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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.

A graph is dense if Of edges the is large (generally quadratic,Typical graphs are not dense Instead, they are sparse

Cov(Ay) = A Cov(y)A'

We therefore have (12.47) where f is a function of q2 since it is the only Lorentz scalar that can be formed from q, but q2 = m; andf(m;) == f." is a constant. Inserting (12.47) into (12.46), the 'TT - -+ J.L - jJ decay amplitude is 0lL =

151 1

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The first thing to consider is how to represent a graph internally Assume that the vertices are sequentially numbered starting from 0, as the graph shown in Figure 151 suggests One simple way to represent a graph is to use a twodimensional array called an adjacency matrix For each edge (v, w), we set a [v][ w ] equal to the edge cost; nonexistent edges can be initialized with a logical INFINITYThe initialization of the graph seems to require that the entire adjacency matrix be initialized to INFINITYThen, as an edge is encountered, an appropriate entry is set In this scenario, the initialization

Adjacency list representation of the graph shown in Figure 151 ; the nodes in list i represent vertices adjacent to iand the cost of the connecting edge

- y5)v(k).

E( ay) = aE( y)

takes O(IVI2) time Although the quadratic initialization cost can be avoided (see Exercise 156), the space cost is still 0(1VI2), which is fine for dense graphs but completely unacceptable for sparse graphs For sparse graphs, a better solution is an adjacency list, which represents a graph by using linear space For each vertex, we keep a list of all adjacent vertices An adjacency list representation of the graph in Figure 151 using a linked list is shown in Figure 152 Because each edge appears in a list node, the number of list nodes equals the number of edges Consequently, O(IE1) space is used to store the list nodes We have IVI lists, so O(IVJ) additional space is also required If we assume that every vertex is in some edge, the number of edges is at least rlV1/21 Hence we may disregard any O(IV1) terms when an O(IE1) term is present Consequently, we say that the space requirement is O(IEI), or linear in the size of the graph The adjacency list can be constructed in linear time from a list of edges We begin by making all the lists empty When we encounter an edge (v, W, c,,,,), we add an entry consisting of w and the cost c , , v's adjato cency list The insertion can be anywhere; inserting it at the front can be done in constant time Each edge can be inserted in constant time, so the entire adjacency list structure can be constructed in linear time Note that when inserting an edge, we do not check whether it is already present That cannot be done in constant time (using a simple linked list), and doing the check would destroy the linear-time bound for construction In most cases, ignoring this check is

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