Data Structures and Algorithm Analysis - Computer Science at ...

400 Chap. 11 Graphs(a) IF an undirected graph is connected and has no simple cycles, THENthe graph has |V| − 1 edges.(b) IF an undirected graph has |V| − 1 edges and no cycles, THEN thegraph is connected.11.3 (a) Draw the adjacency matrix representation for the graph of Figure 11.26.(b) Draw the adjacency list representation for the same graph.(c) If a pointer requires four bytes, a vertex label requires two bytes, andan edge weight requires two bytes, which representation requires morespace for this graph?(d) If a pointer requires four bytes, a vertex label requires one byte, andan edge weight requires two bytes, which representation requires morespace for this graph?11.4 Show the DFS tree for the graph of Figure 11.26, starting at Vertex 1.11.5 Write a pseudocode algorithm to create a DFS tree for an undirected, connectedgraph starting at a specified vertex V.11.6 Show the BFS tree for the graph of Figure 11.26, starting at Vertex 1.11.7 Write a pseudocode algorithm to create a BFS tree for an undirected, connectedgraph starting at a specified vertex V.11.8 The BFS topological sort algorithm can report the existence of a cycle if oneis encountered. Modify this algorithm to print the vertices possibly appearingin cycles (if there are any cycles).11.9 Explain why, in the worst case, Dijkstra’s algorithm is (asymptotically) asefficient as any algorithm for finding the shortest path from some vertex I toanother vertex J.11.10 Show the shortest paths generated by running Dijkstra’s shortest-paths algorithmon the graph of Figure 11.26, beginning at Vertex 4. Show the Dvalues as each vertex is processed, as in Figure 11.19.11.11 Modify the algorithm for single-source shortest paths to actually store andreturn the shortest paths rather than just compute the distances.11.12 The root of a DAG is a vertex R such that every vertex of the DAG can bereached by a directed path from R. Write an algorithm that takes a directedgraph as input and determines the root (if there is one) for the graph. Therunning time of your algorithm should be Θ(|V| + |E|).11.13 Write an algorithm to find the longest path in a DAG, where the length ofthe path is measured by the number of edges that it contains. What is theasymptotic complexity of your algorithm?11.14 Write an algorithm to determine whether a directed graph of |V| verticescontains a cycle. Your algorithm should run in Θ(|V| + |E|) time.11.15 Write an algorithm to determine whether an undirected graph of |V| verticescontains a cycle. Your algorithm should run in Θ(|V|) time.