Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter 9.3, Problem 6E
Program Plan Intro
To describe an
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Suppose we are given two n-element sorted sequences A and B that should not be viewed as sets (that is, A and B may contain duplicate entries). Describe an O(n)-time method (algorithm) for computing a sequence representing the set A È B (with no duplicates).
Consider median-3 quick sort and bubble sort. When using these two algorithms to sort a list of n elements that have already been sorted in increasing order, which of the following is true for the worst-case time complexity?
illustrates the procedure for Fibonacci search. Here n, thenumber of data elements is such that:
i) Fk+1 > (n+1); andii) Fk + m = (n +1) for some m ≥ 0, where Fk+1 and Fk are two consecutiveFibonacci numbers.
Chapter 9 Solutions
Introduction to Algorithms
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- The median m of a sequence of n elements is the element that would fall in the middle if the sequence was sorted. That is, e ≤ m for half the elements, and m ≤ e for the others. Clearly, one can obtain the median by sorting the sequence, but one can do quite a bit better with the following algorithm that finds the kth element of a sequence between a (inclusive) and b (exclusive). (For the median, use k = n/2, a = 0, and b = n.) select(k, a, b)Pick a pivot p in the subsequence between a and b.Partition the subsequence elements into three subsequences: the elements <p, =p, >p Let n1, n2, n3 be the sizes of each of these subsequences.if k < n1 return select(k, 0, n1).else if (k > n1 + n2) return select(k, n1 + n2, n).else return p. c++arrow_forwardSuppose we have the following sorted set of positive integers: 1 2 4 4 6 7 9 9 15 18 22. Also assume that we use linear search (starting from the left as normal) and searching for element 10. How many comparisons are needed at least then before we come to the conclusion that the element is not in the set? 10 true 8 true 9 truearrow_forwardIf a binary search algorithm is applied to a sorted array of 1000 elements, how many comparisons will be needed in the worst case scenario to find an element?arrow_forward
- An array of n elements is almost sorted if and only if every element is at most k spots away from its actual location. Assuming that you can only perform pairwise comparisons, formally prove that any algorithm which can sort almost sorted arrays must have running time S2(n log k), You may assume that n is a multiple of k.arrow_forwardSuppose that L is a sorted list of 4,096 elements. What is the maximum number of comparisons made by binary search to determine if an item is in L?arrow_forward(a) Carry out the Select algorithm on the following set, using k = 19 (return the element of rank 19). Show your steps! Ensure that you run the partition algorithm in such a way that you maintain theelements in their original relative order.56; 78; 34; 19; 67; 32; 13; 12; 90; 92; 50; 51; 30; 1; 99; 58; 43; 42; 24; 65; 21; 25; 68; 69; 101 (b) Carry out the Randomized Select algorithm from class on the set of elements from part (a), usingk = 19. Show your steps! When selecting a random pivot, suppose you always chose the last element inthe array. Ensure that you run the partition algorithm in such a way that you maintain the elements intheir original relative orderarrow_forward
- Using an unsorted array to implement a map would allow an O(log2N) get operation (where N is the number of elements in the map). Group of answer choices True False also explain why?arrow_forwardQuickSort is run for an array A in a manner that PARTITION consistently produces a 5:1 split for the (sub)arrays to be sorted (recursively) next. Picture the recurrence tree for this scenario ... What will be the length of the longest path from the root to a leaf (= deepest depth)in this tree? Group of answer choices log_5/6 (n) ... "log of n with base 5/6" log_6/5 (n) ... "log of n with base 6/5" log_6 (n) ... "log n with base 6" n^log_5(6) ... "n to the power of log_5(6)"arrow_forwardMerge sort algorithm is ________, the worst case time complexity of sorting an array containing N elements with merge sort is . Group of answer choices unstable, O(NlogN) stable, O(NlogN) stable, O(N2) unstable, O(N2)arrow_forward
- Given 2 sorted arrays (in increasing order), find a path through the intersection that produces the maximum sum and return the maximum sum. That is, we can switch from one array to another array only atcommon elements. If no intersection element is present, we need to take the sum of all elements from the array with greater sum. Sample Input:61 5 10 15 20 2552 4 5 9 15Sample Output :81arrow_forwardSuppose you wanted to use a linear search on a sorted data set. What would be the average number of comparisons to find a given element? Why?arrow_forwardWrite a bottom-up mergesort that makes use of the array's order by carrying out the following steps each time it needs to locate two arrays to merge: locate the first element in an array that is smaller than its predecessor, then locate the next, and finally merge them to form a sorted subarray. Consider the array size and the number of maximal ascending sequences in the array while analysing the running time of this method.arrow_forward
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