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Merge Sort Carola Wenk Slides courtesy of Charles Leiserson with - - PowerPoint PPT Presentation

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CS 3343 Fall 2010 Merge Sort Carola Wenk Slides courtesy of Charles Leiserson with small Slides courtesy of Charles Leiserson with small changes by Carola Wenk 9/9/10 CS 3343 Analysis of Algorithms 1 Merge sort Merge sort M ERGE -S ORT

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CS 3343 – Fall 2010

Merge Sort

Carola Wenk Slides courtesy of Charles Leiserson with small

9/9/10 CS 3343 Analysis of Algorithms 1

Slides courtesy of Charles Leiserson with small changes by Carola Wenk

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Merge sort Merge sort

MERGE-SORT (A[1 . . n])

  • 1. If n = 1, done.
  • 2. MERGE-SORT (A[ 1 . . n/2 ])
  • 3. MERGE-SORT (A[ n/2+1 . . n ])

( [ ])

  • 4. “Merge” the 2 sorted lists.

Key subroutine: MERGE

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Merging two sorted arrays Merging two sorted arrays

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Merging two sorted arrays Merging two sorted arrays

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Merging two sorted arrays Merging two sorted arrays

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Merging two sorted arrays Merging two sorted arrays

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Merging two sorted arrays Merging two sorted arrays

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Merging two sorted arrays Merging two sorted arrays

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Merging two sorted arrays Merging two sorted arrays

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Merging two sorted arrays Merging two sorted arrays

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Merging two sorted arrays Merging two sorted arrays

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Merging two sorted arrays Merging two sorted arrays

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Merging two sorted arrays Merging two sorted arrays

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Merging two sorted arrays Merging two sorted arrays

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Merging two sorted arrays Merging two sorted arrays

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Time dn ∈ Θ(n) to merge a total

  • f n elements (linear time)

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  • f n elements (linear time).
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Analyzing merge sort Analyzing merge sort

MERGE-SORT (A[1 . . n])

  • 1. If n = 1, done.

T(n) d0

  • 2. MERGE-SORT (A[ 1 . . n/2 ])
  • 3. MERGE-SORT (A[ n/2+1 . . n ])

T(n/2) T(n/2)

  • 4. “Merge” the 2 sorted lists.

dn Sloppiness: Should be T( n/2 ) + T( n/2 ) ,

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but it turns out not to matter asymptotically.

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Recurrence for merge sort Recurrence for merge sort

d if n = 1; T(n) = d0 if n 1; 2T(n/2) + dn if n > 1.

  • But what does T(n) solve to? I.e., is it

O(n) or O(n2) or O(n3) or …?

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( ) ( ) ( )

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Recursion tree Recursion tree

Solve T(n) = 2T(n/2) + dn, where d > 0 is constant. Solve T(n) 2T(n/2) dn, where d 0 is constant.

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Recursion tree Recursion tree

Solve T(n) = 2T(n/2) + dn, where d > 0 is constant. Solve T(n) 2T(n/2) dn, where d 0 is constant. T(n)

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Recursion tree Recursion tree

Solve T(n) = 2T(n/2) + dn, where d > 0 is constant. Solve T(n) 2T(n/2) dn, where d 0 is constant. dn T(n/2) T(n/2)

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Recursion tree Recursion tree

Solve T(n) = 2T(n/2) + dn, where d > 0 is constant. Solve T(n) 2T(n/2) dn, where d 0 is constant. dn T( /4) T( /4) T( /4) T( /4) dn/2 dn/2 T(n/4) T(n/4) T(n/4) T(n/4)

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Recursion tree Recursion tree

Solve T(n) = 2T(n/2) + dn, where d > 0 is constant. Solve T(n) 2T(n/2) dn, where d 0 is constant. dn d /4 d /4 d /4 d /4 dn/2 dn/2 dn/4 dn/4 dn/4 dn/4 d0

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Recursion tree Recursion tree

Solve T(n) = 2T(n/2) + dn, where d > 0 is constant. Solve T(n) 2T(n/2) dn, where d 0 is constant. dn d /4 d /4 d /4 d /4 dn/2 dn/2 h = log n dn/4 dn/4 dn/4 dn/4 h = log n d0

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Recursion tree Recursion tree

Solve T(n) = 2T(n/2) + dn, where d > 0 is constant. Solve T(n) 2T(n/2) dn, where d 0 is constant. dn dn d /4 d /4 d /4 d /4 dn/2 dn/2 h = log n dn/4 dn/4 dn/4 dn/4 h = log n d0

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Recursion tree Recursion tree

Solve T(n) = 2T(n/2) + dn, where d > 0 is constant. Solve T(n) 2T(n/2) dn, where d 0 is constant. dn dn d /4 d /4 d /4 d /4 dn/2 dn/2 h = log n dn dn/4 dn/4 dn/4 dn/4 h = log n d0

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Recursion tree Recursion tree

Solve T(n) = 2T(n/2) + dn, where d > 0 is constant. Solve T(n) 2T(n/2) dn, where d 0 is constant. dn dn d /4 d /4 d /4 d /4 dn/2 dn/2 h = log n dn d dn/4 dn/4 dn/4 dn/4 h = log n dn … d0 …

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Recursion tree Recursion tree

Solve T(n) = 2T(n/2) + dn, where d > 0 is constant. Solve T(n) 2T(n/2) dn, where d 0 is constant. dn dn d /4 d /4 d /4 d /4 dn/2 dn/2 h = log n dn d dn/4 dn/4 dn/4 dn/4 h = log n dn … d0 #leaves = n d0n …

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Recursion tree Recursion tree

Solve T(n) = 2T(n/2) + dn, where d > 0 is constant. Solve T(n) 2T(n/2) dn, where d 0 is constant. dn dn d /4 d /4 d /4 d /4 dn/2 dn/2 h = log n dn d dn/4 dn/4 dn/4 dn/4 h = log n dn … d0 #leaves = n d0n …

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Total dn log n + d0n

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Conclusions Conclusions

M i Θ( l ) i

  • Merge sort runs in Θ(n log n) time.
  • Θ(n log n) grows more slowly than Θ(n2).
  • Therefore, merge sort asymptotically beats

insertion sort in the worst case.

  • In practice, merge sort beats insertion sort

for n > 30 or so. (Why not earlier?) ( y )

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Recursion-tree method Recursion-tree method

  • A recursion tree models the costs (time) of a
  • A recursion tree models the costs (time) of a

recursive execution of an algorithm.

  • The recursion tree method can be unreliable
  • The recursion-tree method can be unreliable,

just like any method that uses ellipses (…).

  • It is good for generating guesses of what the
  • It is good for generating guesses of what the

runtime could be. But: Need to verify that the guess is right. → Induction (substitution method)

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( )

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Substitution method Substitution method

The most general method to solve a recurrence The most general method to solve a recurrence (prove O and Ω separately):

  • 1. Guess the form of the solution:

(e.g. using recursion trees, or expansion)

  • 2. Verify by induction (inductive step).
  • 3. Solve for O-constants n0 and c (base case of

induction)

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