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CS 3343 -- Spring 2009 Merge sort M ERGE -S ORT A [1 . . n ] 1. If n = 1, done. 2. Recursively sort A [ 1 . . n /2 ] and A [ n /2 +1 . . n ] . 3. Merge the 2 sorted lists. Merge Sort Key subroutine: M ERGE Carola Wenk

  1. CS 3343 -- Spring 2009 Merge sort M ERGE -S ORT A [1 . . n ] 1. If n = 1, done. 2. Recursively sort A [ 1 . .  n /2  ] and A [  n /2  +1 . . n ] . 3. “ Merge ” the 2 sorted lists. Merge Sort Key subroutine: M ERGE Carola Wenk Slides courtesy of Charles Leiserson with small changes by Carola Wenk 1/27/09 CS 3343 Analysis of Algorithms 1 1/27/09 CS 3343 Analysis of Algorithms 2 Merging two sorted arrays Merging two sorted arrays 20 12 20 12 13 11 13 11 7 9 7 9 2 1 2 1 1 1/27/09 CS 3343 Analysis of Algorithms 3 1/27/09 CS 3343 Analysis of Algorithms 4 1

  2. Merging two sorted arrays Merging two sorted arrays 20 12 20 12 20 12 20 12 13 11 13 11 13 11 13 11 7 9 7 9 7 9 7 9 2 1 2 2 1 2 1 1 2 1/27/09 CS 3343 Analysis of Algorithms 5 1/27/09 CS 3343 Analysis of Algorithms 6 Merging two sorted arrays Merging two sorted arrays 20 12 20 12 20 12 20 12 20 12 20 12 13 11 13 11 13 11 13 11 13 11 13 11 7 9 7 9 7 9 7 9 7 9 7 9 2 1 2 2 1 2 1 2 1 2 7 1/27/09 CS 3343 Analysis of Algorithms 7 1/27/09 CS 3343 Analysis of Algorithms 8 2

  3. Merging two sorted arrays Merging two sorted arrays 20 12 20 12 20 12 20 12 20 12 20 12 20 12 20 12 13 11 13 11 13 11 13 11 13 11 13 11 13 11 13 11 7 9 7 9 7 9 9 7 9 7 9 7 9 9 2 1 2 2 1 2 1 2 7 1 2 7 9 1/27/09 CS 3343 Analysis of Algorithms 9 1/27/09 CS 3343 Analysis of Algorithms 10 Merging two sorted arrays Merging two sorted arrays 20 12 20 12 20 12 20 12 20 12 20 12 20 12 20 12 20 12 20 12 13 11 13 11 13 11 13 11 13 11 13 11 13 11 13 11 13 11 13 11 7 9 7 9 7 9 9 7 9 7 9 7 9 9 2 1 2 2 1 2 1 2 7 9 1 2 7 9 11 1/27/09 CS 3343 Analysis of Algorithms 11 1/27/09 CS 3343 Analysis of Algorithms 12 3

  4. Merging two sorted arrays Merging two sorted arrays 20 12 20 12 20 12 20 12 20 12 20 12 20 12 20 12 20 12 20 12 20 12 20 12 13 11 13 11 13 11 13 11 13 11 13 13 11 13 11 13 11 13 11 13 11 13 7 9 7 9 7 9 9 7 9 7 9 7 9 9 2 1 2 2 1 2 1 2 7 9 11 1 2 7 9 11 12 1/27/09 CS 3343 Analysis of Algorithms 13 1/27/09 CS 3343 Analysis of Algorithms 14 Merging two sorted arrays Analyzing merge sort 20 12 20 12 20 12 20 12 20 12 20 12 T ( n ) M ERGE -S ORT A [1 . . n ] 13 11 13 11 13 11 13 11 13 11 13 d 0 1. If n = 1, done. 2 T ( n /2) 7 9 7 9 7 9 9 2. Recursively sort A [ 1 . .  n /2  ] and A [  n /2  +1 . . n ] . 2 1 2 dn 3. “Merge” the 2 sorted lists 1 2 7 9 11 12 Sloppiness: Should be T (  n /2  ) + T (  n /2  ) , Time dn ∈ Θ ( n ) to merge a total but it turns out not to matter asymptotically. of n elements (linear time). 1/27/09 CS 3343 Analysis of Algorithms 15 1/27/09 CS 3343 Analysis of Algorithms 16 4

