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Some Graph Algorithms

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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

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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

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**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

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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

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**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 →

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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

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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

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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

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**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

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**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)

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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

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**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

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**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.

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**Example automaton Example input string: 1 0 0 1 1 1 0 0**

q0 q1 q2 q3 1 Example input string: 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

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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

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**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}

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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”

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The End

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Topological Sort Example This job consists of 10 tasks with the following precedence rules: Must start with 7, 5, 4 or 9. Task 1 must follow 7. Tasks 3.

Topological Sort Example This job consists of 10 tasks with the following precedence rules: Must start with 7, 5, 4 or 9. Task 1 must follow 7. Tasks 3.

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