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”)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) } c b d a

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

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)} c b d a Can also be viewed as symmetric directed graph, replacing each undirected edge by a pair of directed edges.

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

10. Edge List 2 1 3 0 Often corresponds to the input format for contest problems. 01 02 12 12 23 Container (set) of edges may be used by algorithms that add/delete edges.

11. Adjacency List 2 1 3 0 0 1 2 3 4 Can save space and time if graph is sparse. 3 23 12 with pointers & dynamic allocation: 01234 02444 with two arrays: 01234 12233

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

13. Adjacency/Incidence Matrix 2 1 3 0 0 123 00110 10011 20001 30000 A[i][j] = 1 → (i,j) i  E A[i][j] = 0 otherwise 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 AlgorithmFloyd-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 int path[][]; // a 2-D matrix. // At each step, path[i][j] is the (cost of the) shortest path // from i to j using 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. procedure FloydWarshall () 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. 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; } Suppose we use path[i][j] == 0 to indicate lack of connection. Example

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

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]);