Graph Algorithms: Topological Sort

The topological sorting problem: given a directed, acyclic graph G = (V, E) , find a linear ordering of the vertices such that for all (v, w) E, v precedes w in the ordering.

A

BC

F

D E

Graph Algorithms: Topological Sort

The topological sorting problem: given a directed, acyclic graph G = (V, E) , find a linear ordering of the vertices such that for all (v, w) E, v precedes w in the ordering.

A

BC

F

D

EA D

E

FB C

Graph Algorithms: Topological Sort

The topological sorting problem: given a directed, acyclic graph G = (V, E) , find a linear ordering of the vertices such that for all (v, w) E, v precedes w in the ordering.

A

BC

F

D

EA D

E

FB C

Any linear ordering in whichall the arrows go to the right.

Graph Algorithms: Topological Sort

The topological sorting problem: given a directed, acyclic graph G = (V, E) , find a linear ordering of the vertices such that for all (v, w) E, v precedes w in the ordering.

A

BC

F

D

FA D

E

EB C

Any linear ordering in whichall the arrows go to the right.

This is not a topological ordering.

Graph Algorithms: Topological Sort

The topological sorting algorithm: Identify the subset of vertices that have no incoming edge.

A

BC

F

D E

Graph Algorithms: Topological Sort

The topological sorting algorithm: Identify the subset of vertices that have no incoming edge. (In general, this subset must be nonempty—why?)

A

BC

F

D E

Graph Algorithms: Topological Sort

The topological sorting algorithm: Identify the subset of vertices that have no incoming edge. (In general, this subset must be nonempty—because the graph is acyclic.)

A

BC

F

D E

Graph Algorithms: Topological Sort

The topological sorting algorithm: Identify the subset of vertices that have no incoming edge. Select one of them.

A

BC

F

D E

Graph Algorithms: Topological Sort

The topological sorting algorithm: Remove it, and its outgoing edges, and add it to the output.

A

BC

F

D E

Graph Algorithms: Topological Sort

The topological sorting algorithm: Again, identify the subset of vertices that have no incoming edge, . . .

A

BC

F

D E

Graph Algorithms: Topological Sort

The topological sorting algorithm: Again, identify the subset of vertices that have no incoming edge, select one of them, . . .

A

BC

F

D E

Graph Algorithms: Topological Sort

The topological sorting algorithm: Again, identify the subset of vertices that have no incoming edge, select one of them, remove it and any outgoing edges, and put it in the output.

A

BC

F

D E

Graph Algorithms: Topological Sort

The topological sorting algorithm: Again, identify the subset of vertices that have no incoming edge, select one of them, remove it and any outgoing edges, and put it in the output.

A

BC

F

D E

Graph Algorithms: Topological Sort

The topological sorting algorithm: Again, identify the subset of vertices that have no incoming edge, select one of them, remove it and any outgoing edges, and put it in the output.

A B

C

F

D E

Graph Algorithms: Topological Sort

The topological sorting algorithm: Again, identify the subset of vertices that have no incoming edge, select one of them, remove it and any outgoing edges, and put it in the output.

A B

C

F

D E

Graph Algorithms: Topological Sort

The topological sorting algorithm: Again, identify the subset of vertices that have no incoming edge, select one of them, remove it and any outgoing edges, and put it in the output.

A B CF

D E

Graph Algorithms: Topological Sort

The topological sorting algorithm: Again, identify the subset of vertices that have no incoming edge, select one of them, remove it and any outgoing edges, and put it in the output.

A B CF

D E

Graph Algorithms: Topological Sort

The topological sorting algorithm: Again, identify the subset of vertices that have no incoming edge, select one of them, remove it and any outgoing edges, and put it in the output.

A B CF D

E

Graph Algorithms: Topological Sort

The topological sorting algorithm: Again, identify the subset of vertices that have no incoming edge, select one of them, remove it and any outgoing edges, and put it in the output.

A B CF D

E

Graph Algorithms: Topological Sort

The topological sorting algorithm: Again, identify the subset of vertices that have no incoming edge, select one of them, remove it and any outgoing edges, and put it in the output.

A B CF D E

Graph Algorithms: Topological Sort

The topological sorting algorithm: finished!

A B CF D EA

BC

F

D E

Graph Algorithms: Topological Sort

The topological sorting algorithm: Time bound?

A B CF D EA

BC

F

D E

Graph Algorithms: Topological Sort

The topological sorting algorithm: Time bound: Break down into total time to:

Find vertices with no predecessors: ?Remove edges: ?Place vertices in output: ?

A B CF D EA

BC

F

D E

Graph Algorithms: Topological Sort

The topological sorting algorithm: Time bound: Break down into total time to:

Find vertices with no predecessors: ?Remove edges: O(|E|)Place vertices in output: ?

