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unimportant If there are two or more edges of different cost connecting a pair of vertices, any shortest-path algorithm will choose the lower cost edge without resorting to any special processing Note also that v e c t o r s can be used instead of linked lists, with the constant-time push-back operation replacing insertions at the front In most real-life applications the vertices have names, which are unknown at compile time, instead of numbers Consequently, we must provide a way to transform names to numbers The easiest way to do so is to provide a map by which we map a vertex name to an internal number ranging from 0 to IV - 1 (the number of vertices is determined as the program runs) The internal numbers are assigned as the graph is read The first number assigned is 0 As each edge is input, we check whether each of the two vertices has been assigned a number, by looking in the map If it has been assigned an internal number, we use it Otherwise, we assign to the vertex the next available number and insert the vertex name and number in the map With this transformation, all the graph algorithms use only the internal numbers Eventually, we have to output the real vertex names, not the internal numbers, so for each internal number we must also record the corresponding vertex name One way to do so is to keep a string for each vertex We use this technique to implement a G r a p h class The class and the shortest path algorithms require several data structures-namely, list, a queue a map, and a priority queue The # i n c l u d e directives for system headers are shown in Figure 153 The queue (implemented with a linked list) and priority queue are used in various shortest-path calculations The adjacency list is represented with v e c t o r s A map is also used to represent the graph When we write an actual C++ implementation, we do not need internal vertex numbers Instead, each vertex is stored in a V e r t e x object, and instead of using a number, we can use the address of the v e r t e x object as its (uniquely identifying) number As a result, the code makes frequent use.

of vertex* variables However, when describing the algorithms, assuming that vertices are numbered is often convenient, and we occasionally do so Before we show the Graph class interface, let us examine Figures 154 and 155, which show how our graph is to be represented Figure 154 shows the representation in which we use internal numbers Figure 155 replaces the internal numbers with vertex* variables, as we do in our code Although this simplifies the code it greatly complicates the picture Because the two figures represent identical inputs, Figure 154 can be used to follow the complications in Figure 155 we As indicated in the part labeled I n p ~ ~ t can expect the user to provide a list of edges, one per line At the start of the algorithm, we do not know the names of any of the vertices how many vertices there are, or how many edges there are We use two basic data structures to represent the graph As we mentioned in the preceding paragraph, for each vertex we maintain a vertex object that stores some information We describe the details of vertex (in particular, how different vertex objects interact with each other) last As mentioned earlier the first major data structure is a map that allows us to find, for any vertex name, a pointer to the vertex object that represents it This map is shown in Figure 155 as vertexMap (Figure 154 maps the name to an i n t in the component labeled Dictionan)

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Var(ay) = a Z Var(y) These latter equations are the special cases of (a) and (b) when m = 1 and n=l. To prove these rules in general would involve a lot of subscripts, but we can convince ourselves further by looking at another special case. Let m = 1 and n = 2. Then

A 19

(12.48)

An abstract scenario of the data structures used in a shortest-path calculation, with an input graph taken from a file The shortest weighted path from A to C is A to B to E to D to C (cost is 76)

Legend: Dark-bordered boxes are vertex objects The unshaded portion in each box contains the name and adjacency list and does not change when shortest-path computation is performed Each adjacency list entry contains an Edge that stores a pointer to another vertex object and the edge cost Shaded portion is d i s t and prev, Jilled in after shortest path computation runs Dark pointers emanate from ver t emap Light pointers are adjacency list entries Dashed-pointers are the prev data member that results from a shortest path computation

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