Design a bottom-up decrease-and-conquer algorithm for the following task. For any even n, mark n cells on an infinite sheet of graph paper so that each marked cell has an odd number of marked neighbors. Two cells are considered neighbors if they are next to each other either horizontally or vertically but not diagonally. The marked cells must form a contiguous region, that is a region in which there is a path between any pair of marked cells that goes through a sequence of marked neighbors. Below are examples of solutions for n=2 and n=4 respectively.

Design a bottom-up decrease-and-conquer algorithm for the following task. For any even n, mark n cells on an infinite sheet of graph paper so that each marked cell has an odd number of marked neighbors. Two cells are considered neighbors if they are next to each other either horizontally or vertically but not diagonally. The marked cells must form a contiguous region, that is a region in which there is a path between any pair of marked cells that goes through a sequence of marked neighbors. Below are examples of solutions for n=2 and n=4 respectively.

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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In this question you will explore Graph Colouring algorithms. Given a graph G, we say that G is k-colourable if every vertex of G can be assigned one of k colours so that for every pair u, v of adjacent vertices, u and v are assigned different colours. The chromatic number of a graph G, denoted by χ(G), is the smallest integer k for which graph G is k-colorable. To show that χ(G) = k, you must show that the graph is k-colourable and that the graph is not (k − 1)-colourable. Question: It is NP-complete to determine whether an arbitrary graph has chromatic number k, where k ≥ 3. However, determining whether an arbitrary graph has chromatic number 2 is in P. Given a graph G on n vertices, create an algorithm that will return TRUE if χ(G) = 2 and FALSE if χ(G) 6= 2. Clearly explain how your algorithm works, why it guarantees the correct output, and determine the running time of your algorithm.
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Draw the graph represented by the following adjacency matrix on a piece of paper, take a photo of your work, and upload it here. Also, answer the following question about this graph: in what order do we visit the vertices if we run a BFS starting from vertex o? (vertices are labelled o to 3) 0011 1011 0101 1010
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When faced with a difficult problem in mathematics, it often helps to draw a picture. If the problem involves a discrete collection of interrelated objects, it is natural to sketch the objects and draw lines between them to indicate the relationships. A graph (composed of dots called vertices connected by lines or curves called edges) is the mathematical version of such a sketch. The edges of a graph may have arrows on them; in this case, the graph is called a directed graph. When we draw a graph, it doesn’t really matter where we put the vertices or whether we draw the edges as curved or straight; rather, what matters is whether or not two given vertices are connected by an edge (or edges). The degree of a vertex is the number of edges incident to it (i.e., the number of times an edge touches it). This is different than the number of edges touching it, because an edge my form a loop; for instance, vertex ? in graph ? (above) has degree 5. In a directed graph, we can speak of the…
When faced with a difficult problem in mathematics, it often helps to draw a picture. If the problem involves a discrete collection of interrelated objects, it is natural to sketch the objects and draw lines between them to indicate the relationships. A graph (composed of dots called vertices connected by lines or curves called edges) is the mathematical version of such a sketch. The edges of a graph may have arrows on them; in this case, the graph is called a directed graph. When we draw a graph, it doesn’t really matter where we put the vertices or whether we draw the edges as curved or straight; rather, what matters is whether or not two given vertices are connected by an edge (or edges). The degree of a vertex is the number of edges incident to it (i.e., the number of times an edge touches it). This is different than the number of edges touching it, because an edge my form a loop; for instance, vertex ? in graph ? (above) has degree 5. In a directed graph, we can speak of the…
A grid must have a mechanism for numbering the tiles in order to support random-access search.A square grid, for example, contains rows and columns that provide a natural numbering for the tiles. Create designs for triangular and hexagonal grids. Define a rule for finding the neighbourhood (i.e. nearby tiles) of a particular tile in the grid using the numbering scheme. For example, if we have a four-connected square grid with indices I for rows and j for columns, we may define the neighbourhood of tile I j as neighbourhood(i, j) = I 1, j,i, j 1.
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