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 18.2, Problem 7E
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To explain theminimization of the B-tree search time and also the minimum value of t when
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Iterative lengthening search is an iterative analogue of uniform-cost search. The basic idea is to use increasing limits on path cost. If a node is generated whose path cost exceeds the current limit, it is immediately discarded. For each new iteration, the limit is set to the lowest path cost of any node discarded in the previous iteration. (a) Show that this algorithm is optimal for general path costs. You may assume that all costs are integers (this is not a loss of generality if the search space is finite). You may wish to consider the minimal path cost C; what happens when we set the path cost to be some limit l < C? (b) Consider a uniform tree with branching factor b, solution depth d, and unit step costs (each action costs one unit). How many iterations will iterative lengthening require? (c) (7 points) Now consider the…
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- I have spent hours on internet to find an example of iterative lengthening search algorithm but found nothing. I need it ASAP. Can anyone help me? Please. It is its definition: Iterative lengthening search is an iterative analogue of uniform-cost search. The basic idea is to use increasing limits on path cost. If a node is generated whose path cost exceeds the current limit, it is immediately discarded. For each new iteration, the limit is set to the lowest path cost of any node discarded in the previous iteration. I dont understand many things in this definition. 1-What is the first limit before we start? 2-Do we delete the discarded node from the graph or what? 3-What is the generated node? Is it expanded or is it just found as neigbor? 4-What is "lowest path cost of any node discarded"?? Path from where to where?? From start to that node or what? 5-What does path cost mean in this problem? Is it from source to the new node or between current node and next chosen node? I am literally…arrow_forwardConsider a Skip List with n nodes. Each node k has a random "height" hg, which is the number of linked lists that includes node k. The heights nodes are statistically independent, where P[hx = m] = 2-m, for m 2 1 and P[h = m] = 0 , for m m] 0 such that P[H > m] <= for m = c log, narrow_forwardGiven three groups of boxes A, B, and C of n boxes each, where the shapes of the boxes are different. The capacity of each box is measured in milliliter (ml). The list of boxes' capacities in Group A is exactly randomly repeated in Group B and C. This means that for each box in Group A there exist a corresponding box in Group B and C that hold the same capacity, but we do not know which box would match with the other in group A, B and C. Your mission is to find a smart way to match these boxes. For example: Group A Input: capacity in milliliter Group 1 A B C 140 170 120 2 120 150 90 Output: A[1] with B[3] with C[6] A[2] with B[6] with C[1] A[3] with B[2] with C[4] ... and so on 3 150 140 200 Group B 4 100 90 150 5 170 100 180 6 200 120 140 7 7 90 180 100 Group C c) Implement your efficient algorithm using Python d) Prepare a brief report (250 words) comparing the two algorithms 8 180 200 170 a) Using a brute-force approach, design an algorithm to solve this problem, and analyze its…arrow_forward
- Dijkstra’s algorithm computes the least-cost path from one node (the source, which we will refer to as ?u) to all other nodes in the network. Dijkstra’s algorithm is iterative and has the property that after the k-th iteration of the algorithm, the least-cost paths are known to k destination nodes, and among the least-cost paths to all destination nodes, these k paths will have the k smallest costs. Let - D(v): cost of the least-cost path from the source node to destination ?v as of this iteration of the algorithm. - p(v): previous node (neighbor of v) along the current least-cost path from the source to v. - N′: a subset of nodes; v is in N′ if the least-cost path from the source to ?v is definitely known. The centralized routing algorithm consists of an initialization step followed by a loop. The number of times the loop is executed is equal to the number of nodes in the network. Upon termination, the algorithm will have calculated the shortest paths from the source…arrow_forwardNeed help on this problem. One important application of the Longest Common Subsequences problem is to compare DNA sequences. consider the following two small DNA sequences: X = TAGTC and Y = AGACT. we use Dynamic programming to find the LCS of X and y.arrow_forwardGiven algorithm to count least common multiplicator (LCM) from two real numbers a and b. For example, lcm(10,12) will return 60. lcm(a, b): result = a; while(result mod b != 0):result = result + a;endwhile return result Show that the algorithm is correct using loop invariant. What is the loop invariant? Determine correctness in initialization, maintenance and its effects on termination.arrow_forward
- Given a jungle matrix NxM: jungle = [ [1, 0, 0, 0], [1, 1, 0, 1], [0, 1, 0, 0], [1, 1, 1, 1,] ] Where 0 means the block is dead end and 1 means the block can be used in the path from source to destination. Task: Starting at position (0, 0), the goal is to reach position (N-1, M-1). Your program needs to build and output the solution matrix – a 4x4 matrix with 1’s in positions used to get from the starting position (0,0) to the ending position (N-1,M-1) with the following constraints: You can only move one space at a time You can only in two directions: forward and down. You can only pass thru spaces on the jungle matrix mark ' 1 ' If you cannot reach the ending position - print a message that you're trapped in the jungle Algorithm: If destination is reached print the solution matrix Else Mark current cell in the solution matrix Move forward horizontally and recursively check if this leads to a solution If there is no solution, move down and recursively check if this leads to…arrow_forwardIn the sequence alignment problem, suppose you knew that the input strings were relatively similar, in the sense that there is an optimal alignment that uses at most k gaps, where k is much smaller than the lengths m and n of the strings. Show how to compute the NW score in O((m+n)k) time.arrow_forwardIn this question, we consider the operation of the Ford-Fulkerson algorithm on the network shown overleaf: 0/16 0/12 0/8 0/4 0/8 0/5 19 0/11 174 0/13 0/14 0/2 0/11 0/10 Each edge is annotated with the current flow (initially zero) and the edge's capacity. In general, a flow of x along an edge with capacity y is shown as x/y. (a) Show the residual graph that will be created from this network with the given (empty) flow. In drawing a residual graph, to show a forward edge with capacity x and a backward edge with capacity y, annotate the original edge *;ỹ. (b) What is the bottleneck edge of the path (s.₁,vs,t) in the residual graph you have given in answer to part (a)?arrow_forward
- Consider the LCS problem discussed in class. Suppose we only want to compute the length of the LCS of x[1..m] and y[1..n] (i.e. we do not care about the actual subsequence itself). This means we do not need to “traceback” to find the LCS. In this setting, show how the algorithm can be altered so that it only needs min(m,n) + O(1) space rather than m · n space (note that it must be min(m,n) and not O(min(m, n))).arrow_forwardA subsequence of a sequence is obtained by deleting zero or more values in the sequence. For example, acdfh is a subsequence of abcdefghi. The order must be preserved. The longest common subsequence problem takes two strings as input and outputs the length of the longest string that is a subsequence of both input strings. Describe an efficient dynamic programming algorithm for the longest common subsequence problem. Start by writing recursive equations using the shop around principle. Concentrate on the last values in the input strings. If they are the same, keep them and look for a longest common subsequence of the two strings with their last characters removed. If the last two characters are different, you can only keep one of them. Try keeping one, and throwing the other one away. Then keep the other one. Take a maximum, since you are looking for a longest common subsequence. Then implement the equations using memoization.arrow_forwardGiven an undirected weighted graph G with n nodes and m edges, and we have used Prim’s algorithm to construct a minimum spanning tree T. Suppose the weight of one of the tree edge ((u, v) ∈ T) is changed from w to w′, design an algorithm to verify whether T is still a minimum spanning tree. Your algorithm should run in O(m) time, and explain why your algorithm is correct. You can assume all the weights are distinct. (Hint: When an edge is removed, nodes of T will break into two groups. Which edge should we choose in the cut of these two groups?)arrow_forward
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