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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Question
Chapter 25, Problem 1P
(a)
Program Plan Intro
To show that the updation procedure of transitive closer G * ( V , E *) of graph G ( V, E ) will take O ( V2 ) time, while an edge is added to the graph G .
(a)
Expert Solution
Explanation of Solution
Here, in the graph G , if the updation of transitive closure takes O ( V2 ) time. To understand this scenario, suppose, addition of an edge ( x1 , x2 ) is performed in the graph G . Now, take the set of vertices ( u, v ) in order to get the path from u to x1 and x2 to v . If there is a possibility of having an edge that contains vertices in such manner like ( u , x1 ) and ( x2 , v ).
Therefore, perform addition of an edge ( u, v ) in transitive closure only if the closure holds the edges in ( u, x1 ) and ( x2, v ) manner. So, the consideration of pair required only once and the total run time of this procedure will be O ( V2 ).
(b)
Program Plan Intro
To give an example that the update of transitive closer will take O ( V2 ) time, when a new edge ‘ e’ is added to the graph G .
(b)
Expert Solution
Explanation of Solution
Consider the condition where there aretwo strongly connected components with sizes | V |/2 and there is no common edge between them. The transitive closure can be computed by adding these two connected components of the graph.
Now, perform addition of a single edge between two connected components that provide connectivity between two separate components of the graph. Here, it is clearly visible that the total no. of edges will be increased by | V |/4 and the total no of edges will be [| V |/2 + | V |/4]. So, every time while adding a new edge required a constant time at least.
Therefore, the update of transitive closer will take O ( V2 ) time, when a new edge ‘ e’ is added to the graph G .
(c)
Program Plan Intro
To define an
(c)
Expert Solution
Explanation of Solution
Consider a set of vertices in the graphG , and there is a path between every pair of vertices. Now, while performing an addition of an edge ( u, v ), look at the ancestor of vertex u and add it there. The reason of looking at the ancestor of an edge is to explore the branches of the tree. This procedure will also applicable for the vertex v. through this, it will be very easy to get the already added edges and will only consider those edges in n times mostly.
Since, the edges in the tree required O ( V2 ) time for consideration. Therefore, the total time taken by the algorithm will be O ( V3 ).
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Students have asked these similar questions
I need help to solve a simple problem using Grover’s algorithm, where the solution is not necessarily known beforehand. The problem is a 2×2 binary sudoku with two rules:
• No column may contain the same value twice.
• No row may contain the same value twice.
Each square in the sudoku is assigned to a variable as follows:
We want to design a quantum circuit that outputs a valid solution to this sudoku. While using Grover’s algorithm for this task is not necessarily practical, the goal is to demonstrate how classical decision problems can be converted into oracles for Grover’s algorithm.
Turning the Problem into a Circuit
To solve this, an oracle needs to be created that helps identify valid solutions. The first step is to construct a classical function within a quantum circuit that checks whether a given state satisfies the sudoku rules.
Since we need to check both columns and rows, there are four conditions to verify:
v0 ≠ v1 # Check top row
v2 ≠ v3 # Check bottom row…
using r language
I need help to solve a simple problem using Grover’s algorithm, where the solution is not necessarily known beforehand. The problem is a 2×2 binary sudoku with two rules:
• No column may contain the same value twice.
• No row may contain the same value twice.
Each square in the sudoku is assigned to a variable as follows:
We want to design a quantum circuit that outputs a valid solution to this sudoku. While using Grover’s algorithm for this task is not necessarily practical, the goal is to demonstrate how classical decision problems can be converted into oracles for Grover’s algorithm.
Turning the Problem into a Circuit
To solve this, an oracle needs to be created that helps identify valid solutions. The first step is to construct a classical function within a quantum circuit that checks whether a given state satisfies the sudoku rules.
Since we need to check both columns and rows, there are four conditions to verify:
v0 ≠ v1 # Check top row
v2 ≠ v3 # Check bottom row…
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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