Quick-Sort 2/25/2019 2:22 AM Quick-Sort  4 2  2 4 7 9  7 9 2  2

Slides:



Advertisements
Similar presentations
AVL Trees1 Part-F2 AVL Trees v z. AVL Trees2 AVL Tree Definition (§ 9.2) AVL trees are balanced. An AVL Tree is a binary search tree such that.
Advertisements

Goodrich, Tamassia Sorting, Sets and Selection1 Merge Sort 7 2  9 4   2  2 79  4   72  29  94  4.
Data Structures Lecture 9 Fang Yu Department of Management Information Systems National Chengchi University Fall 2010.
Merge Sort 7 2  9 4   2  2 79  4   72  29  94  4.
Merge Sort 4/15/2017 4:30 PM Merge Sort 7 2   7  2  2 7
Quicksort Quicksort     29  9.
Merge Sort1 Part-G1 Merge Sort 7 2  9 4   2  2 79  4   72  29  94  4.
Merge Sort1 7 2  9 4   2  2 79  4   72  29  94  4.
Insertion Sort. Selection Sort. Bubble Sort. Heap Sort. Merge-sort. Quick-sort. 2 CPSC 3200 University of Tennessee at Chattanooga – Summer 2013 © 2010.
CSC2100B Quick Sort and Merge Sort Xin 1. Quick Sort Efficient sorting algorithm Example of Divide and Conquer algorithm Two phases ◦ Partition phase.
© 2004 Goodrich, Tamassia Quick-Sort     29  9.
© 2004 Goodrich, Tamassia QuickSort1 Quick-Sort     29  9.
Quick-Sort     29  9.
© 2004 Goodrich, Tamassia Quick-Sort     29  9.
© 2004 Goodrich, Tamassia Selection1. © 2004 Goodrich, Tamassia Selection2 The Selection Problem Given an integer k and n elements x 1, x 2, …, x n, taken.
TTIT33 Algorithms and Optimization – Dalg Lecture 2 HT TTIT33 Algorithms and optimization Lecture 2 Algorithms Sorting [GT] 3.1.2, 11 [LD] ,
© 2004 Goodrich, Tamassia Merge Sort1 Quick-Sort     29  9.
Divide-and-Conquer1 7 2  9 4   2  2 79  4   72  29  94  4 Modified by: Daniel Gomez-Prado, University of Massachusetts Amherst.
1 CSC401 – Analysis of Algorithms Lecture Notes 9 Radix Sort and Selection Objectives  Introduce no-comparison-based sorting algorithms: Bucket-sort and.
Selection1. 2 The Selection Problem Given an integer k and n elements x 1, x 2, …, x n, taken from a total order, find the k-th smallest element in this.
© 2004 Goodrich, Tamassia Merge Sort1 Chapter 8 Sorting Topics Basics Merge sort ( 合併排序 ) Quick sort Bucket sort ( 分籃排序 ) Radix sort ( 基數排序 ) Summary.
MergeSort Source: Gibbs & Tamassia. 2 MergeSort MergeSort is a divide and conquer method of sorting.
Sorting.
CSC 213 Lecture 12: Quick Sort & Radix/Bucket Sort.
© 2004 Goodrich, Tamassia Quick-Sort     29  9.
Sorting Fun1 Chapter 4: Sorting     29  9.
Lecture10: Sorting II Bohyung Han CSE, POSTECH CSED233: Data Structures (2014F)
Randomized Algorithms CSc 4520/6520 Design & Analysis of Algorithms Fall 2013 Slides adopted from Dmitri Kaznachey, George Mason University and Maciej.
© 2004 Goodrich, Tamassia Quick-Sort     29  9.
1 Merge Sort 7 2  9 4   2  2 79  4   72  29  94  4.
Towers of Hanoi Move n (4) disks from pole A to pole B such that a larger disk is never put on a smaller disk A BC ABC.
QuickSort by Dr. Bun Yue Professor of Computer Science CSCI 3333 Data.
Merge Sort 1/12/2018 5:48 AM Merge Sort 7 2   7  2  2 7
Advanced Sorting 7 2  9 4   2   4   7
Chapter 11 Sorting Acknowledgement: These slides are adapted from slides provided with Data Structures and Algorithms in C++, Goodrich, Tamassia and Mount.
Merge Sort 7 2  9 4   2   4   7 2  2 9  9 4  4.
Merge Sort 1/12/2018 9:39 AM Presentation for use with the textbook Data Structures and Algorithms in Java, 6th edition, by M. T. Goodrich, R. Tamassia,
Merge Sort 1/12/2018 9:44 AM Presentation for use with the textbook Data Structures and Algorithms in Java, 6th edition, by M. T. Goodrich, R. Tamassia,
Quick-Sort 2/18/2018 3:56 AM Selection Selection.
Quick Sort and Merge Sort
COMP9024: Data Structures and Algorithms
COMP9024: Data Structures and Algorithms
Quick-Sort 9/12/2018 3:26 PM Presentation for use with the textbook Data Structures and Algorithms in Java, 6th edition, by M. T. Goodrich, R. Tamassia,
Quick-Sort 9/13/2018 1:15 AM Quick-Sort     2
Quick-Sort To understand quick-sort, let’s look at a high-level description of the algorithm 1) Divide : If the sequence S has 2 or more elements, select.
Selection Selection 1 Quick-Sort Quick-Sort 10/30/16 13:52
MergeSort Source: Gibbs & Tamassia.
Radish-Sort 11/11/ :01 AM Quick-Sort     2 9  9
Objectives Introduce different known sorting algorithms
Quick-Sort 11/14/2018 2:17 PM Chapter 4: Sorting    7 9
Quick-Sort 11/19/ :46 AM Chapter 4: Sorting    7 9
Merge Sort 12/4/ :32 AM Merge Sort 7 2   7  2   4  4 9
1 Lecture 13 CS2013.
Sorting Recap & More about Sorting
Sorting, Sets and Selection
Quick-Sort 2/23/2019 1:48 AM Chapter 4: Sorting    7 9
Merge Sort 2/23/ :15 PM Merge Sort 7 2   7  2   4  4 9
Merge Sort 2/24/2019 6:07 PM Merge Sort 7 2   7  2  2 7
Copyright © Aiman Hanna All rights reserved
Copyright © Aiman Hanna All rights reserved
Quick-Sort 4/8/ :20 AM Quick-Sort     2 9  9
Merge Sort 4/10/ :25 AM Merge Sort 7 2   7  2   4  4 9
Quick-Sort 4/25/2019 8:10 AM Quick-Sort     2
Divide & Conquer Sorting
Quick-Sort 5/7/2019 6:43 PM Selection Selection.
Quick-Sort 5/25/2019 6:16 PM Selection Selection.
Merge Sort 5/30/2019 7:52 AM Merge Sort 7 2   7  2  2 7
CS203 Lecture 15.
Divide-and-Conquer 7 2  9 4   2   4   7
Chapter 11 Sets, and Selection
Presentation transcript:

