort and the working of recursion. The time and space complexity are also dealt with.

Understand mergesort and the working of recursion. The time and space complexity are also dealt with.

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    Mergesort without Animation Mergesort without Animation Presentation Transcript

    • MERGE SORT - HOW RECURSION WORKS 10 4 126152 2 4 6 10 12 15 Dr.S.Lovelyn Rose PSGCollege ofTechnology Coimbatore, India
    • Splitting Solve subproblems Combine solution ALGORITHM MergeSort(low,high) { if(low<high) { mid=(low+high)/2 MergeSort(low,mid) MergeSort(mid+1,high) Merge(low,mid,high) } }
    • Merge(low,mid,high) { h=low; i=low; j=mid+1 while((h<=mid) and (j<=high)) { if(a[h]<=a[j]) { b[i]=a[h] h=h+1 } else { b[i]=a[j] j=j+1 } i=i+1 }//End of while temp array 2nd sub array1st sub array
    • temp to original array for k=low to high a[k]=b[k] } If(h>mid) { for k=j to high { b[i]=a[k] i=i+1 } } else { for k=h to mid { b[i]=a[k] i=i+1 } }
    • Recursion Order 8 3 2 9 7 1 5 4 8 3 2 9 7 1 5 4 8 3 2 9 7 1 5 4 8 3 2 9 7 1 5 4 3 8 4 51 72 9 2 3 8 9 1 4 5 7 1 2 3 4 5 7 8 9 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 Note :The numbers in red help to understand the order in which recursion takes place
    • Formulating the Recurrence Relation T(n)=2T(n/2)+(n-1) Size of a sublist Number of sublists (n-1)  Number of times basic operation is performed when a function is called  Inside Merge() a minimum of (n-1) comparisons is performed. Eg.  If 1 element no comparison  If 2 elements->max 1 comparison  If 4 elements->3 comparisons
    • Solving Recurrence MastersTheorem a=2,b=2,f(n)=n-1 log b a=1 By case 2,f(n)=Ѳ(n1 )=O(n) T(n)= Ѳ(n log n)
    • By Substitution Method T(n)=2T(n/2)+(n-1) =2(2T(n/22 )+((n/2)-1))+(n-1) =22 T(n/22 )+(n-2)+(n-1) =22 (2T(n/23 )+((n/4)-1))+(2n-(1+2)) =23T (n/23 )+3n-(1+2+22 ) . . =2k T(n/2k )+kn-(1+2+…+2k-1 ) Let n=2k =2k T(1 )+n log n-(2k -1 ) =n log n-n+1 = Ѳ(n log n)
    • Space Complexity = n+n = 2n = Ѳ(n) Additional array ‘b’ of size ‘n’ Input Array of size ‘n’
    • My Blogs http://datastructuresinterview.blogspot.in/ http://talkcoimbatore.blogspot.in/ http://simpletechnical.blogspot.in/