6. a) Write the binomial coefficient in factorial form. Simplify so that your final answer does NOT contain any factorials. b) Explain why the number of edges in the complete graph K, is for all integers n2 2. The answer cannot be just examples of specific choices of n, but must apply to all integers n 2 2.

6. a) Write the binomial coefficient in factorial form. Simplify so that your final answer does NOT contain any factorials. b) Explain why the number of edges in the complete graph K, is for all integers n2 2. The answer cannot be just examples of specific choices of n, but must apply to all integers n 2 2.

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

See similar textbooks

Similar questions

5. Fleury's algorithm is an optimisation solution for finding a Euler Circuit of Euler Path in a graph, if they exist. Describe how this algorithm will always find a path or circuit if it exists. Describe how you calculate if the graph is connected at each edge removal. Fleury's Algorithm: The algorithm starts at a vertex of v odd degree, or, if the graph has none, it starts with an arbitrarily chosen vertex. At each step it chooses the next edge in the path to be one whose deletion would not disconnect the graph, unless there is no such edge, in which case it picks the remaining edge (a bridge) left at the current vertex. It then moves to the other endpoint of that edge and adds the edge to the path or circuit. At the end of the algorithm there are no edges left ( or all your bridges are burnt). (NOTE: Please elaborate on the answer and explain. Please do not copy-paste the answer from the internet or from Chegg.)
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…
36. Let G be a simple graph on n vertices and has k components. Then the number m of edges of G satisfies n-k ≤m if G is a null graph. This statement is A. sometimes true B. always true C. never true D. Neither true nor false
Part 2: Random GraphsA tournament T is a complete graph whose edges are all oriented. Given a completegraph on n vertices Kn, we can generate a random tournament by orienting each edgewith probability 12 in each direction.Recall that a Hamiltonian path is a path that visits every vertex exactly once. AHamiltonian path in a directed graph is a path that follows the orientations of thedirected edges (arcs) and visits every vertex exactly once. Some directed graphs havemany Hamiltonian paths.In this part, we give a probabilistic proof of the following theorem:Theorem 1. There is a tournament on n vertices with at least n!2n−1 Hamiltonian paths.For the set up, we will consider a complete graph Kn on n vertices and randomlyorient the edges as described above. A permutation i1i2 ...in of 1,2,...,n representsthe path i1 −i2 −···−in in Kn. We can make the path oriented by flipping a coin andorienting each edge left or right: i1 ←i2 →i3 ←···→in.(a) How many permutations of the vertices…
All of the following statements are false. Provide a counterexample for each one. iii. If it can be shown that there is not a proper 3-coloring of a graph G, then χ(G) = 4. iv. If G is a graph with χ(G) ≤ 4 then G is planar
Is it true or false? If it is true, include a (short, but clear) argument why it is true, and if it is false, include a concrete graph which shows that the claim is false.a) If all vertices in a connected graph have even degree, then for whichever two vertices u and v in the graph you choose, there is an Eulerian trail between u and v. b) Given a graph G we construct a new graph H by adding a new vertex v and edges between v and every vertex of G. If G is Hamiltonian, then so is H.c) We know that if a graph has a walk between u and v it also has a path between u and v, for any two vertices u and v. Is it always true that if a graph has a circuit containing u and v it also has a cycle containing u and v? d) The complete bipartite graph K?,?(lowered indicies) is Hamiltonian if and only if m = n ≥ 2.
What is the transitive closure of the following graph? P.
The following table presents the implementation of Dijkstra's algorithm on the evaluated graph G with 8 vertices. a) What do the marks (0, {a}) and (∞, {x}) in the 1st row of the table mean? b) What do the marks marked in blue in the table mean? c) Reconstruct all edges of the graph G resulting from the first 5 rows of the table of Dijkstra's algorithm. d) How many different shortest paths exist in the graph G between the vertices a and g?
A randomly matchable graph: if every matching of the graph can be extended to a 1-factor.Give examples of the following: a) an infinite class of bipartite graphs that is randomly matchable; b) an infinite class of non-bipartite graphs that is randomly matchable; c) an example of a graph that has a 1-factor, but is not randomly matchabl
3 In hill-climbing algorithms there are steps that make lots of progressand steps that make very little progress. For example, the first iteration on the inputgiven in Figure 15.2 might find a path through the augmentation graph through whicha flow of 30 can be added. It might, however, find the path through which only a flow of2 can be added. How bad might the running time be when the computation is unluckyenough to always take the worst legal step allowed by the algorithm? Start by taking thestep that increases the flow by 2 for the input given in Figure 15.2. Then continue to takethe worst possible step. You could draw out each and every step, but it is better to usethis opportunity to use loop invariants. What does the flow look like after i iterations?Repeat this process on the same graph except that the four edges forming the squarenow have capacities 1,000,000,000,000,000 and the crossover edge has capacity 1. (Alsomove t to c or give that last edge a large capacity.)1. What is…
A. Let G be a graph on 17 vertices. We know that G contains at most one cycle. How many edges can G contain at the most? B. The expression r is written in post-order notation (also referred to as RPN). What value does r evaluate to? r = 1 6 7 + - 5 3 + 2 * +
EX.S. Write a pseudo-code for the algorithm given a directed graph without cireles G=(V, E) and vertex veV, returns foc each vertex vE V i (1) the length of the longest trajectory between s and Vin G. (2) the previous of v in the longest trajectory between s. and v in G. The algorithm must run in time o IVI+IE). EXplain in detail in words how the algorithm works, briefly explain its correctness (no formal proof is needed) and analyze its runtime.