Quick-Sort 2/25/2019 2:22 AM Quick-Sort 7 4 9 6 2  2 4 6 7 9 4 2  2 4 7 9  7 9 2  2 9  9 Quick-Sort

Outline and Reading Quick-sort (§4.3) Analysis of quick-sort (4.3.1) Algorithm Partition step Quick-sort tree Execution example Analysis of quick-sort (4.3.1) In-place quick-sort (§4.8) Summary of sorting algorithms Quick-Sort

Quick-Sort Quick-sort is a randomized sorting algorithm based on the divide-and-conquer paradigm: Divide: pick a random element x (called pivot) and partition S into L elements less than x E elements equal x G elements greater than x Recur: sort L and G Conquer: join L, E and G x x L G E x Quick-Sort

Partition We partition an input sequence as follows: We remove, in turn, each element y from S and We insert y into L, E or G, depending on the result of the comparison with the pivot x Each insertion and removal is at the beginning or at the end of a sequence, and hence takes O(1) time Thus, the partition step of quick-sort takes O(n) time Algorithm partition(S, p) Input sequence S, position p of pivot Output subsequences L, E, G of the elements of S less than, equal to, or greater than the pivot, resp. L, E, G  empty sequences x  S.remove(p) while S.isEmpty() y  S.remove(S.first()) if y < x L.insertLast(y) else if y = x E.insertLast(y) else { y > x } G.insertLast(y) return L, E, G Quick-Sort

Quick-Sort Tree An execution of quick-sort is depicted by a binary tree Each node represents a recursive call of quick-sort and stores Unsorted sequence before the execution and its pivot Sorted sequence at the end of the execution The root is the initial call The leaves are calls on subsequences of size 0 or 1 7 4 9 6 2  2 4 6 7 9 4 2  2 4 7 9  7 9 2  2 9  9 Quick-Sort

Execution Example Pivot selection 7 2 9 4 3 7 6 1  1 2 3 4 6 7 8 9 7 2 9 4 3 7 6 1  1 2 3 4 6 7 8 9 7 2 9 4  2 4 7 9 3 8 6 1  1 3 8 6 2  2 9 4  4 9 3  3 8  8 9  9 4  4 Quick-Sort

