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 1.1, Problem 1E
Program Plan Intro
To describe a real world situation that uses sorting and also give a situation that requires computation of convex hull.
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You are organizing a programming competition, where contestants implement Dijkstra's algorithm. Given adirected graph G = (V, E) with integer-weight edges and a starting vertex s ∈ V , their programs are supposedto output triplets (v, v.d, v.π) for each vertex v ∈ V . Design an O(V +E) time algorithm that takes as inputthe original graph G in both adjacency matrix (G.M) and adjacency list (G.Adj) representations, startingvertex s, and the output of a contestant's program (given as an array A of triplets), and returns whetherA is the correct output for G. Write down the pseudocode for your algorithm, explain why it correctlyveri es the output, and analyze your algorithm's running time. You may assume that all edge weights of the input graph provided to the contestantsare nonnegative and A (the output of their programs) is in the valid format, i.e., you don't need to verifythat A is actually an array of triplets, with v and v.π being valid vertices and v.d being an integer.Can you…
The graph-coloring problem is usually stated as the vertex-coloring problem: assign the smallest number of colors to vertices of a given graph so that no two adjacent vertices are the same color. Consider the edge-coloring problem: assign the smallest number of colors possible to edges of a given graph so that no two edges with the same end point are the same color. Explain how the edge-coloring problem can be polynomial reduced to a vertex-coloring problem. Give an example.
Chapter 1 Solutions
Introduction to Algorithms
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- Correct answer will be upvoted else downvoted. Today we will play a red and white shading game (no, this isn't the Russian Civil War; these are only the shades of the Canadian banner). You are given a n×m matrix of "R", "W", and "." characters. "R" is red, "W" is white and "." is clear. The neighbors of a cell are those that share an edge with it (those that main offer a corner don't count). Your responsibility is to shading the clear cells red or white so every red cell just has white neighbors (and no red ones) and each white cell just has red neighbors (and no white ones). You are not permitted to recolour currently hued cells. Input The primary line contains t (1≤t≤100), the number of experiments. In each experiment, the principal line will contain n (1≤n≤50) and m (1≤m≤50), the tallness and width of the network separately. The following n lines will contain the matrix. Each character of the matrix is either 'R', 'W', or '.'. Output For each experiment,…arrow_forward3) The graph k-coloring problem is stated as follows: Given an undirected graph G = (V,E) with N vertices and M edges and an integer k. Assign to each vertex v in Va color c(v) such that 1< c(v)arrow_forward3) The graph k-coloring problem is stated as follows: Given an undirected graph G= (V,E) with N vertices and M edges and an integer k. Assign to each vertex v in V a color c(v) such that 1arrow_forwardThe minimum vertex cover problem is stated as follows: Given an undirected graph G = (V, E) with N vertices and M edges. Find a minimal size subset of vertices X from V such that every edge (u, v) in E is incident on at least one vertex in X. In other words you want to find a minimal subset of vertices that together touch all the edges. For example, the set of vertices X = {a,c} constitutes a minimum vertex cover for the following graph: a---b---c---g d e Formulate the minimum vertex cover problem as a Genetic Algorithm or another form of evolutionary optimization. You may use binary representation, OR any repre- sentation that you think is more appropriate. you should specify: • A fitness function. Give 3 examples of individuals and their fitness values if you are solving the above example. • A set of mutation and/or crossover and/or repair operators. Intelligent operators that are suitable for this particular domain will earn more credit. • A termination criterion for the…arrow_forwardImplement constraint satisfaction through map colouring problem in python.Graph coloring problem is to assign colors to certain elements of a graph subject to certain constraints. Vertexcoloring is the most common graph coloring problem. The problem is, given m colors, find a way of coloring thevertices of a graph such that no two adjacent vertices are colored using same color. The other graph coloringproblems like Edge Coloring (No vertex is incident to two edges of same color) and Face Coloring(Geographical Map Coloring) can betransformed into vertex coloring.Chromatic Number: The smallest number of colors needed to color a graph G is called its chromatic number. Write the code in pythonarrow_forwardConsider eight points on the Cartesian two-dimensional xx-yy plane. For each pair of vertices uu and vv, the weight of edge uvuv is the Euclidean (Pythagorean) distance between those two points. For example, dist(a,h) = \sqrt{4^2 + 1^2} = \sqrt{17}dist(a,h)=42+12=17 and dist(a,b) = \sqrt{2^2 + 0^2} = 2dist(a,b)=22+02=2. Using the algorithm of your choice, determine one possible minimum-weight spanning tree and compute its total distance, rounding your answer to one decimal place. 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