how to make barcode in c#.net Decompositions of graphs in Software

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A wide range of problems can be expressed with clarity and precision in the concise pictorial language of graphs For instance, consider the task of coloring a political map What is the minimum number of colors needed, with the obvious restriction that neighboring countries should have different colors One of the dif culties in attacking this problem is that the map itself, even a stripped-down version like Figure 31(a), is usually cluttered with irrelevant information: intricate boundaries, border posts where three or more countries meet, open seas, and meandering rivers Such distractions are absent from the mathematical object of Figure 31(b), a graph with one vertex for each country (1 is Brazil, 11 is Argentina) and edges between neighbors It contains exactly the information needed for coloring, and nothing more The precise goal is now to assign a color to each vertex so that no edge has endpoints of the same color Graph coloring is not the exclusive domain of map designers Suppose a university needs to schedule examinations for all its classes and wants to use the fewest time slots possible The only constraint is that two exams cannot be scheduled concurrently if some student will be taking both of them To express this problem as a graph, use one vertex for each exam and put an edge between two vertices if there is a con ict, that is, if there is somebody taking both endpoint exams Think of each time slot as having its own color Then, assigning time slots is exactly the same as coloring this graph! Some basic operations on graphs arise with such frequency, and in such a diversity of contexts, that a lot of effort has gone into nding ef cient procedures for them This chapter is devoted to some of the most fundamental of these algorithms those that uncover the basic connectivity structure of a graph Formally, a graph is speci ed by a set of vertices (also called nodes) V and by edges E between select pairs of vertices In the map example, V = {1, 2, 3, , 13} and E includes, among many other edges, {1, 2}, {9, 11}, and {7, 13} Here an edge between x and y speci cally means x shares a border with y This is a symmetric relation it implies also that y shares a border with x and we denote it using set notation, e = {x, y} Such edges are undirected and are part of an undirected graph Sometimes graphs depict relations that do not have this reciprocity, in which case it is 83
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Figure 31 (a) A map and (b) its graph (a) (b)
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necessary to use edges with directions on them There can be directed edges e from x to y (written e = (x, y)), or from y to x (written (y, x)), or both A particularly enormous example of a directed graph is the graph of all links in the World Wide Web It has a vertex for each site on the Internet, and a directed edge (u, v) whenever site u has a link to site v: in total, billions of nodes and edges! Understanding even the most basic connectivity properties of the Web is of great economic and social interest Although the size of this problem is daunting, we will soon see that a lot of valuable information about the structure of a graph can, happily, be determined in just linear time
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How is a graph represented
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We can represent a graph by an adjacency matrix; if there are n = |V | vertices v 1 , , vn , this is an n n array whose (i, j)th entry is aij = 1 if there is an edge from vi to vj 0 otherwise
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For undirected graphs, the matrix is symmetric since an edge {u, v} can be taken in either direction The biggest convenience of this format is that the presence of a particular edge can be checked in constant time, with just one memory access On the other hand the matrix takes up O(n2 ) space, which is wasteful if the graph does not have very many edges An alternative representation, with size proportional to the number of edges, is the adjacency list It consists of |V | linked lists, one per vertex The linked list for vertex u holds the names of vertices to which u has an outgoing edge that is, vertices v for which (u, v) E 84
Therefore, each edge appears in exactly one of the linked lists if the graph is directed or two of the lists if the graph is undirected Either way, the total size of the data structure is O(|E|) Checking for a particular edge (u, v) is no longer constant time, because it requires sifting through u s adjacency list But it is easy to iterate through all neighbors of a vertex (by running down the corresponding linked list), and, as we shall soon see, this turns out to be a very useful operation in graph algorithms Again, for undirected graphs, this representation has a symmetry of sorts: v is in u s adjacency list if and only if u is in v s adjacency list
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