Knight's tour problem. Consider the chessboard shown in Figure 3.13. Note that some squares are shaded. We wish to determine a knight's tour, if one exists, that starts at the square designated by s and, after visiting the minimum number of squares, ends at the square designated by t. The tour must not visit any shaded square, Formułate this problem as a reachability problem on an appropriately defined graph. Figure 3.13 Chessboard.

Knight's tour problem. Consider the chessboard shown in Figure 3.13. Note that some squares are shaded. We wish to determine a knight's tour, if one exists, that starts at the square designated by s and, after visiting the minimum number of squares, ends at the square designated by t. The tour must not visit any shaded square, Formułate this problem as a reachability problem on an appropriately defined graph. Figure 3.13 Chessboard.

Fundamentals of Information Systems

8th Edition

ISBN:9781305082168

Author:Ralph Stair, George Reynolds

Publisher:Ralph Stair, George Reynolds

Chapter3: Database Systems And Applications

Section: Chapter Questions

Problem 2.1DQ

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Q1) how many graphs are there on 20 nodes? (To make this question precise, we have to make sure we known what it mean that two graphs are the same . For the purpose of this exercise,we consider the nodes given and labeled,say,asAlice ,Bob,...... The graph consisting of a single edge connecting Alice and Bob is different from the graph consisting of a single edge connecting Eve and frank.) Q2) Formulate the following assertion as a theorem about graphs and prove it :At every party one can find two people who know the same number of other people (like Bob and Eve in our first example).
(1) Like divide and conquer, dynamic programming (DP) solves problems by combiningsolutions to subproblems: true or false?_______; unlike divide and conquer, DPsubproblems are independent: true or false?_______. (2) For a graph G=<V, E>, the total running time of BFS is ?(____); the total running timeof DFS is ?(____). If G is an undirected graph, ∑? ∈ ???????(?) = ; if G is adirected graph, ∑? ∈ ???????(?) = __________. (3) Heap sort algorithm is stable or not? _____ , in-place or not? ______. Its worst case timecomplexity is ?(____). The height of a heap A[1..n] is ____. (4) For a sorting algorithm for an array of n elements, in the worst case the number ofinversions is _______ , the average number of inversions is _______.Shell sort algorithm is stable or not? ______ , in-place or not? _______ . (5) The worst case time complexity of BuildMaxHeap is ?(_______). The worst case timecomplexity of MaxHeapify is ?(______). The worst case time complexity of Heap-Extract-Max…
True or False (If your answer to the question is "False", explain why, and provide correction when possible). (a) Let h(n) be the heuristics for the node n, h(m) be the heuristics for the node m, d(m,n) be the actual minimal cost from node m to n in a graph. A* satisfies the monotone restriction iff d(m,n) <= |h(n)-h(m)|. (b) If an A* heuristics is admissible then it satisfies the monotone restriction. (c) Best-first search guarantees optimality in its returned solution. (d) Least-cost-first search guarantees optimality in its returned solution. (e) If all edges are with unit cost, then Breadth-first search guarantees optimality in its returned solution.
Let G = (V, E) be an undirected graph and each edge e ∈ E is associated with a positive weight ℓ(e).For simplicity we assume weights are distinct. Is the following statement true or false? Let T be a minimum spanning tree for the graph with the original weight. Suppose we replace eachedge weight ℓ(e) with ℓ(e)^2, then T is still a minimum spanning tree.
Consider the navigation problem shown in Figure 1. The number next to each edge is the cost of the performing the action corresponding to that edge. You start from A and your goal is to get to F. List the order in which nodes are expanded, which nodes are added to the fringe and which states are added to the closed set when performing Graph Search using: breadth-first search. depth-first search. iterative deepening search. uniform cost search
a) Write the adjacency matrix representing the graph above. b)Would you prefer to represent this graph (specifically the one above) as an adjacency list or as an adjacency matrix? Justify your answer. c) Topological sorting is an algorithm on graphs used to produce a schedule. Assuming each node in this graph represents an item, what is ONE schedule associated with the above graph? If the topological sorting algorithm cannot be run on the above graph, justify your answer.
Given a vertex set W = {1, 2, 3, 4}, solve for the following questions: (i) Knowing that edges are the same if and only if they have the same endpoint, how many different edges are possible? (ii) Suppose for this question that graphs are different if they have different sets of edges (they do not depend on if the graphs are isomorphic or not). How many simple graphs are there with the vertex set W? (iii) Draw the eleven nonisomorphic simple graphs that have four vertices.
Let G = (V, E) be an undirected graph and each edge e ∈ E is associated with a positive weight ℓ(e).For simplicity we assume weights are distinct. Is the following statement true or false? Prove by contradiction or counterexample. Let T be a minimum spanning tree for the graph with the original weight. Suppose we replace eachedge weight ℓ(e) with ℓ(e)^2, then T is still a minimum spanning tree.
A manufacturer of chemical goods is faced with the problem that certain goods cannot be stored at the same place due to the danger of unwanted reactions. What he seeks is a storage scheme such that goods that cannot be located at the same place are indeed separated. Provide a graph model to solve this problem. Do not write details of any algorithms. Simply describe your graph vertices, edges, and the classic graph theory problem that is useful for the design of the storage scheme here.
a) Draw the connected subgraph of the given graph above which contains only four nodes ACGB and is also a minimum spanning tree with these four nodes. What is its weighted sum? Draw the adjacency matrix representation of this subgraph (use boolean matrix with only 0 or 1, to show its adjacency in this case).b) Find the shortest path ONLY from source node D to destination node G of the given graph above, using Dijkstra’s algorithm. Show your steps with a table as in our course material, clearly indicating the node being selected for processing in each step.c) Draw ONLY the shortest path obtained above, and indicate the weight in each edge in the diagram. Also determine the weighted sum
3.Consider this graph data:- Jessica's buddy list: Meghan, Alan, Martin.- Meghan's buddy list: Alan, Lori.- Toni's buddy list: Lori, Meghan.- Martin's buddy list: Lori, Meghan.- Alan's buddy list: Martin, Jessica.- Lori's buddy list: Meghan.a. Compute the in/out degree of each vertex. Is the graph connected?b. Who is the most popular? Least? Who is the most antisocial?c. If we're having a party and want to distribute the message the most quickly, who should we tell first?
Vertex S denotes the start state and vertices G1 and G2 denote the two goal states. Directed edges are labelled with the actual costs of traversing the edge. What path from S to a goal would be found by Iterative deepening Depth-First Search? If all else is equal, consider states in alphabetic order. options are : a. S, A, B, C, D, E, F, G1 b. S, A, B, C, E, F, D, G1 c. S, A, B, C, E, F, G1 d. S, A, B, C, G1 e. S, A, C, G1 f. S, A, E, G1 g. S, B, A, C, E, G1 h. S, B, F, D, G2 i. S, C, B, F, A, D, E, G1 j. S, C, G1