By Hang T. Lau

Due to its portability and platform-independence, Java is the appropriate desktop programming language to take advantage of while engaged on graph algorithms and different mathematical programming difficulties. accumulating one of the most well known graph algorithms and optimization methods, A Java Library of Graph Algorithms and Optimization presents the resource code for a library of Java courses that may be used to resolve difficulties in graph thought and combinatorial optimization. Self-contained and principally self sufficient, every one subject starts off with an issue description and an summary of the answer technique, through its parameter record specification, resource code, and a try out instance that illustrates using the code. The publication starts off with a bankruptcy on random graph new release that examines bipartite, ordinary, attached, Hamilton, and isomorphic graphs in addition to spanning, categorized, and unlabeled rooted timber. It then discusses connectivity systems, by means of a paths and cycles bankruptcy that includes the chinese language postman and touring salesman difficulties, Euler and Hamilton cycles, and shortest paths. the writer proceeds to explain try out techniques related to planarity and graph isomorphism. next chapters care for graph coloring, graph matching, community circulate, and packing and protecting, together with the project, bottleneck project, quadratic project, a number of knapsack, set masking, and set partitioning difficulties. the ultimate chapters discover linear, integer, and quadratic programming. The appendices supply references that provide extra info of the algorithms and comprise the definitions of many graph concept phrases utilized in the ebook.

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Parent: int[n+1]; exit: parent[i] is the previous node which was visited just before node i; parent[i]=0 if node i is the first node being visited in the component, for i=1,2,…,n. sequence: int[n+1]; exit: sequence[i] is the order in which node i was visited in the search, for i=1,2,…,n. public static void breadthFirstSearch(int n, int m, int nodei[], int nodej[], int parent[], int sequence[]) { int i,j,k,enqueue,dequeue,queuelength,p,q,u,v; int queue[] = new int[n+1]; int firstedges[] = new int[n+2]; int endnode[] = new int[m+1]; boolean mark[] = new boolean[m+1]; boolean iterate,found; // set up the forward star representation of the graph for (j=1; j<=m; j++) mark[j] = true; firstedges[1] = 0; k = 0; for (i=1; i<=n; i++) { for (j=1; j<=m; j++) if (mark[j]) { if (nodei[j] == i) { k++; endnode[k] = nodej[j]; mark[j] = false; } else { if (nodej[j] == i) { k++; endnode[k] = nodei[j]; mark[j] = false; } } © 2007 by Taylor & Francis Group, LLC Chapter 2: Connectivity } firstedges[i+1] = k; } for (i=1; i<=n; i++) { sequence[i] = 0; parent[i] = 0; } k = 0; p = 1; enqueue = 1; dequeue = 1; queuelength = enqueue; queue[enqueue] = p; k++; sequence[p] = k; parent[p] = 0; iterate = true; // store all descendants while (iterate) { for (q=1; q<=n; q++) { // check if p and q are adjacent if (p < q) { u = p; v = q; } else { u = q; v = p; } found = false; for (i=firstedges[u]+1; i<=firstedges[u+1]; i++) if (endnode[i] == v) { // p and q are adjacent found = true; break; } if (found && sequence[q] == 0) { enqueue++; if (n < enqueue) enqueue = 1; queue[enqueue] = q; k++; parent[q] = p; sequence[q] = k; } } // process all nodes of the same height if (enqueue >= dequeue) { if (dequeue == queuelength) { queuelength = enqueue; } © 2007 by Taylor & Francis Group, LLC 45 A Java Library of Graph Algorithms and Optimization 46 p = queue[dequeue]; dequeue++; if (n < dequeue) dequeue = 1; iterate = true; // process other components } else { iterate = false; for (i=1; i<=n; i++) if (sequence[i] == 0) { dequeue = 1; enqueue = 1; queue[enqueue] = i; queuelength = 1; k++; sequence[i] = k; parent[i] = 0; p = i; iterate = true; break; } } } } Example: Apply a breadth-first search to the following graph.

NextDouble() * (maxweight + 1 - minweight)); } } return 0; } Example: Generate a random undirected connected simple graph of 8 nodes and 10 edges with edge weights in the range of [90, 99]. 9 Random Hamilton Graph The following procedure [JK91] generates a random simple Hamilton graph with some given number of nodes and edges. The graph can be directed or undirected. The method generates a random permutation of n objects (perm[1], perm[2], …, perm[n]). The graph is initialized with the Hamilton cycle (perm[1], perm[2]), (perm[2], perm[3]), …, (perm[n−1], perm[n]), (perm[n], perm[1]).

The node i of the first random graph corresponds to the node perm[i] in the second graph. Procedure parameters: int randomIsomorphicGraphs (n, m, seed, simple, directed, firsti, firstj, secondi, secondj, map) randomIsomorphicGraph: int; exit: the method returns the following error code: 0: solution found with normal execution 1: value of m is too large, should be at most n∗(n−1)/2 for simple undirected graph, and n∗(n−1) for simple directed graph. n: int; entry: number of nodes of each graph. Nodes of each graph are labeled from 1 to n.