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6.4: Discussion and Exercises

  • Page ID
    47914
  • The running times of the depth-first-search and breadth-first-search algorithms are somewhat overstated by the Theorems 6.3.1 and 6.3.2. Define \(\mathtt{n}_{\mathtt{r}}\) as the number of vertices, \(\mathtt{i}\), of \(G\), for which there exists a path from \(\mathtt{r}\) to \(\mathtt{i}\). Define \(\mathtt{m}_\mathtt{r}\) as the number of edges that have these vertices as their sources. Then the following theorem is a more precise statement of the running times of the breadth-first-search and depth-first-search algorithms. (This more refined statement of the running time is useful in some of the applications of these algorithms outlined in the exercises.)

    Theorem \(\PageIndex{1}\)

    When given as input a Graph, \(\mathtt{g}\), that is implemented using the AdjacencyLists data structure, the \(\mathtt{bfs(g,r)}\), \(\mathtt{dfs(g,r)}\) and \(\mathtt{dfs2(g,r)}\) algorithms each run in \(O(\mathtt{n}_{\mathtt{r}}+\mathtt{m}_{\mathtt{r}})\) time.

    Breadth-first search seems to have been discovered independently by Moore [52] and Lee [49] in the contexts of maze exploration and circuit routing, respectively.

    Adjacency-list representations of graphs were presented by Hopcroft and Tarjan [40] as an alternative to the (then more common) adjacency-matrix representation. This representation, as well as depth-first-search, played a major part in the celebrated Hopcroft-Tarjan planarity testing algorithm that can determine, in \(O(\mathtt{n})\) time, if a graph can be drawn, in the plane, and in such a way that no pair of edges cross each other [41].

    In the following exercises, an undirected graph is one in which, for every \(\mathtt{i}\) and \(\mathtt{j}\), the edge \((\mathtt{i},\mathtt{j})\) is present if and only if the edge \((\mathtt{j},\mathtt{i})\) is present.

    Exercise \(\PageIndex{1}\)

    Draw an adjacency list representation and an adjacency matrix representation of the graph in Figure \(\PageIndex{1}\).

    img4780.png
    Figure \(\PageIndex{1}\): An example graph.

    Exercise \(\PageIndex{2}\)

    The incidence matrix representation of a graph, \(G\), is an \(\mathtt{n}\times\mathtt{m}\) matrix, \(A\), where

    \[\displaystyle A_{i,j} =
    \begin{cases}
    -1 & \text{if vertex $i$ the source of edge $j$} \\
    +1 & \text{if vertex $i$ the target of edge $j$} \\
    0 & \text{otherwise.}
    \end{cases}\nonumber\]

    1. Draw the incident matrix representation of the graph in Figure \(\PageIndex{1}\).
    2. Design, analyze and implement an incidence matrix representation of a graph. Be sure to analyze the space, the cost of \(\mathtt{addEdge(i,j)}\), \(\mathtt{removeEdge(i,j)}\), \(\mathtt{hasEdge(i,j)}\), \(\mathtt{inEdges(i)}\), and \(\mathtt{outEdges(i)}\).

    Exercise \(\PageIndex{3}\)

    Illustrate an execution of the \(\mathtt{bfs(G,0)}\) and \(\mathtt{dfs(G,0)}\) on the graph, \(\mathtt{G}\), in Figure \(\PageIndex{1}\).

    Exercise \(\PageIndex{4}\)

    Let \(G\) be an undirected graph. We say \(G\) is connected if, for every pair of vertices \(\mathtt{i}\) and \(\mathtt{j}\) in \(G\), there is a path from \(\mathtt{i}\) to \(\mathtt{j}\) (since \(G\) is undirected, there is also a path from \(\mathtt{j}\) to \(\mathtt{i}\)). Show how to test if \(G\) is connected in \(O(\mathtt{n}+\mathtt{m})\) time.

    Exercise \(\PageIndex{5}\)

    Let \(G\) be an undirected graph. A connected-component labelling of \(G\) partitions the vertices of \(G\) into maximal sets, each of which forms a connected subgraph. Show how to compute a connected component labelling of \(G\) in \(O(\mathtt{n}+\mathtt{m})\) time.

    Exercise \(\PageIndex{6}\)

    Let \(G\) be an undirected graph. A spanning forest of \(G\) is a collection of trees, one per component, whose edges are edges of \(G\) and whose vertices contain all vertices of \(G\). Show how to compute a spanning forest of of \(G\) in \(O(\mathtt{n}+\mathtt{m})\) time.

    Exercise \(\PageIndex{7}\)

    We say that a graph \(G\) is strongly-connected if, for every pair of vertices \(\mathtt{i}\) and \(\mathtt{j}\) in \(G\), there is a path from \(\mathtt{i}\) to \(\mathtt{j}\). Show how to test if \(G\) is strongly-connected in \(O(\mathtt{n}+\mathtt{m})\) time.

    Exercise \(\PageIndex{8}\)

    Given a graph \(G=(V,E)\) and some special vertex \(\mathtt{r}\in V\), show how to compute the length of the shortest path from \(\mathtt{r}\) to \(\mathtt{i}\) for every vertex \(\mathtt{i}\in V\).

    Exercise \(\PageIndex{9}\)

    Give a (simple) example where the \(\mathtt{dfs(g,r)}\) code visits the nodes of a graph in an order that is different from that of the \(\mathtt{dfs2(g,r)}\) code. Write a version of \(\mathtt{dfs2(g,r)}\) that always visits nodes in exactly the same order as \(\mathtt{dfs(g,r)}\). (Hint: Just start tracing the execution of each algorithm on some graph where \(\mathtt{r}\) is the source of more than 1 edge.)

    Exercise \(\PageIndex{10}\)

    A universal sink in a graph \(G\) is a vertex that is the target of \(\mathtt{n}-1\) edges and the source of no edges.1 Design and implement an algorithm that tests if a graph \(G\), represented as an AdjacencyMatrix, has a universal sink. Your algorithm should run in \(O(\mathtt{n})\) time.


    Footnotes

    1A universal sink, \(\mathtt{v}\), is also sometimes called a celebrity: Everyone in the room recognizes \(\mathtt{v}\), but \(\mathtt{v}\) doesn't recognize anyone else in the room.

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