A B CF D EA

BC

F

D E

Graph Algorithms: Topological Sort

The topological sorting algorithm: Time bound: Break down into total time to:

Find vertices with no predecessors: ?Remove edges: O(|E|)Place vertices in output: O(|V|)

A B CF D EA

BC

F

D E

Graph Algorithms: Topological Sort

Find vertices with no predecessors: ?

A

BC

F

D E

A

B

C

D

E

F

B D

E

ED

C

Assume an adjacency list representation:

Graph Algorithms: Topological Sort

The topological sorting algorithm:

…and initialize and maintain for each vertex its no. of predecessors.

A

BC

F

D E

A

B

C

D

E

F

B D

E

ED

C00

1

2

1

2

0

1

0

2

2

1

Graph Algorithms: Topological Sort

Find vertices with no predecessors: ?

A

BC

F

D E

Time for each vertex: O(|V|)

A

B

C

D

E

F

B D

E

ED

C

0

1

0

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2

1

Graph Algorithms: Topological Sort

Find vertices with no predecessors: ?

A

BC

F

D E

Total time: O(|V| )2

A

B

C

D

E

F

B D

E

ED

C

0

1

0

2

2

1

Graph Algorithms: Topological Sort

The topological sorting algorithm: Time bound: Break down into total time to:

Find vertices with no predecessors: O(|V| )Remove edges: O(|E|)Place vertices in output: O(|V|)

2

Graph Algorithms: Topological Sort

The topological sorting algorithm: Time bound: Break down into total time to:

Find vertices with no predecessors: O(|V| )Remove edges: O(|E|)Place vertices in output: O(|V|)

2

Total: O(|V| + |E|)2

Graph Algorithms: Topological Sort

The topological sorting algorithm: Time bound: Break down into total time to:

Find vertices with no predecessors: O(|V| )Remove edges: O(|E|)Place vertices in output: O(|V|)

2

Total: O(|V| + |E|)2

Too much!

Graph Algorithms: Topological Sort

The topological sorting algorithm:We need a faster way to do this step: Find vertices with no predecessors.

Graph Algorithms: Topological Sort

The topological sorting algorithm:Key idea: initialize and maintain a queue (or stack)holding pointers to the vertices with 0 predecessors

A

BC

F

D E

A

B

C

D

E

F

B D

E

ED

C00

1

2

1

2

0

1

0

2

2

1

Graph Algorithms: Topological Sort

The topological sorting algorithm:As each vertex is removed, update the predecessorcounts, and for any vertex whose count has becomezero, put it in the queue.

A

BC

F

D E

A

B

C

D

E

F

B D

E

ED

C00

1

2

1

2

0

1

0

2

2

1

Graph Algorithms: Topological Sort

The topological sorting algorithm:As each vertex is removed, update the predecessorcounts, and for any vertex whose count has becomezero, put it in the queue.

BC

D E

A

B

C

D

E

F

E

ED

C

0

0

1

1

2

0

0

0

1

2

1

F

Output: A

No scan is required, so O(1).

Graph Algorithms: Topological Sort

The topological sorting algorithm:As each vertex is removed, update the predecessorcounts, and for any vertex whose count has becomezero, put it in the queue.

BC

D E

A

B

C

D

E

F

E

ED

C0

1

1

2

0

0

0

1

2

1

Output: A F

Graph Algorithms: Topological Sort

The topological sorting algorithm:As each vertex is removed, update the predecessorcounts, and for any vertex whose count has becomezero, put it in the queue.

C

D E

A

B

C

D

E

F

E

ED

1

0

2

0

0

0

1

2

0

Output: A F B

Graph Algorithms: Topological Sort

The topological sorting algorithm:As each vertex is removed, update the predecessorcounts, and for any vertex whose count has becomezero, put it in the queue.

D E

A

B

C

D

E

F

E0 1

0

0

0

0

1

0

Output: A F B C

Graph Algorithms: Topological Sort

The topological sorting algorithm:As each vertex is removed, update the predecessorcounts, and for any vertex whose count has becomezero, put it in the queue.

E

A

B

C

D

E

F

0

0

0

0

0

0

0

Output: A F B C D

Graph Algorithms: Topological Sort

The topological sorting algorithm:As each vertex is removed, update the predecessorcounts, and for any vertex whose count has becomezero, put it in the queue.

Finished!

A

B

C

D

E

F

0

0

0

0

0

0

Output: A F B C D E

Graph Algorithms: Topological Sort

The topological sorting algorithm: Time bound: Now the time for each part is

Find vertices with no predecessors: O(|V|)Remove edges: O(|E|)Place vertices in output: O(|V|)

Total: O(|V|+|E|)

Linear in |V|+|E|.Much better!