Type of Graph Problem

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Running Time

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Unweighted

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Breadth-first search O(E O ( E O(IE)

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1 Weighted no negative edges 1 1 Weighted, negative edges

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I Weighted, acyclic 1

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Dijkstra's algorithm

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1 Belman-Ford algorithm

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I Uses topological sort

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Figure 1539 Worst-case running times of various graph algorithms

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activity-node graph A graph of vertices as activities and edges as precedence relationships (p 522) adjacency lists An array of lists used to represent a graph, using linear space (p 492) adjacency matrix A matrix representation of a graph that uses quadratic space (p 49 1 ) adjacent vertices Vertex w is adjacent to vertex v if there is an edge from v to w (p 489) Bellman-Ford algorithm An algorithm that is used to solve the negative-weighted, shortest-path problem (p 5 16) breadth-first search A search procedure that processes vertices in layers: Those closest to the start are evaluated first, and those most distant are evaluated last (p 505) critical-path analysis A form of analysis used to schedule tasks associated with a project (p 522) cycle In a directed graph, a path that begins and ends at the same vertex and contains at least one edge (p 490) dense and sparse graphs A dense graph has a large number of edges (generally quadratic) Typical graphs are not dense but are sparse (P 491) Dijkstra's algorithm An algorithm that is used to solve the positiveweighted, shortest-path problem (p 509) directed acyclic graph (DAG) A type of directed graph having no cycles (p 490) directed graph A graph in which edges are ordered pairs of vertices (P 489) edge cost (weight) The third component of an edge that measures the cost of traversing the edge (p 489) event-node graph A graph that consists of event vertices that correspond to the completion of an activity and all its dependent activities Edges show what activity must be completed to advance from one vertex to the next The earliest completion time is the longest path (p 522) graph A set of vertices and a set of edges that connect the vertices (P 489) indegree The number of incoming edges of a vertex (p 5 19)

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negative-cost cycle A cycle whose cost is less than zero and makes most, if not all, paths undefined because we can loop around the cycle arbitrarily many times and obtain an arbitrarily small weighted path length (p 5 16) path A sequence of vertices connected by edges (p 490) path length The number of edges on a path (p 490) positive-cost cycle In a longest-path problem, the equivalent of a negative-cost cycle in a shortest-path problem (p 524) simple path A path in which all vertices are distinct, except that the first and last vertices can be the same (p 490) single-source algorithms Algorithms that compute the shortest paths from some starting point to all vertices in a graph (p 496) slack time The amount of time that an activity can be delayed without delaying overall completion (p 525) topological sort A process that orders vertices in a directed acyclic graph such that if there is a path from u to v, then v appears after u in the ordering A graph that has a cycle cannot have a topological order (p 5 17) unweighted path length The number of edges on a path (p 503) weighted path length The sum of the edge costs on a path (p 509)

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1 A common error is failing to ensure that the input graph satisfies the requisite conditions for the algorithm being used (ie, acyclic or positive weighted) 2 For Path,the comparison function compares the c o s t data member only If the dest data member is used to drive the comparison function, the algorithm may appear to work for small graphs, but for larger graphs, it is incorrect and gives slightly suboptimal answers It never produces a path that does not exist, however Thus this error is difficult to track down 3 The shortest-path algorithm for negative-weighted graphs must have a test for negative cycles; otherwise, it runs forever

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