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04 - Randomized Partition & Randomize Selection

04 - Randomized Partition & Randomize Selection -...

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Part I: Divide and Conquer Lecture 4: Randomized Partition and Randomized Selection Lecture 4: Randomized Partition and Randomized Selection Part I: Divide and Conquer
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Objective and Outline Objective : Show two examples that use both (1) divide and conquer and (2) randomization Reference : Chapter 7, 9 of CLRS Outline: Partition Basic partition Randomized partition and randomized quicksort Analysis of the randomized quicksort Selection The selection problem First solution: Selection by sorting A divide-and-conquer algorithm Analysis of the divide-and-conquer algorithm Lecture 4: Randomized Partition and Randomized Selection Part I: Divide and Conquer
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Partition Given : An array of distinct numbers Partition : Rearrange the array A [ p .. r ] into two (possibly empty) subarrays A [ p .. q - 1] and A [ q + 1 .. r ] such that A [ u ] A [ q ] < A [ v ] , for any p u q - 1 and q + 1 v r x p r q x x x = A[r] Quicksort works by: 1 calling partition first 2 recursively sorting A [ p .. q - 1] and A [ q + 1 .. r ] Lecture 4: Randomized Partition and Randomized Selection Part I: Divide and Conquer
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The Idea of Partition( A , p , r ) Remark : In COMP 171, you learned one way to do partition Now, we discussed another approach. Use A [ r ] as the pivot , and grow partition from left to right i p j r x x x unrestricted 1 Initially ( i , j ) = ( p - 1 , p ) 2 Increase j by 1 each time to find a place for A [ j ] At the same time increase i when necessary 3 Stops when j = r Lecture 4: Randomized Partition and Randomized Selection Part I: Divide and Conquer
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One Iteration of the Procedure Partition Increase j by 1 each time to find a place for A [ j ] At the same time increase i when necessary i p j r x x x >x j p i x x x r r p x x i j x < x p i j r x x (A) A[j] > x (B) A[j] < x 1 Only increase j by 1 2 i i + 1. A [ i ] A [ j ]. j j + 1 Lecture 4: Randomized Partition and Randomized Selection Part I: Divide and Conquer
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Example: The Operation of Partition( A , p , r ) i j j j p i j p i j p i j p i j, r p i j, r r r r r r r r p, j p, i p, i p, i 2 8 7 4 1 3 5 6 2 2 2 2 2 2 2 2 1 1 1 1 1 3 3 3 3 8 8 8 8 8 8 8 7 7 7 7 7 7 5 5 5 6 6 8 7 1 3 5 6 4 7 1 3 5 6 4 1 3 5 6 4 3 5 6 4 5 6 4 6 4 4 4 (1) (2) (3) (4) (5) (6) (9) (7) (8) Lecture 4: Randomized Partition and Randomized Selection Part I: Divide and Conquer
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The Partition( A , p , r ) Algorithm Partition(A, p, r) begin x = A[r]; // A[r] is the pivot element i = p-1; for j = p to r-1 do if A[j] x then i = i+1; exchange A[i] and A[j]; end end exchange A[i+1] and A[r]; // put pivot in position return i+1 // q = i+1 end Lecture 4: Randomized Partition and Randomized Selection Part I: Divide and Conquer
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Running Time of Partition( A , p , r ) Partition( A , p , r ): x = A [ r ] i = p - 1 for j = p to r - 1 if A [ j ] x ( r - p ) i = i + 1 exchange A [ i ] A [ j ] exchange A [ i + 1] A [ r ] return i + 1 Total: ( r - p ) Running time is Θ( r - p ) linear in the length of the array A [ p .. r ] Lecture 4: Randomized Partition and Randomized Selection Part I: Divide and Conquer
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Outline Outline: Partition Randomized partition and randomized quicksort Analysis of the randomized quicksort The selection problem First solution: Selection by sorting A divide-and-conquer algorithm Analysis of the divide-and-conquer algorithm Lecture 4: Randomized Partition and Randomized Selection Part I: Divide and Conquer
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