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 25.1, Problem 8E
Program Plan Intro
To modify the fastest all pair shortest path
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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 :81
Given a matrix of size N x M where N is the number of rows and M is the number of columns, write an algorithm to find the shortest path from the top-left cell to the bottom-right cell that passes through all the cells with a prime value and avoids cells with composite values. The algorithm should have a time complexity of O(NM log(max(N,M))).
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++
Chapter 25 Solutions
Introduction to Algorithms
Ch. 25.1 - Prob. 1ECh. 25.1 - Prob. 2ECh. 25.1 - Prob. 3ECh. 25.1 - Prob. 4ECh. 25.1 - Prob. 5ECh. 25.1 - Prob. 6ECh. 25.1 - Prob. 7ECh. 25.1 - Prob. 8ECh. 25.1 - Prob. 9ECh. 25.1 - Prob. 10E
Ch. 25.2 - Prob. 1ECh. 25.2 - Prob. 2ECh. 25.2 - Prob. 3ECh. 25.2 - Prob. 4ECh. 25.2 - Prob. 5ECh. 25.2 - Prob. 6ECh. 25.2 - Prob. 7ECh. 25.2 - Prob. 8ECh. 25.2 - Prob. 9ECh. 25.3 - Prob. 1ECh. 25.3 - Prob. 2ECh. 25.3 - Prob. 3ECh. 25.3 - Prob. 4ECh. 25.3 - Prob. 5ECh. 25.3 - Prob. 6ECh. 25 - Prob. 1PCh. 25 - Prob. 2P
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- Give a MPI program segment to convert a n-by-n matrix distributed on a n-by-n 2D mesh such that allits rows and columns get sorted in ascending order. Show only the iterative loop. Only communicationsallowed are to the four direct neighbors. (Hint: You may employ a variant of odd-even transpositionsort, but the whole matrix does not need to be sorted.)arrow_forwardSuppose that, in a divide-and-conquer algorithm, we always divide aninstance of size n of a problem into 10 subinstances of size n/3, and thedividing and combining steps take a time in Θ(n2) in n>1, and it performs one basic operation if n = 1. Write a recurrenceequation for the running time of T(n), and solve the equation.arrow_forwardFor the problem, give pseudocode for your solution, and remember to include a proof of correctness and runtime. Note that in general, faster algorithms will receive more credit, so a brute force O(n 2 ) algorithm will not get many points if there is a faster O(n) or O(n log n) alterative. In IR2 , we define a slab to be a pair of parallel lines. Given a set of points P in IR2 , find the narrowest slab containing P, where the width of the slab is the vertical distance between its bounding lines.arrow_forward
- Given a sorted array of n comparable items A, and a search value key, return the position (array index) of key in A if it is present, or -1 if it is not present. If key is present in A, your algorithm must run in order O(log k) time, where k is the location of key in A. Otherwise, if key is not present, your algorithm must run in O(log n) time.arrow_forwardWrite an algorithm called MatrixRowMultiple(A[0…n – 1, 0… n – 1]) which takes an n x n matrix A and outputs if true, if each element in the first row is twice the value of the corresponding element of row 2. Otherwise, it returns false. (ii) Determine the worst case complexity of the algorithm which you have developed.arrow_forwardGiven an n×n matrix M in which every entry is either a 0 or 1. Present an algorithm that determines if ∃i, 1 ≤ i ≤ n, such that M[i, j] = 0 and M[j, i] = 1, ∀j, 1 ≤ j ≤ n ∧ j 6= i, using examining an entry of M as the key operation. Your algorithm must examine at most 3n − ⌊lg n⌋ − 3 entries of M.arrow_forward
- Suppose we are given two sorted arrays A and B which each contain n elements. Give an O(log n) time divide-and-conquer algorithm which finds the median of A ∪ B.arrow_forwardFor a string P of length m, define a function ∆P :{1,...,m}→{1,...,m} as follows: ∆P[q] is the length of the shortest actual suffix of Pq which is also a prefix of P. If there is no actual suffix of Pq which is also a prefix of P then ∆P[q]=0. Give an efficient algorithm for calculating ∆P. (The algorithm should output the entire array ∆P .)arrow_forwardWe argued in class that if the input is not uniformally distributed, the worst-case running time of Bucket Sort (using Insertion Sort in each bucket) in O(n^2). The reason: all those numbers could fall in the same bucket, and then the time of Insertion sorting then dominates. How many out of the total n numbers could fall in one bucket and the algorithm still run in linear time if the rest are evenly distributed? (Choose one option, no points if you choose more than one.) A. O(1)B. O(√n)C. O(On)D. O(log n)E. None of the abovearrow_forward
- Suppose we are given a sequence S of n elements, each of which is an integer in the range [0, n2-1]. Describe a simle method for sorting S in O(n) time. Hint: Think of alternate ways of viewing the elements.arrow_forwardSuppose we are to sort a set of 'n' integers using an algorithm with time complexity O(n) that takes 1 ms (milliseconds) in the worst case when n = 5000. How large an element can the algorithm then handle at least 0.2 ms?arrow_forwardTo have random-access lookup, a grid should have a scheme for numbering the tiles.For example, a square grid has rows and columns, which give a natural numberingfor the tiles. Devise schemes for triangular and hexagonal grids. Use the numberingscheme to define a rule for determining the neighbourhood (i.e. adjacent tiles) of agiven tile in the grid. For example, if we have a four-connected square grid, wherethe indices are i for rows and j for columns, the neighbourhood of tile i, j can bedefined asneighbourhood(i, j) = {i ± 1, j,i, j ± 1}arrow_forward
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