  5. Recurrence for merge sort Recursion tree Solve T ( n ) = 2 T ( n /2) + dn , where d > 0 is constant. d 0 if n = 1; T ( n ) = 2 T ( n /2) + dn if n > 1. • But what does T ( n ) solve to? I.e., is it O(n) or O(n 2 ) or O(n 3 ) or …? 1/27/09 CS 3343 Analysis of Algorithms 17 1/27/09 CS 3343 Analysis of Algorithms 18 Recursion tree Recursion tree Solve T ( n ) = 2 T ( n /2) + dn , where d > 0 is constant. Solve T ( n ) = 2 T ( n /2) + dn , where d > 0 is constant. dn T ( n ) T ( n /2) T ( n /2) 1/27/09 CS 3343 Analysis of Algorithms 19 1/27/09 CS 3343 Analysis of Algorithms 20 5

  6. Recursion tree Recursion tree Solve T ( n ) = 2 T ( n /2) + dn , where d > 0 is constant. Solve T ( n ) = 2 T ( n /2) + dn , where d > 0 is constant. dn dn dn /2 dn /2 dn /2 dn /2 T ( n /4) T ( n /4) T ( n /4) T ( n /4) dn /4 dn /4 dn /4 dn /4 … d 0 1/27/09 CS 3343 Analysis of Algorithms 21 1/27/09 CS 3343 Analysis of Algorithms 22 Recursion tree Recursion tree Solve T ( n ) = 2 T ( n /2) + dn , where d > 0 is constant. Solve T ( n ) = 2 T ( n /2) + dn , where d > 0 is constant. dn dn dn dn /2 dn /2 dn /2 dn /2 h = log n h = log n dn /4 dn /4 dn /4 dn /4 dn /4 dn /4 dn /4 dn /4 … … d 0 d 0 1/27/09 CS 3343 Analysis of Algorithms 23 1/27/09 CS 3343 Analysis of Algorithms 24 6

  7. Recursion tree Recursion tree Solve T ( n ) = 2 T ( n /2) + dn , where d > 0 is constant. Solve T ( n ) = 2 T ( n /2) + dn , where d > 0 is constant. dn dn dn dn dn /2 dn dn /2 dn dn /2 dn /2 h = log n h = log n dn /4 dn /4 dn /4 dn /4 dn /4 dn /4 dn /4 dn /4 dn … … … d 0 d 0 1/27/09 CS 3343 Analysis of Algorithms 25 1/27/09 CS 3343 Analysis of Algorithms 26 Recursion tree Recursion tree Solve T ( n ) = 2 T ( n /2) + dn , where d > 0 is constant. Solve T ( n ) = 2 T ( n /2) + dn , where d > 0 is constant. dn dn dn dn dn /2 dn /2 dn dn /2 dn /2 dn h = log n h = log n dn /4 dn /4 dn /4 dn /4 dn dn /4 dn /4 dn /4 dn /4 dn … … … … d 0 #leaves = n d 0 n d 0 #leaves = n d 0 n Total n log n + d 0 n 1/27/09 CS 3343 Analysis of Algorithms 27 1/27/09 CS 3343 Analysis of Algorithms 28 7

  8. Conclusions Recursion-tree method • A recursion tree models the costs (time) of a • Merge sort runs in Θ ( n log n ) time. recursive execution of an algorithm. • Θ ( n log n ) grows more slowly than Θ ( n 2 ). • The recursion-tree method can be unreliable, • Therefore, merge sort asymptotically beats just like any method that uses ellipses (…). insertion sort in the worst case. • It is good for generating guesses of what the • In practice, merge sort beats insertion sort runtime could be. for n > 30 or so. (Why not earlier?) But: Need to verify that the guess is right. → Induction (substitution method) 1/27/09 CS 3343 Analysis of Algorithms 29 1/27/09 CS 3343 Analysis of Algorithms 30 Substitution method 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 n 0 and c (base case of induction) 1/27/09 CS 3343 Analysis of Algorithms 31 8

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