Check the attached three problems.

**Subject Computer Science Data Structures and Algorithms**

Check the attached three problems.

This material may consist of step-by-step explanations on how to solve a problem or examples of proper writing, including the use of citations, references, bibliographies, and formatting. This material is made available for the sole purpose of studying and learning - misuse is strictly forbidden.

Problem 34.5-1

The first part is to prove that our problem (subgraph-isomorphism) is in NP class.

For an instance (G1,G2) it can be considered a map from the vertices of G1 to those of G2. This will show which vertices of the graph G2 correspond to nodes of G1. So if f: V1->V2 is the mapping, where G1(V1, E1) and G2(V2, E2), the certificate will verify if f is one-to-one function and whether for any two vertices a, b from V1, the edge (a,b) belongs to G1 if and only if the edge (f(a),f(b)) belongs to G2. Since we have at most n^2 pairs, it means the certificate runs in quadratic polynomial time. In conclusion the isomorphism between G1 and a specified subgraph of G2 can be verified in polynomial time => subgraph isomorphism is in NP....

The first part is to prove that our problem (subgraph-isomorphism) is in NP class.

For an instance (G1,G2) it can be considered a map from the vertices of G1 to those of G2. This will show which vertices of the graph G2 correspond to nodes of G1. So if f: V1->V2 is the mapping, where G1(V1, E1) and G2(V2, E2), the certificate will verify if f is one-to-one function and whether for any two vertices a, b from V1, the edge (a,b) belongs to G1 if and only if the edge (f(a),f(b)) belongs to G2. Since we have at most n^2 pairs, it means the certificate runs in quadratic polynomial time. In conclusion the isomorphism between G1 and a specified subgraph of G2 can be verified in polynomial time => subgraph isomorphism is in NP....

This is only a preview of the solution. Please use the purchase button to see the entire solution

Matrix Chain Multiplication Using Dynamic Programming Algorithm

$18.00

Matrix

Chain

Multiplication

Dynamic

Programming

Algorithm

Optimal

Product

Dimension

Matrix-chain-order

Technique

Matrix

Chain

Multiplication

Dynamic

Programming

Algorithm

Optimal

Product

Dimension

Matrix-chain-order

Technique

Algorithm Analysis, Correctness and Sorted Linked List Algorithm

$10.00

Algorithm

Linked

List

Loop

Invariant

Worst

Case

Analysis

Complexity

Correctness

Ascending

Initialization

Maintenance

Termination

Algorithm

Linked

List

Loop

Invariant

Worst

Case

Analysis

Complexity

Correctness

Ascending

Initialization

Maintenance

Termination

Approximation Algorithm Counterexample - Related to Set Cover Problem

$8.00

Set

Cover

Instance

Optimal

Fractional

NP

NP-Hard

NP-Complete

Element

Integral

Approximation

Algorithm