Exercise d. Coloring, with the oracle's help. (Textbook problem 4.2) Analogous to the previous problem, but a little trickier: suppose we have an oracle for the decision problem GRAPH k-COLORING. Show that by asking a polynomial number of questions, we can find a k-coloring if one exists. Hint: You want to iteratively compute a coloring, where partway through, some nodes have colors assigned already and others do not. You need to ask the oracle about a modified version of the original graph to learn if this partial coloring can be finished; if not, make a different choice when coloring the next node.

Exercise d. Coloring, with the oracle's help. (Textbook problem 4.2) Analogous to the previous problem, but a little trickier: suppose we have an oracle for the decision problem GRAPH k-COLORING. Show that by asking a polynomial number of questions, we can find a k-coloring if one exists. Hint: You want to iteratively compute a coloring, where partway through, some nodes have colors assigned already and others do not. You need to ask the oracle about a modified version of the original graph to learn if this partial coloring can be finished; if not, make a different choice when coloring the next node.

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Definition: We define a path in a graph to be short if it contains ≤ 100 edges. Note that we are looking at the number of edges, not the number of vertices. The Problem: •INPUT – An unweighted DAG G = (V, E) – Two specific vertices s, t ∈V •OUTPUT: the number of different short paths in G from s to t. You don’t have to output the actual paths; you just have to figure out how many of them there are. Show a dynamic programming algorithm that solves the above problem in O(|E|) time. You only need to write pseudocode, nothing else. NOTE: a reminder that you can assume in this class that all arithmetic operations (multiplication, addition, etc.) take O(1) time, even if the numbers are very big.
If there is an Euler path that starts at AA and ends at some other vertex, give it. Otherwise enter DNE.This should be a list of letters, from A to the other vertex: e.g. ACBDEAD (if that were even possible!). Path: Since this is a planar graph, the number of regions - arcs + nodes equals what number?
Do some outside research on depth-first traversal as it relates to traversing graphs. Then answer the following questions: a. Suppose you have an arbitrary connected graph G, shown in the image below. Use the vertex A as your starting point. Write out the order in which the algorithm could traverse the graph with a depth-first search, and explain your reasoning (there are multiple correct answers, hence the need for an explanation). b. Use a proof by induction to prove that when a depth-first traversal is performed, every vertex v in your graph G will have been visited at least one time. B D H E A G с I FL
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.
Please Answer this in Python language: You're given a simple undirected graph G with N vertices and M edges. You have to assign, to each vertex i, a number C; such that 1 ≤ C; ≤ N and Vi‡j, C; ‡ Cj. For any such assignment, we define D; to be the number of neighbours j of i such that C; < C₁. You want to minimise maai[1..N) Di - mini[1..N) Di. Output the minimum possible value of this expression for a valid assignment as described above, and also print the corresponding assignment. Note: The given graph need not be connected. • If there are multiple possible assignments, output anyone. • Since the input is large, prefer using fast input-output methods. Input 1 57 12 13 14 23 24 25 35 Output 2 43251 Q
True or false: For graphs with negative weights, one workaround to be able to use Dijkstra’s algorithm (instead of Bellman-Ford) would be to simply make all edge weights positive; for example, if the most negative weight in a graph is -8, then we can simply add +8 to all weights, compute the shortest path, then decrease all weights by -8 to return to the original graph. Select one: True False
*Discrete Math In the graph above, let ε = {2, 3}, Let G−ε be the graph that is obtained from G by deleting the edge {2,3}. Let G∗ be the graph that is obtain from G − ε by merging 2 and 3 into a single vertex w. (As in the notes, v is adjacent to w in the new if and only if either {2,v} or {3,v is an edge of G.) (a) Draw G − ε and calculate its chromatic polynomial. (b) Give an example of a vertex coloring that is proper for G − ε, but not for G. (c) Explain, in own words, why no coloring can be proper for G but not proper for G − ε. (d) Draw G∗ and calculate its chromatic polynomial. (e) Verify that, for this example,PG(k) = PG−ε(k) − PG∗ (k).
The maze is described as a graph with a start, goal, edge lengths, and two types of edges: regular paths in the maze, and hedges which one can crawl through. We are only allowed to crawl through edge once. (Some parts of the maze are too thick to crawl through.) Design an algorithm which finds the shortest path to the goal, as quickly as possible. Please do not use the modified version of Dijkstra. Instead modify the graph and use regular version of Dijkstra
Programming with Python: v Implement Depth First Search (Traversal) v Implement Breadth First Traversal For the implementation: 1. Create a directed graph with 10 Vertices 2. Add arbitrary edges among vertices to satisfy the directed graph 3. Run your program and show appropriate output (Traversal order) 4. Your implementation should be fully commented Please, submit your source code and output (Screen Capture) too.
Correct answer will be upvoted else Multiple Downvoted. Don't submit random answer. Computer science. each individual has a rating chart portrayed by a variety of integers an of length n. You are currently refreshing the foundation, so you've made a program to pack these diagrams. The program functions as follows. Given an integer boundary k, the program takes the base of each adjoining subarray of length k in a. All the more officially, for a cluster an of length n and an integer k, characterize the k-pressure exhibit of an as a cluster b of length n−k+1, to such an extent that bj=minj≤i≤j+k−1ai For instance, the 3-pressure cluster of [1,3,4,5,2] is [min{1,3,4},min{3,4,5},min{4,5,2}]=[1,3,2]. A stage of length m is an exhibit comprising of m unmistakable integers from 1 to m in subjective request. For instance, [2,3,1,5,4] is a stage, however [1,2,2] isn't a change (2 shows up twice in the exhibit) and [1,3,4] is likewise not a stage (m=3 but rather there is 4 in the…
The edge-coloring problem is to color the edges of a graph with the fewest number of colors in such a way any two edges that share a vertex have different colors . You are given the algorithm that colors a graph with at most d+1 colors if the graph has a vertex with maximum degree d. You do not need to know how the algorithm works. Prove that this algorithm is a 2-approximation to the edge coloring problem. You may assume that d≥1. There are no decision problems in NP-hard class. True or False
The Graph Data Structure is made up of nodes and edges. (A Tree Data Structure is a special kind of a Graph Data Structure). A Graph may be represented by an Adjacency Matrix or an Adjacency List. Through this exercise, you should be able to have a better grasp the Adjacency Matrix concept. You are expected to read about the Adjacency Matrix concept as well as the Adjacency List concept. Suppose the vertices A, B, C, D, E, F, G and H of a Graph are mapped to row and column indices(0,1,2,3,4,5,6,and 7) of a matrix (i.e. 2-dimensional array) as shown in the following table. Vertex of Graph Index in the 2-D Array Adjacency Matrix Representation of Graph A B 2 F 6. H 7 Suppose further, that the following is an Adjacency Matrix representing the Graph. 3 4 5. 6. 7 0. 1 1 1 1 01 1 01 1. 3 14 1 1 1 6. 1 Exercise: Show/Draw the Graph that is represented by the above Adjacency matrix. Upload the document that contains your result. (Filename: AdjacencyMatrixExercise.pdf) Notes: -The nodes of the…