Basic Graph Algorithms

1. Basic Graph Algorithms Programming Puzzles and Competitions CIS 4900 / 5920 Spring 2009

2. Outline • Introduction/review of graphs • Some basic graph problems & algorithms • Start of an example question from ICPC’07 (“Tunnels”)

3. Relation to Contests • Many programming contest problems can be viewed as graph problems. • Some graph algorithms are complicated, but a few are very simple. • If you can find a way to apply one of these, you will do well.

4. How short & simple? int [][] path = new int[edge.length][edge.length]; for (int i =0; i < n; i++) for (int j = 0; j < n; j++) path[i][j] = edge[i][j]; for (int k = 0; k < n; k++) for (int i =0; i < n; i++) for (int j = 0; j < n; j++) if (path[i][k] != 0 && path[k,j] != 0) { x = path[i][k] + path[k][j]; if ((path[i,j] == 0) || path[i][j] > x) path[i][j] = x; }

5. Directed Graphs • G = (V, E) • V = set of vertices (a.k.a. nodes) • E = set of edges (ordered pairs of nodes)

6. Directed Graph • V = { a, b, c, d } • E = { (a, b), (c, d), (a, c), (b, d), (b, c) } b d a c

7. Undirected Graph • V = { a, b, c, d } • E = { {a, b}, {c, d}, {a, c}, {b, d}, {b, c} } b d a c

8. Undirected Graph as Directed • V = { a, b, c, d } • E = { (a, b), (b,a),(c,d),(d,c),(a,c),(c,a), (b,d),(d,b),(b,c)(c,b)} b d a c Can also be viewed as symmetric directed graph, replacingeach undirected edge by a pair of directed edges.

9. Computer Representations • Edge list • Hash table of edges • Adjacency list • Adjacency matrix

10. Edge List 1 3 0 2 Often corresponds to the input format for contest problems. Container (set) of edges may be used by algorithms that add/delete edges.

11. Adjacency List with two arrays: 1 3 0 with pointers & dynamic allocation: 2 Can save space and time if graph is sparse.

12. Hash Table (Associative Map) H(0,1) H(1,2) etc. 1 1 3 0 1 2 good for storing information about nodes or edges, e.g., edge weight

13. Adjacency/Incidence Matrix A[i][j] = 1 → (i,j) i EA[i][j] = 0 otherwise 1 3 0 2 a very convenient representation for simple coding of algorithms,although it may waste time & space if the graph is sparse.

14. Some Basic Graph Problems • Connectivity, shortest/longest path • Single source • All pairs: Floyd-Warshall Algorithm • dynamic programming, efficient, very simple • MaxFlow (MinCut) • Iterative flow-pushing algorithms

15. Floyd-Warshall Algorithm Assume edgeCost(i,j) returns the cost of the edge from i to j (infinity if there is none), n is the number of vertices, and edgeCost(i,i) = 0 intpath[][]; // a 2-D matrix. // At each step, path[i][j] is the (cost of the) shortest path // from i to jusing intermediate vertices (1..k-1). // Each path[i][j] is initialized to edgeCost (i,j) // or ∞ if there is no edge between i and j. procedureFloydWarshall () for k in 1..n for each pair (i,j) in {1,..,n}x{1,..,n} path[i][j] = min ( path[i][j], path[i][k]+path[k][j] ); * Time complexity: O(|V|3 ).

16. Details • Need some value to represent pairs of nodes that are not connected. • If you are using floating point, there is a value ∞ for which arithmetic works correctly. • But for most graph problems you may want to use integer arithmetic. • Choosing a good value may simplify code When and why to use F.P. vs. integers is an interesting side discussion.

17. Example Suppose we use path[i][j] == 0 to indicate lack of connection. if (path[i][k] != 0 && path[k,j] != 0) { x = path[i][k] + path[k][j]; if ((path[i,j] == 0) || path[i][j] > x) path[i][j] = x; }

18. How it works path[i][j] i j paths that go though only nodes 0..k-1 path[i][k] path[k,j] k

19. Correction In class, I claimed that this algorithm could be adapted to find length of longest cycle-free path, and to count cycle-free paths. That is not true. However there is a generalization to find the maximum flow between points, and the maximum-flow path: for k in 1,..,n for each pair (i,j) in {1,..,n}x{1,..,n} maxflow[i][j] = max (maxflow[i][j] min (maxflow[i][k], maxflow[k][j]);