Some Graph Algorithms

1. Topological sort Suppose a project involves doing a number of tasks, but some of the tasks cannot be done until others are done Your job is to find a legal order in which to do the tasks For example, assume: A must be done before D B must be done before C, D, or E D must be done before H E must be done before D or F F must be done before C H must be done before G or I I must be done before F This is a partial ordering of the tasks Some possible total orderings: A B E D H I F C G B A E D H G I F C A B E D H I G F C

Informal algorithm Extracting a total ordering from a partial ordering is called a topological sort Here’s the basic idea: Repeatedly, Choose a node all of whose “predecessors” have already been chosen A B C D E F G H I A B C F D E I G H Example: Only A or B can be chosen. Choose A. Only B can be chosen. Choose B. Only E can be chosen. Choose E. Continue in this manner until all nodes have been chosen. If all your remaining nodes have predecessors, then there is a cycle in the data, and no solution is possible

Implementing topological sort The graph structure can be implemented in any convenient way We need to keep track of the number of in-edges at each node Whenever we choose a node, we need to decrement the number of in-edges at each of its successors Since we always want a node with the fewest (zero) in-edges, a priority queue seems like a good idea To remove an element from a priority queue and reheap it takes O(log n) time There is a better way

Using buckets We can start with an array of linked lists; array[n] points to the linked list of nodes with n in-edges At each step, Remove a node N from array[0] For each node M that N points to, Get the in-degree d of node M Remove node M from bucket array[d] Add node M to bucket array[d-1] Quit when bucket array[0] is empty As always, it doesn’t make sense to use a high efficiency (but more complex) algorithm if the problem size is small

Bucket example Buckets: 0 → A B 1 → E G H I 2 → C F 3 → D Buckets after choosing B: 0 → A E 1 → G H I C 2 → F D 3 →

2. Connectivity Suppose you want to find out quickly (O(1) time) whether it is possible to get from one node to another in a directed graph You can use an adjacency matrix to represent the graph A B C D E F G A B C D E F G A B C D E F G A B C D E F G A B G E F D C A connectivity table tells us whether it is possible to get from one node to another by following one or more edges

Transitive closure Reachability is transitive: If you can get from A to E, and you can get from E to G, then you can get from A to G A B C D E F G A B C D E F G A B C D E F G A B C D E F G new Warshall’s algorithm is a systematic method of finding the transitive closure of a graph

Warshall’s algorithm Transitivity: If you can get from A to B, and you can get from B to C, then you can get from A to C Warshall’s observation: If you can get from A to B using only nodes with indices less than B, and you can get from B to C, then you can get from A to C using only nodes with indices less than B+1 Warshall’s observation makes it possible to avoid most of the searching that would otherwise be required

Warshall’s algorithm: Implementation for (i = 1; i <= N; i++) { for (j = 1; j <= N; j++) { if (a[j][i]) { for (k = 1; k <= N; k++) { if (a[i][k]) a[j][k] = true; } } } } It’s easy to see that the running time of this algorithm* is O(N3) *Algorithm adapted from Algorithms in C by Robert Sedgewick

3. All-pairs shortest path Closely related to Warshall’s algorithm is Floyd’s algorithm Idea: If you can get from A to B at cost c1, and you can get from B to C with cost c2, then you can get from A to C with cost c1+c2 Of course, as the algorithm proceeds, if you find a lower cost you use that instead The running time of this algorithm is also O(N3)

State graphs The next couple of algorithms are for state graphs, in which each node represents a state of the computation, and the edges between nodes represent state transitions Example: Thread states in Java ready waiting running dead start The edges should be labelled with the causes of the state transitions, but in this example they are too verbose

4. Automata Automata are a formalization of the notion of state graphs Each automaton has a start state, one or more final states, and transitions between states The start state A state transition a A final state

Operation of an automaton An automation represents a “program” to accept or reject a sequence of inputs It operates as follows: Start with the “current state” set to the start state and a “read head” at the beginning of the input string; While there are still characters in the string: Read the next character and advance the read head; From the current state, follow the arc that is labeled with the character just read; the state that the arc points to becomes the next current state; When all characters have been read, accept the string if the current state is a final state, otherwise reject the string.

Example automaton Example input string: 1 0 0 1 1 1 0 0 q0 q1 q2 q3 1 Example input string: 1 0 0 1 1 1 0 0 Sample trace: q0 1 q1 0 q3 0 q1 1 q0 1 q1 1 q0 0 q2 0 q0 Since q0 is a final state, the string is accepted

Example automaton A “hard-wired” automaton is easy to implement in a programming language state := q0; loop case state of q0 : read char; if eof then accept string; if char = 0 then state := q2; if char = 1 then state := q1; q1 : read char; if eof then reject string; if char = 0 then state := q3; if char = 1 then state := q0; q2 : read char; if eof then reject string; if char = 0 then state := q0; if char = 1 then state := q3; q3 : read char; if eof then reject string; if char = 0 then state := q1; if char = 1 then state := q2; end case; end loop; q0 q1 q2 q3 1 • A non-hard-wired automaton can be implemented as a directed graph

5. Nondeterministic automata A nondeterministic automaton is one in which there may be more than one out-edge with the same label A C B a etc. A nondeterministic automaton accepts a sequence of inputs if there is any way to use that string to reach a final state There are two basic ways to implement a nondeterministic automaton: Do a depth-first search, using the inputs to choose the next state Keep a set of all the states you could be in; for example, starting from {A} with input a, you would go to {B, C}

6. String searching Automata can be used to implement efficient string searching Example: Search ABAACAAABCAB for ABABC 1 2 5 3 A B A B C A * 4 The “*” stands for “everything else”

The End