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I Graphs and Paths

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In this chapter we examine the graph and show how to solve a particular kind of problem-namely, calculation of shortest paths The computation of shortest paths is a fundamental application in computer science because many interesting situations can be modeled by a graph Finding the fastest routes for a mass transportation system, and routing electronic mail through a network of computers are but a few examples We examine variations of the shortest path problems that depend on an interpretation of shortest and the graph's propertjes Shortest-path problems are interesting because, although the algorithms are fairly simple, they are slow for large graphs unless careful attention is paid to the choice of data structures In this chapter, we show: formal definitjons of a graph and its components the data structures used to represent a graph, and algorithms for solving several variations of the shortest-path problem, with complete C++ implementations

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Definitions

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A graph consists of a set of vertices and a set of edges that connect the vertices That is, G = E),where V is the set of v e k c e s and E is the set of edges Each edge is a pair (v, w), where v, w E \! Vertices are sometimes called nodes, and edges are sometimes called arcs If the edge pair is ordered, the graph is called a directed graph Directed graphs aresometimes called digraphs In a digraph, vertex w is adjacent to vertex v if and only if (v, w ) E ESometimes an edge has a third component, called the edge cost (or weight) that measures the cost of traversing the edge In this chapter, all graphs are directed

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A graph consists of a Set Of vertices and a set of edges that connect the vertices, ~ftheedge pair is ordered7thegraph is a directed graph Vertex w i s adjacent to vertex vif there is an edge from v to w

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Graphs and Paths

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A directed graph

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The graph shown in Figure 151 has seven vertices,

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and 12 edges,

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= { v o , v , , v,, v,, v,,

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v,, V6 I-,

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A path is a sequence of vertices connected by edges

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The unweighted path length measures the number of edges on a path The weighted path length is the sum of the edge costs on a path

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A cycle in a directed graph is a path that begins and ends at the same vertex and contains at least one edge

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The following vertices are adjacent to V3: VZV4, V,, and V6 Note that V, and V, are not adjacent to V3 For this graph, / VI = 7 and IEl = 12; here, I 1 S represents the size of set S A path in a graph is a sequence of vertices connected by edges In other w2, words, w , , , wh,the sequence of vertices is such that ( w , , w i E E for I 5 i < N The path length is the number of edges on the path-namely, N - I-also called the unweighted path length The weighted path length is the sum of the costs of the edges on the path For example, Vo, V,, V5 is a path from vertex Vo to V 5 The path length is two edges-the shortest path between Vo and V,, and the weighted path length is 9 However, if cost is important, the weighted shortest path between these vertices has cost 6 and is Vo, V,, V,, V, A path may exist from a vertex to itself If this path contains no edges, the path length is 0, which is a convenient way to define an otherwise special case A simple path is a path in which all vertices are distinct, except that the first and last vertices can be the same A cycle in a directed graph is a path that begins and ends at the same vertex and contains at least one edge That is, it has a length of at least 1 such that w , = w , ; , , this cycle is simple if the path is simple A directed acyclic graph (DAG) is a type of directed graph having no cycles

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An example of a real-life situation that can be modeled by a graph is the airport system Each airport is a vertex If there is a nonstop flight between two airports, two vertices are connected by an edge The edge could have a weight, representing time, distance, or the cost of the flight In an undirected graph, an edge ( v , w) would imply an edge (w, v) However, the costs of the edges might be different because flying in different directions might take longer (depending on prevailing winds) or cost more (depending on local taxes) Thus we use a directed graph with both edges listed, possibly with different weights Naturally, we want to determine quickly the best flight between any two airports; best could mean the path with the fewest edges or one, or all, of the weight measures (distance, cost, and so on) A second example of a real-life situation that can be modeled by a graph is the routing of electronic mail through computer networks Vertices represent computers, the edges represent links between pairs of computers, and the edge costs represent communication costs (phone bill per megabyte), delay costs (seconds per megabyte), or combinations of these and other factors For most graphs, there is likely at most one edge from any vertex v to any other vertex w (allowing one edge in each direction between v and w) Consequently, 1E 6 ( ~ 1 ' When most edges are present, we have 1El = O ( / V / Such a graph is considered to be a dense graph-that is, it ') has a large number of edges, generally quadratic In most applications, however, a sparse graph is the norm For instance, in the airport model, we do not expect direct flights between every pair of airports Instead, a few airports are very well connected and most others have relatively few flights In a complex mass transportation system involving buses and trains, for any one station we have only a few other stations that are directly reachable and thus represented by an edge Moreover, in a computer network most computers are attached to a few other local computers So, in most cases, the graph is relatively sparse, where IEl = @(IV ) or perhaps slightly more (there is no standard definition of sparse) The algorithms that we develop, then, must be efficient for sparse graphs

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