A strict order relation can be represented by a directed acyclic graph (or DAG), which is a directed graphwith no cycles. For example, the following directed graph is a DAG:Figure 2: Example of DAGA topological sort of a DAG is an ordering of the vertices that is consistent with the edges of the graph.That is, if there is an edge (u,v) in the graph, then u should appear before v in the topological sort. Forexample, 1,2,3 and 1,3,2 are topological sorts of the DAG shown above, but 2,1,3 is not a topological sortbecause 2 cannot be listed before 1.CS 111CS 141CS 151CS 261CS 362CS 211CS 361CS 401CS 251CS 342CS 301Figure 3: UIC CS Coursesa Is the following ordering a topological sort of G? Justify your answer. CS 111, CS 141, CS 211, CS261, CS 362, CS 361, CS 151, CS 251, CS 401, CS 342, CS 301.b Give two different topological sorts of G.c If a student can take an unlimited number of courses per semester, what is the fewest number ofsemesters required to complete these courses? Justify your answer.

Step 1: for question 3) see the directed edges. First complete the course CS111 then from there we…

Answered: A strict order relation can be…

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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This is a discrete math (graph) problem. Please explain each step in detail, no cursive writing.
Additional Topics: Topological Sort A strict order relation can be represented by a directed acyclic graph (or DAG), which is a directed graph with no cycles. For example, the following directed graph is a DAG: A topological sort of a DAG is an ordering of the vertices that is consistent with the edges of the graph. That is, if there is an edge (u, v) in the graph, then u should appear before v in the topological sort. For example, 1,2,3 and 1,3, 2 are topological sorts of the DAG shown above, but 2,1,3 is not a topological sort because 2 cannot be listed before 1. 7. Given the following DAG G representing UIC's Computer Science courses and prerequisites (note that edges implied by the transitive property are omitted): CS 261 CS 362 CS 141 CS 11 CS 211 Cs 361 CS 401 CS 151 CS 251 CS 342 Cs 301 Answer the following questions: a. Give two different topological sorts of G. b. Is each of the following orderings a topological sort of G? Justify your answer. CS111, CS151, CS301, CS251,…
5. (This question goes slightly beyond what was covered in the lectures, but you can solve it by combining algorithms that we have described.) A directed graph is said to be strongly connected if every vertex is reachable from every other vertex; i.e., for every pair of vertices u, v, there is a directed path from u to v and a directed path from v to u. A strong component of a graph is then a maximal subgraph that is strongly connected. That is all vertices in a strong component can reach each other, and any other vertex in the directed graph either cannot reach the strong component or cannot be reached from the component. (Note that we are considering directed graphs, so for a pair of vertices u and v there could be a path from u to v, but no path path from v back to u; in that case, u and v are not in the same strong component, even though they are connected by a path in one direction.) Given a vertex v in a directed graph D, design an algorithm for com- puting the strong connected…
In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices of a graph such that no two adjacent vertices share the same color; this is called a vertex coloring. The chromatic number of a graph is the least mumber of colors required to do a coloring of a graph. Example Here in this graph the chromatic number is 3 since we used 3 colors The degree of a vertex v in a graph (without loops) is the number of edges at v. If there are loops at v each loop contributes 2 to the valence of v. A graph is connected if for any pair of vertices u and v one can get from u to v by moving along the edges of the graph. Such routes that move along edges are known by different names: edge progressions, paths, simple paths, walks, trails, circuits, cycles, etc. a. Write down the degree of the 16 vertices in the graph below: 14…
Discrete Mathmatics The graph intersection of a collection of sets A1, A2, · · · , An is the graph that has a vertex for each of these sets and has an edge connecting the vertices representing two sets if these sets have a nonempty intersection. Construct the intersection graph for of the following collection of sets. A1 = {0, 2, 4, 6, 8} A2 = {0, 1, 2, 3, 4} A3 = {1, 3, 5, 7, 9} A4 = {5, 6, 7, 8, 9} A5 = {0, 1, 8, 9}
An undirected graph, G = {V, E}, defined as follows: V = {a, b, c, d, e, f, g, h, i, j, k, l, m} , E = {(a, b), (a, d), (a, h), (b, c), (b, d), (b, e), (c, f), (d, h), (d, i), (e, f), (e, i), (e, j), (f, g), (f, j), (f, k), (g, k), (h, i), (h, l), (i, j), (i, m), (j, m), (k, m), (l, m)} Please give the Graph figure (10%) and its Adjacency List with increasing order by alphabet (10%). After that, present its DFS Spanning Tree (15%) and BFS Spanning Tree (15%) from Vertex C. You can insert your hand-drawn figure into the replying web page or insert a hyperlink, which connects to a public cloud space saving your answer in hand-drawn figures or other types presentations.
Let G be a graph. We say that a set of vertices C form a vertex cover if every edge of G is incident to at least one vertex in C. We say that a set of vertices I form an independent set if no edge in G connects two vertices from I. For example, if G is the graph above, C = [b, d, e, f, g, h, j] is a vertex cover since each of the 20 edges in the graph has at least one endpoint in C, and I = = [a, c, i, k] is an independent set because none of these edges appear in the graph: ac, ai, ak, ci, ck, ik. 2a In the example above, notice that each vertex belongs to the vertex cover C or the independent set I. Do you think that this is a coincidence? 2b In the above graph, clearly explain why the maximum size of an independent set is 5. In other words, carefully explain why there does not exist an independent set with 6 or more vertices.
Given a graph data structure: G = (V,E) where, V = {A, B, C, D, E } E = { (A,B), (A,D), (B,D), (B,C), (D,E) } a) Draw the graph G. b) Draw all spanning trees of the graph G, which are also “linear” (each graph node has no more than two neighbors).
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…
One can manually count path lengths in a graph using adjacency matrices. Using the simple example below, produces the following adjacency matrix: A B A 1 1 B 1 0 This matrix means that given two vertices A and B in the graph above, there is a connection from A back to itself, and a two-way connection from A to B. To count the number of paths of length one, or direct connections in the graph, all one must do is count the number of 1s in the graph, three in this case, represented in letter notation as AA, AB, and BA. AA means that the connection starts and ends at A, AB means it starts at A and ends at B, and so on. However, counting the number of two-hop paths is a little more involved. The possibilities are AAA, ABA, and BAB, AAB, and BAA, making a total of five 2-hop paths. The 3-hop paths starting from A would be AAAA, AAAB, AABA, ABAA, and ABAB. Starting from B, the 3-hop paths are BAAA, BAAB, and BABA. Altogether, that would be eight 3-hop paths within this graph. Write a program…
Given a graph that is a tree (connected and acyclic). (1) Pick any vertex v. (II) Compute the shortest path from v to every other vertex. Let w be the vertex with the largest shortest path distance. (III) Compute the shortest path from w to every other vertex. Let x be the vertex with the largest shortest path distance. Consider the path p from w to x. Which of the following are true a. p is the longest path in the graph b. p is the shortest path in the graph c. p can be calculated in time linear in the number of edges/vertices a,c a,b a,b,c b.c
2. Let G = (V, E) be a directed weighted graph with the vertices V = {A, B, C, D, E, F) and the edges E= {(A, B, 12), (A, D, 17), (B, C, 8), (B, D, 13), (B, E, 15), (B, F, 13), (C, E, 12), (C, F, 25)}, where the third components is the cost. (a) Write down the adjacency list representation the graph G = (V, E).

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