Quicksort Lecture 4 Quicksort Divide and Conquer Partitioning

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Quicksort Lecture 4

Quicksort Lecture 4

Quicksort

Quicksort

Divide and Conquer

Divide and Conquer

Partitioning Subroutine

Partitioning Subroutine

Example of Partitioning

Example of Partitioning

Example of Partitioning

Example of Partitioning

Example of Partitioning

Example of Partitioning

Example of Partitioning

Example of Partitioning

Example of Partitioning

Example of Partitioning

Example of Partitioning

Example of Partitioning

Example of Partitioning

Example of Partitioning

Example of Partitioning

Example of Partitioning

Example of Partitioning

Example of Partitioning

Example of Partitioning

Example of Partitioning

Example of Partitioning

Example of Partitioning

Example of Partitioning

Example of Partitioning

Running time for PARTITION The running time of PARTITION on the subarray A[p. .

Running time for PARTITION The running time of PARTITION on the subarray A[p. . . r] is where n=r-p+1

Pseudo code for Quicksort

Pseudo code for Quicksort

Analysis of Quicksort

Analysis of Quicksort

Worst-case of quicksort

Worst-case of quicksort

Worst-case decision tree

Worst-case decision tree

Worst-case decision tree

Worst-case decision tree

Worst-case decision tree

Worst-case decision tree

Worst-case decision tree

Worst-case decision tree

Worst-case decision tree

Worst-case decision tree

Worst-case decision tree

Worst-case decision tree

Worst-case analysis For the worst case, we can write the recurrence equation as We

Worst-case analysis For the worst case, we can write the recurrence equation as We guess that

Worst-case analysis This expression achieves a maximum at either end points: q=0 or q=n-1.

Worst-case analysis This expression achieves a maximum at either end points: q=0 or q=n-1. Using the maximum of T(n) we have Thus

Worst-case analysis We can also show that the recurrence equation as Has a solution

Worst-case analysis We can also show that the recurrence equation as Has a solution of We guess that

Worst-case analysis Using the maximum of T(n) we have We can pick the constant

Worst-case analysis Using the maximum of T(n) we have We can pick the constant c 1 large enough so that and Thus the worst case running time of quicksort is

Best-case analysis

Best-case analysis

Analysis of almost best-case

Analysis of almost best-case

Analysis of almost best-case

Analysis of almost best-case

Analysis of almost best-case

Analysis of almost best-case

Analysis of almost best-case

Analysis of almost best-case

Analysis of almost best-case

Analysis of almost best-case

Best-case analysis For the best case, we can write the recurrence equation as We

Best-case analysis For the best case, we can write the recurrence equation as We guess that

Best-case analysis

Best-case analysis

Best-case analysis This expression achieves a minimum at Using the minimum of T(n) we

Best-case analysis This expression achieves a minimum at Using the minimum of T(n) we have

More intuition

More intuition

Randomized quicksort

Randomized quicksort

Randomized quicksort Standard Problematic Algorithm :

Randomized quicksort Standard Problematic Algorithm :

Randomized quicksort RANDOMIZED-PARTITION (A, p, r) 1 2 3 i←RANDOM(p, r) exchange A[r]↔A[i] return

Randomized quicksort RANDOMIZED-PARTITION (A, p, r) 1 2 3 i←RANDOM(p, r) exchange A[r]↔A[i] return PARTITION(A, p, r) RANDOMIZED-QUICKSORT (A, p, r) 1 if p<r 2 then q←RANDOMIZED-PARTITION (A, p, r) RANDOMIZED-QUICKSORT (A, p, q-1) RANDOMIZED-QUICKSORT (A, q+1, r)

Randomized quicksort

Randomized quicksort

Randomized quicksort analysis

Randomized quicksort analysis

Calculating Expectation

Calculating Expectation

Calculating Expectation

Calculating Expectation

Calculating Expectation

Calculating Expectation

Calculating Expectation

Calculating Expectation

Calculating Expectation

Calculating Expectation

Calculating Expectation

Calculating Expectation

Calculating Expectation

Calculating Expectation

Substitution Method

Substitution Method

Substitution Method

Substitution Method

Substitution Method

Substitution Method

Substitution Method

Substitution Method

Quicksort in practice

Quicksort in practice

ALGORITMA DIVIDE AND CONQUER DIVIDE AND CONQUER DivideALGORITMA DIVIDE AND CONQUER DIVIDE AND CONQUER Divide
Divide and Conquer Divide and Conquer algorithms consistDivide and Conquer Divide and Conquer algorithms consist
Module 2 Divide and Conquer Divide and ConquerModule 2 Divide and Conquer Divide and Conquer
Algoritma Divide and Conquer Divide and Conquer dulunyaAlgoritma Divide and Conquer Divide and Conquer dulunya
Divide and Conquer Divide and Conquer algorithms consistDivide and Conquer Divide and Conquer algorithms consist
Divide and Conquer Merge Sort Divide and ConquerDivide and Conquer Merge Sort Divide and Conquer
Module 2 Divide and Conquer Divide and ConquerModule 2 Divide and Conquer Divide and Conquer
Divide and Conquer Algorithms Divide and Conquer GenericDivide and Conquer Algorithms Divide and Conquer Generic
Algoritma Divide and Conquer Divide and Conquer dulunyaAlgoritma Divide and Conquer Divide and Conquer dulunya
Algoritma Divide and Conquer Divide and Conquer dulunyaAlgoritma Divide and Conquer Divide and Conquer dulunya
Divide and Conquer Merge Sort Divide and conquerDivide and Conquer Merge Sort Divide and conquer
Partitioning Divide and Conquer Partitioning Dividing the problemPartitioning Divide and Conquer Partitioning Dividing the problem
7 Quicksort 1 Contents Quicksort Randomized quicksort 27 Quicksort 1 Contents Quicksort Randomized quicksort 2
Divide and Conquer Mergesort Quicksort Binary Search SelectionDivide and Conquer Mergesort Quicksort Binary Search Selection
Quicksort Quicksort QuicksortAp r Divide Ap r ApQuicksort Quicksort QuicksortAp r Divide Ap r Ap
Partitioning Lets Divide and Conquer Gavin Soorma SeniorPartitioning Lets Divide and Conquer Gavin Soorma Senior
Partitioning Lets Divide and Conquer Gavin Soorma SeniorPartitioning Lets Divide and Conquer Gavin Soorma Senior
Divide and Conquer Merge Sort Partitioning numbers QuickDivide and Conquer Merge Sort Partitioning numbers Quick
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Divide and conquer Partitioning OSPF networks with SDNDivide and conquer Partitioning OSPF networks with SDN
3 Way Partitioning 3 way partitioning Choose partitioning3 Way Partitioning 3 way partitioning Choose partitioning
3 Way Partitioning 3 way partitioning Choose partitioning3 Way Partitioning 3 way partitioning Choose partitioning
3 Way Partitioning 3 way partitioning Choose partitioning3 Way Partitioning 3 way partitioning Choose partitioning
Algoritma Divide and Conquer Bagian 1 Divide andAlgoritma Divide and Conquer Bagian 1 Divide and