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 15.5, Problem 4E
Program Plan Intro
To modify the procedure of OPTIMAL-BSTusing Knuth statement so it runs the
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Which of the following is true?
Lütfen birini seçin:
O A. In a binary search tree (BST), the minimum element greater
than a given element can be found in O(1) time.
O B. If a dynamic programming problem provides optimal
substructure property, then local optimal solution is global
optimal
C. Merge Sort sorts an array in place.
OD. Minimum Spanning Tree ( MST ) is a tree with the minimum
cost and connects all the vertices together.
O E. Dijkstra's algorithm is a divide and conquer algorithm.
Run Skiena’s algorithm by hand to solve the Partial Digest Problem for
? = {1,2,3,3,4,4,5,6,7,9}
to find a set ? such that ∆? = ?. Assume that the smallest element of ? is 0
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…
Chapter 15 Solutions
Introduction to Algorithms
Ch. 15.1 - Prob. 1ECh. 15.1 - Prob. 2ECh. 15.1 - Prob. 3ECh. 15.1 - Prob. 4ECh. 15.1 - Prob. 5ECh. 15.2 - Prob. 1ECh. 15.2 - Prob. 2ECh. 15.2 - Prob. 3ECh. 15.2 - Prob. 4ECh. 15.2 - Prob. 5E
Ch. 15.2 - Prob. 6ECh. 15.3 - Prob. 1ECh. 15.3 - Prob. 2ECh. 15.3 - Prob. 3ECh. 15.3 - Prob. 4ECh. 15.3 - Prob. 5ECh. 15.3 - Prob. 6ECh. 15.4 - Prob. 1ECh. 15.4 - Prob. 2ECh. 15.4 - Prob. 3ECh. 15.4 - Prob. 4ECh. 15.4 - Prob. 5ECh. 15.4 - Prob. 6ECh. 15.5 - Prob. 1ECh. 15.5 - Prob. 2ECh. 15.5 - Prob. 3ECh. 15.5 - Prob. 4ECh. 15 - Prob. 1PCh. 15 - Prob. 2PCh. 15 - Prob. 3PCh. 15 - Prob. 4PCh. 15 - Prob. 5PCh. 15 - Prob. 6PCh. 15 - Prob. 7PCh. 15 - Prob. 8PCh. 15 - Prob. 9PCh. 15 - Prob. 10PCh. 15 - Prob. 11PCh. 15 - Prob. 12P
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- 3. (i) Solve the following recurrence relation by expansion (substitution): for n ≥ 2, n a power of 2 T(n) = 8T() + n², T(1) = 1 (ii) Express T(n) in order, i.e., T(n) = (f(n)) for n ≥ 1, n a power of 2. (iii) Check your solution by plugging it back into the recurrence relation.arrow_forwardProblem 4: Let S = {s1, s2, . . . , sn} and T = {t1, t2, . . . , tm}, n ≤ m, be two sets of integers. (i) Describe a deterministic algorithm that checks whether S is a subset of T. What is the running time of your algorithm? (ii) Devise an algorithm that uses a hash table of length n to test whether S is a subset of T. What is the expected running time of your algorithm?arrow_forwardProblem 3. Consider the following recurrence. T(n) = {(n) = 37(n T(n) = 3T(n/2) + n² if n=1 otherwise. (a) Solve this recurrence exactly by the method of substitution. You may assume n is a power of 2. (b) Solve it using the recursion tree method.arrow_forward
- Problem 5: Let S be a set of n positive integers. (i) Design an O(n log n) algorithm to verify that: (VTS)(Σt ≥ |T|³). tET In other words, if there is some subset TCS such that the sum of the elements in T is less than |T|³, then the algorithm should return FALSE. Otherwise, it should return TRUE. (ii) In addition to S, you are given an integer 1 ≤ k ≤n. Design an O(n) algorithm to verify that: (VTS)(|T| = k→Σt ≥ |T|³). tETarrow_forward3. Consider the following algorithm, called sorted greedy makespan algorithm. On input k and a list t1, ..., tm, it sorts the lengths of the tasks non-increasingly and applies the greedy makespan algorithm. We can combine 3 true statements in the above list to prove that this algorithm is a p-approximation for some p > 1. What is the smallest p that you get as a function of k?arrow_forwardProblem 1. Question: Give an efficient greedy algorithm that finds an optimal vertex cover for a tree in linear time. Give pseudocode, and try to argue that your algorithm can be implemented in linear time. Remarks: A tree with n≥2 vertices always has a "leaf", a vertex touching one edge. Think about a greedy rule for covering that edge. There is even a linear time "dynamic programming" algorithm that can handle weighted vertices, but you don't have to do that.arrow_forward
- Write the algorithm that finds and returns how many paths in k units of length between any given two nodes (source node, destination node; source and target nodes can also be the same) in a non-directional and unweighted line of N nodes represented as a neighborhood matrix. (Assume that each side in the unweighted diagram is one unit long.) Note: By using the problem reduction method of the Transform and Conquer strategy, you have to make the given problem into another problem. Algorithm howManyPath (M [0..N-1] [0..N-1], source, target, k)// Input: NxN neighborhood matrix, source, target nodes, k value.// Ouput: In the given line, there are how many different paths of k units length between the given source and target node.arrow_forwardProve that(Generic shortest-paths algorithm) Proposition Q Set distTo[s] to 0 and all other distTo[] values to infinity, then do the following:Continue to relax any edge in G until no edge is eligible.The value of distTo[w] after this computation is the length of the shortest path from s to w (and the value of edgeTo[] is the last edge on that path) for all vertices w reachable from s.arrow_forward11. Given is a sequence X of n keys k1, k2, ..., kin. For each key k; (1 < i< n), its position in sorted order differs from i by at most d. Present an O(n log d)-time sequential algorithm to sort X. Prove the correctness of your algorithm using the zero-one principle. Implement the same algorithm on a yn x Vn mesh. What is the resultant run time?arrow_forward
- Problem 4: Let S = {s1, s2, . . . , sn} and T = {t1, t2, . . . , tm}, n ≤ m, be two sets of integers. (ii) Devise an algorithm that uses a hash table of length n to test whether S is a subset of T. What is the expected running time of your algorithm?arrow_forwardConsider the following algorithm for the maximum cut problem, based on the technique of local search. Given a partition of V into sets, the basic step of the algorithm, called flip, is that of moving a vertex from one side of the partition to the other. The following algorithm finds a locally optimal solution under the flip operation, i.e., a solution which cannot be improved by a single flip. The algorithm starts with an arbitrary partition of V. While there is a vertex such that flipping it increases the size of the cut, the algorithm flips such a vertex. (Observe that a vertex qualifies for a flip if it has more neighbors in its own partition than in the other side.) The algorithm terminates when no vertex qualifies for a flip. Show that this algorithm terminates in polynomial time, and achieves an approximation guarantee of 1/2.arrow_forwardIn the minimum spanning tree problem formulation, if n is the total number of nodes, and the edges between any two nodes i and j is rij- The variables Tij are defined in the model as: Select one: O Continous O Integer O None of the mentioned Binary O Integer or Continousarrow_forward
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