CS代考 COMP 3711 − Design and Analysis of Algorithms 2022 Spring Semester − Writte

COMP 3711 − Design and Analysis of Algorithms 2022 Spring Semester − Written Assignment # 4
Distributed: April 3, 2022 Due: April 17, 2022, 11:59 PM
Your solution should contain
(i) your name, (ii) your student ID #, and (iii) your email address at the

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top of its first page. Some Notes:
• Please write clearly and briefly. In particular, your solutions should be written or printed on clean white paper with no watermarks, i.e., student society paper is not allowed.
• Please follow the guidelines on doing your own work and avoiding plagiarism as described in the class policies. You must acknowledge individuals who assisted you, or sources where you found solutions. Failure to do so will be considered plagiarism.
• Please make a copy of your assignment before submitting it. If we can¡ ̄t find your submission, we will ask you to resubmit the copy.
• Submit a SOFTCOPY of your assignment to Canvas by the deadline. The softcopy should be one PDF file (no word or jpegs permitted, nor multiple files). If your submission is a scan of a handwritten solution, make sure that it is of high enough resolution to be easily read. At least 300dpi and possible denser.

Problem 1 [25 pts] Number of Contiguous Sub-arrays with sum k
Give an O(n) algorithm for finding the number of contiguous sub-arrays of an array A[1…n] with sum equal to k. Note that a contiguous sub-array includes all the elements between two indices i and j. For example, if A = [2, 4, 6, 8, 10] the sub-array [4, 6, 8] is contiguous, while [4, 8] is not.
For example, let A = [1,1,1,2,1] and k = 3, then there are 3 contiguous sub-arrays that got a sum equal to 3, e.g. [1, 1, 1], [1, 2], and [2, 1].
Problem 2 [25 pts] Distinct subsequences
Given two strings A[1…n] and B[1…m], such that n >= m, propose an al- gorithm to find the number of distinct subsequences that can be formed from A such that they are identical to B.
For example, let A = caaat and B = cat, then the answer is 3, since the subsequences ca t, c a t, and c at are identical to B.
Problem 3 [25 pts] Unique Paths
You are given a binary matrix of size m × n, where 1s indicate objects and 0s indicate empty cells. The matrix is like a maze, meaning that we can move to empty cells (0s) but cannot move on the objects (1s). Assuming that the starting position is at (1,1) top-left corner, the target position is at (m,n) bottom-right corner and you can move only right (x, y + 1) or down (x + 1, y), propose an algorithm that finds all the unique paths from the start to the end cell.
For example, given the matrix 0 1 0 the number of unique paths are 2.
Either go right twice (→) and then down twice (↓) or go down twice (↓) and
then right twice (→).
Problem 4 [25 pts] Change Permutations
Given an integer W and a set of coins C, propose an algorithm to find the number of ways the coins can sum to W assuming you have an infinite number of each one of the coins.
For example, let C = [1,2,4] and W = 4, the result is 4, because there are 4 different ways such the sum of the coins is equal to W = 4: {1,1,1,1}, {1, 1, 2}, {2, 2}, and {4}.

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