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Basic Graph Algorithms: Connectivity, Cycle Detection, Shortest Paths, and More

This chapter covers essential graph algorithms focusing on both unweighted and weighted graphs or digraphs. Key topics include determining connectivity using depth-first traversal, detecting cycles with back edges, and finding the shortest path using breadth-first traversal for unweighted and weighted cases. Additionally, concepts like spanning trees, minimum spanning trees, and dynamic graph modifications (adding/removing vertices and edges) are discussed. These foundational algorithms are critical for efficient data structure manipulation and problem-solving in Java.

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Basic Graph Algorithms: Connectivity, Cycle Detection, Shortest Paths, and More

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  1. Modified Chapter 15 Graphs Part C – Graph Algorithms 1

  2. Basic Graph Algorithms • Un-weighted graphs/digraphs • Connectivity • Cycle detection • Shortest path • Spanning tree • Topological sort Java Software Structures, 4th Edition, Lewis/Chase

  3. Basic Graph Algorithms • Weighted graphs/digraphs • Minimum spanning tree • Cheapest path • Traveling salesman problem Java Software Structures, 4th Edition, Lewis/Chase

  4. Dynamic graphs • Adding and removing vertices • Adding and removing edges Java Software Structures, 4th Edition, Lewis/Chase

  5. Java Software Structures, 4th Edition, Lewis/Chase

  6. Connectivity • For a graph (undirected) how can we determine if it is connected? • Answer: Do a depth first traversal and see if all of the vertices are reached. • For a digraph, how do can we determine if it is connected? • Answer: Do a depth first traversal from every vertex and see if all other vertices are reached. Java Software Structures, 4th Edition, Lewis/Chase

  7. Java Software Structures, 4th Edition, Lewis/Chase

  8. Cycle detection • For a graph, how can we detect if it contains any cycles? • Do a depth first traversal and look for “back edges”, edges from current node to a node already visited. • For a digraph – • Do the same, for every vertex as starting vertex. Java Software Structures, 4th Edition, Lewis/Chase

  9. Finding shortest path (un-weighted) • For a graph, how do we find the shortest path (fewest edges) from vertex A to vertex B? • Do a breadth first traversal starting at A and going until B is reached, recording the path length and predecessor at each node visited. • Need more info stored at each node visited. • For a digraph – • Same Java Software Structures, 4th Edition, Lewis/Chase

  10. Shortest path form A to B create a queue Q visit A, mark Aas visited, set path length=0, set pred=null,and put A into Q repeat until B is visited remove the head element u of Q for each w: (unvisited) neighbors of u visit w, mark w, set path length of w, set pred of w=u, and enqueue w Follow the preds backwards from B to A Java Software Structures, 4th Edition, Lewis/Chase

  11. Java Software Structures, 4th Edition, Lewis/Chase

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