By Hang T. Lau

Due to its portability and platform-independence, Java is the best laptop programming language to take advantage of while engaged on graph algorithms and different mathematical programming difficulties. amassing one of the most renowned graph algorithms and optimization approaches, A Java Library of Graph Algorithms and Optimization offers the resource code for a library of Java courses that may be used to resolve difficulties in graph idea and combinatorial optimization. Self-contained and principally self reliant, each one subject begins with an issue description and an overview of the answer strategy, by means of its parameter checklist specification, resource code, and a try out instance that illustrates using the code. The e-book starts off with a bankruptcy on random graph new release that examines bipartite, common, attached, Hamilton, and isomorphic graphs in addition to spanning, classified, and unlabeled rooted timber. It then discusses connectivity tactics, by way of a paths and cycles bankruptcy that comprises the chinese language postman and touring salesman difficulties, Euler and Hamilton cycles, and shortest paths. the writer proceeds to explain try systems concerning planarity and graph isomorphism. next chapters take care of graph coloring, graph matching, community movement, and packing and protecting, together with the project, bottleneck task, quadratic task, a number of knapsack, set protecting, and set partitioning difficulties. the ultimate chapters discover linear, integer, and quadratic programming. The appendices supply references that supply extra info of the algorithms and comprise the definitions of many graph idea phrases utilized in the ebook.

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**Extra resources for A Java Library of Graph Algorithms and Optimization**

**Example text**

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.

Procedure parameters: int randomConnectedGraph (n, m, seed, weighted, minweight, maxweight, nodei, nodej, weight) randomConnectedGraph: int; exit: the method returns the following error code: 0: solution found with normal execution 1: value of m is too small, should be at least n−1 2: value of m is too large, should be at most n∗(n−1)/2 n: int; entry: number of nodes of the graph. Nodes of the graph are labeled from 1 to n. m: int; entry: number of edges of the graph. If m is less than n−1 then m will be set to n−1.

Let p(i) denote the unique predecessor of each node i in the tree, d(i) be the distance from node i to the root of T, b(i) be the label assigned to the edge (i,p(i)), and h(j) be a Boolean variable for marking edge j. For each component of the given graph, perform the following: Step 1. Set b(i) = 0, h(i) = false, for all i. Choose an arbitrary node r as the root. Initially T consists of the single node r, d(r) = 0, X is empty, Y = T, Z = V – {r}. Step 2. If Y is nonempty then select the most recent member of Y, say u, delete u from Y and continue with Step 3.