Execution Example (cont.) Partition, recursive call, pivot selection 7 2 9 4 3 7 6 1  1 2 3 4 6 7 8 9 2 4 3 1  2 4 7 9 3 8 6 1  1 3 8 6 2  2 9 4  4 9 3  3 8  8 9  9 4  4 Quick-Sort

Execution Example (cont.) Partition, recursive call, base case 7 2 9 4 3 7 6 1   1 2 3 4 6 7 8 9 2 4 3 1  2 4 7 3 8 6 1  1 3 8 6 1  1 9 4  4 9 3  3 8  8 9  9 4  4 Quick-Sort

Execution Example (cont.) Recursive call, …, base case, join 7 2 9 4 3 7 6 1  1 2 3 4 6 7 8 9 2 4 3 1  1 2 3 4 3 8 6 1  1 3 8 6 1  1 4 3  3 4 3  3 8  8 9  9 4  4 Quick-Sort

Execution Example (cont.) Recursive call, pivot selection 7 2 9 4 3 7 6 1  1 2 3 4 6 7 8 9 2 4 3 1  1 2 3 4 7 9 7 1  1 3 8 6 1  1 4 3  3 4 8  8 9  9 9  9 4  4 Quick-Sort

Execution Example (cont.) Partition, …, recursive call, base case 7 2 9 4 3 7 6 1  1 2 3 4 6 7 8 9 2 4 3 1  1 2 3 4 7 9 7 1  1 3 8 6 1  1 4 3  3 4 8  8 9  9 9  9 4  4 Quick-Sort

Execution Example (cont.) Join, join 7 2 9 4 3 7 6 1  1 2 3 4 6 7 7 9 2 4 3 1  1 2 3 4 7 9 7  17 7 9 1  1 4 3  3 4 8  8 9  9 9  9 4  4 Quick-Sort

Worst-case Running Time The worst case for quick-sort occurs when the pivot is the unique minimum or maximum element One of L and G has size n - 1 and the other has size 0 The running time is proportional to the sum n + (n - 1) + … + 2 + 1 Thus, the worst-case running time of quick-sort is O(n2) depth time n 1 n - 1 … … Quick-Sort

Expected Running Time Consider a recursive call of quick-sort on a sequence of size s Good call: the sizes of L and G are each less than 3s/4 Bad call: one of L and G has size greater than 3s/4 A call is good with probability 1/2 1/2 of the possible pivots cause good calls: 7 2 9 4 3 7 6 1 9 7 2 9 4 3 7 6 1 2 4 3 1 7 9 7 1  1 1 7 2 9 4 3 7 6 Good call Bad call 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Bad pivots Good pivots Bad pivots Quick-Sort

Expected Running Time, Part 2 Probabilistic Fact: The expected number of coin tosses required in order to get k heads is 2k For a node of depth i, we expect i/2 ancestors are good calls The size of the input sequence for the current call is at most (3/4)i/2n Therefore, we have For a node of depth 2log4/3n, the expected input size is one The expected height of the quick-sort tree is O(log n) The amount or work done at the nodes of the same depth is O(n) Thus, the expected running time of quick-sort is O(n log n) Quick-Sort

In-Place Quick-Sort Quick-sort can be implemented to run in-place In the partition step, we use replace operations to rearrange the elements of the input sequence such that the elements less than the pivot have rank less than h the elements equal to the pivot have rank between h and k the elements greater than the pivot have rank greater than k The recursive calls consider elements with rank less than h elements with rank greater than k Algorithm inPlaceQuickSort(S, l, r) Input sequence S, ranks l and r Output sequence S with the elements of rank between l and r rearranged in increasing order if l  r return i  a random integer between l and r x  S.elemAtRank(i) (h, k)  inPlacePartition(x) inPlaceQuickSort(S, l, h - 1) inPlaceQuickSort(S, k + 1, r) Quick-Sort

In-Place Partitioning Perform the partition using two indices to split S into L and E U G (a similar method can split E U G into E and G). Repeat until j and k cross: Scan j to the right until finding an element > x. Scan k to the left until finding an element < x. Swap elements at indices j and k j k 3 2 5 1 0 7 3 5 9 2 7 9 8 9 7 6 9 (pivot = 6) j k 3 2 5 1 0 7 3 5 9 2 7 9 8 9 7 6 9 Quick-Sort

Summary of Sorting Algorithms Time Notes selection-sort O(n2) in-place slow (good for small inputs) insertion-sort quick-sort O(n log n) expected in-place, randomized fastest (good for large inputs) heap-sort O(n log n) fast (good for large inputs) merge-sort sequential data access fast (good for huge inputs) Quick-Sort