COMP 352 FALL Tutorial 10

Transcription

1 1 COMP 352 FALL 2016 Tutorial 10

2 SESSION OUTLINE Divide-and-Conquer Method Sort Algorithm Properties Quick Overview on Sorting Algorithms Merge Sort Quick Sort Bucket Sort Radix Sort Problem Solving 2

3 DIVIDE-AND-CONQUER Divide and Conquer is a method of algorithm design. This method has three distinct steps: Divide: Divide the data into two or more disjoint subsets. Recur: Use divide and conquer to solve the subproblems associated with the subsets. Conquer: combine the subproblem solutions into a solution for the original problem. 3

4 SORT ALGORITHM PROPERTIES IN-PLACE SORT A sorting algorithm is in-place if it uses no auxiliary data structures (however, O(1) auxiliary variables are allowed) it updates the input sequence only by means of operations replaceelement and swapelements 4

5 SORT ALGORITHM PROPERTIES STABLE SORT A sorting algorithm where the order of elements having the same key is not changed in the final sequence 5

6 MERGE-SORT ALGORITHM Divide: If S has at least two elements (nothing needs to be done if S has zero or one elements), remove all the elements from S and put them into two sequences, S 1 and S 2, each containing about half of the elements of S. (i.e. S 1 contains the first n/2 elements and S 2 contains the remaining n/2 elements. Recur: Recursive sort sequences S 1 and S 2. Conquer: Put back the elements into S by merging the sorted sequences S 1 and S 2 into a unique sorted sequence. 6

7 MERGE-SORT EXAMPLE 7

8 RUNNING TIME OF MERGE-SORT At each level in the binary tree created for Merge-Sort O(n) time is spent The height of the tree is O(log n) Therefore, the time complexity is O(nlog n) 8

9 QUICK-SORT Divide : If the sequence S has 2 or more elements, select an element x from S to be your pivot. Any arbitrary element, like the last, will do. Remove all the elements of S and divide them into 3 sequences: L, holds S s elements less than x E, holds S s elements equal to x G, holds S s elements greater than x Recurse: Recursively sort L and G Conquer: Finally, to put elements back into S in order, first inserts the elements of L, then those of E, and those of G. 9

10 IDEA OF QUICK SORT 1) Select: pick an element 2) Divide: rearrange elements so that x goes to its final position E 3) Recur and Conquer: recursively sort 10

11 QUICK-SORT TREE 11

12 QUICK SORT RUNNING TIME Worst case: when the pivot does not divide the sequence in two At each step, the length of the sequence is only reduced by 1 Total running time 1 i n length ( S General case: Time spent at level i in the tree is O(n) Running time: O(n) * O(height) Average case: O(n log n) i ) O( n 2 ) 12

13 BUCKET SORT Assumption: the keys are in the range [0, N) Basic idea: Create N linked lists (buckets) to divide interval [0,N) into subintervals of size 1 Add each input element to appropriate bucket Concatenate the buckets Expected total time is O(n + N), with n = size of original sequence if N is O(n) sorting algorithm in O(n) 13

14 BUCKET SORT Each element of the array is put in one of the N buckets 14

15 RADIX SORT Assumption: input has d digits ranging from 0 to k Basic idea: Sort elements by digit starting with least significant Use a stable sort (like bucket sort) for each stage 15

16 RADIX SORT In general, radix sort based on bucket sort. 16

17 PROBLEM SOLVING PROBLEM 1 A company wants to offer a free CD to its customers. However, it wishes to offer 1 CD per household. Customers of the same household are defined as those that share the same address. Customers are kept in an unsorted linked list, where each element contains a name and an address. Describe an efficient algorithm, in pseudocode or in English, to prune the company s customers list. Make sure that you are doing better than the naïve, inefficient O(n^2) method that compares every possible pair of records. 17

18 PROBLEM SOLVING - PROBLEM 2 You are given a set of n real numbers and another real number x. Describe an O(nlogn) time algorithm that determines whether or not there exists 2 elements in S whose sum is exactly x. 18

19 PROBLEM SOLVING - PROBLEM 3: C Suppose we are given an n-element sequence S such that each element in S represents a different vote for president, where each vote is given as an integer representing a particular candidate. Design an O(nlogn) time algorithm to see who wins the election S represents, assuming the candidate with the most votes wins (even if there are O(n) candidates). 19

20 PROBLEM SOLVING - PROBLEM 4: C Given an array A of n entries with keys equal to 0 or 1, describe an in-place method for ordering A so that all the 0's are before every 1. 20

21 PROBLEM SOLVING - PROBLEM 5 Given an array A[1...n] of integers, give O(nlogn) algorithm to find the most commonly occurring element. If there is more than one such element, you should return the larger element. For example, if the array is [1,9,5,9,4,7,9], then you should return 9 since it is the only element that appears twice. If the array is [9,5,6,5,3,9,15], both 5 and 9 appear twice, but since 9>5, you should return 9. You must give a pseudocode for your algorithm as well as explain why it is O(nlogn). 21