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Topological SortingChapter 9

CPTR 318

1

DAG

• A directed acyclic graph

– A directed graph

– No cycles

• A DAG “flows” in only one direction

• Vertices form a partial order

– reflexive

– antisymmetric

– transitive

2

Example DAG

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Topological Sort

• A linear ordering of the vertices within a DAG

• A vertex appears before any other vertex connected by an outbound edge

• All DAGs have one or more topological orderings

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Topological Sort

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C

1: A, B, C, D, H, E, F, G, I

2: H, A, B, C, E, F, D, I, G

3: A, B, H, C, E, D, F, G, I

Give a topological sort of this graph

H

I

Applications

• Project planning

– PERT

– Weighted edges: compute critical path

• Scheduling courses with prerequisites

6

2

Topological Sort Algorithm

• An application of depth-first search

7

dfs(Vertex v){

v.mark = visited;do_something(v); for ( each vertex w adjacent to v )

if ( w.mark != visited ) dfs(w);

}

Topological Sort Algorithm (1)

8

// Builds a list in topological order

List topo_dfs(Vertex v, List L)

{

v.mark = visited;

for ( each vertex w adjacent to v )

if ( w.mark != visited )

L = topo_dfs(w, L);

L.push_front(v); // Put v on front of list

return L;

}

Topological Sort Algorithm (2)

9

// Calls topo_dfs on all source vertices

// (vertices with no incoming edges)

List topo_sort(Graph G)

{

list = Ø; // List initially empty

for ( each vertex v in G )

if ( v has no incoming edges )

list = topo_dfs(v, list);

return list;

}

Topological Sort in Action

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Topological Sort in Action

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E

Topological Sort in Action

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E

Topological Sort in Action

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Topological Sort in Action

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Topological Sort in Action

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Topological Sort in Action

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Topological Sort in Action

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Topological Sort in Action

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Topological Sort in Action

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GFE

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Topological Sort in Action

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GFEH

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Topological Sort in Action

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GFEH

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Topological Sort in Action

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GFEH

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Topological Sort in Action

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GFEH

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Topological Sort in Action

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GFEH

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Topological Sort in Action

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GFEH

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H E F

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Topological Sort in Action

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GFEH

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Topological Sort in Action

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GFEH

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Topological Sort in Action

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GFEH

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Topological Sort in Action

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GFEH

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Topological Sort in Action

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GFEH

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Topological Sort in Action

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GFEH

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Topological Sort in Action

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GFEH

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Topological Sort in Action

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Topological Sort in Action

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Topological Sort in Action

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Topological Sort in Action

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GFEHID

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H E F

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Topological Sort in Action

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GFEHIDC

H E F

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Topological Sort in Action

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GFEHIDC

H E F

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Topological Sort in Action

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GFEHIDCB

H E F

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Topological Sort in Action

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GFEHIDCBA

H E F

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Topological Sort Result

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GFEHIDCBA

Alternate Topological Sort

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Alternate Topological Sort

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Alternate Topological Sort

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Alternate Topological Sort

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Alternate Topological Sort

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Alternate Topological Sort

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Alternate Topological Sort

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Alternate Topological Sort

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Alternate Topological Sort

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Alternate Topological Sort

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Alternate Topological Sort

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Alternate Topological Sort

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Alternate Topological Sort

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Alternate Topological Sort

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Alternate Topological Sort

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GFE

Alternate Topological Sort

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GFE

Alternate Topological Sort

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GFE

Alternate Topological Sort

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GFE

Alternate Topological Sort

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GFE

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Alternate Topological Sort

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GFE

Alternate Topological Sort

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GFE

Alternate Topological Sort

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GFE

Alternate Topological Sort

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GFEI

Alternate Topological Sort

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GFEID

Alternate Topological Sort

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GFEIDC

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Alternate Topological Sort

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GFEIDC

Alternate Topological Sort

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GFEIDCB

Alternate Topological Sort

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GFEIDCBA

Alternate Topological Sort

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GFEIDCBA

Alternate Topological Sort

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GFEIDCBAH

Alternate Result

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GFEIDCBAH

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Topological Sort by Numbering

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// Assigns numbers to vertices accessible

// from v in reverse topological order

topo_dfs(Vertex v)

{

for ( each vertex w adjacent to v )

if ( w.number < 0 )

topo_dfs(w);

v.number = N++; // Global N

}

Topological Sort by Numbering

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// Calls topo_dfs on all source vertices

// (vertices with no incoming edges)

topo_sort(Graph G)

{

list = Ø; // List initially empty

N = 0; // Reset global N

for ( each vertex v in G )

v.number = -1; // Each node un-numbered

for ( each vertex v in G )

if ( v has no incoming edges )

topo_dfs(v);

}

Topological Sort Numbering

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-1

-1

-1

-1

-1

-1

-1

-1 -1

Alternate Topological Sort

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-1

-1

-1

-1

-1

-1

-1

-1 -1

Alternate Topological Sort

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-1

-1

-1

-1

-1

-1

-1

-1 -1

Alternate Topological Sort

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A D

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-1

-1

-1

-1

-1

-1

-1

-1 -1

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Alternate Topological Sort

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A D

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-1

-1

-1

-1

-1

-1

-1

-1 -1

Alternate Topological Sort

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A D

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-1

-1

-1

-1

-1

-1

-1

-1 -1

Alternate Topological Sort

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-1

-1

-1

-1

-1

-1

-1

-1 -1

Alternate Topological Sort

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-1

-1

-1

-1

-1

-1

-1

-1 -1

Alternate Topological Sort

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-1

-1

-1

-1

-1

-1

-1

-1 -1

Alternate Topological Sort

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-1

-1

-1

-1

-1

-1

-1

-1 -1

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Alternate Topological Sort

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-1

-1

-1

-1

-1

-1

-1

-1 -1

Alternate Topological Sort

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-1

-1

-1

-1

-1

-1

-1

-1 -1

Alternate Topological Sort

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G

-1

0

-1

-1

-1

-1

-1

-1 -1

0

Alternate Topological Sort

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GF

-1

0

1

-1

-1

-1

-1

-1 -1

01

Alternate Topological Sort

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GFE

-1

0

1

-1

-1

2

-1

-1 -1

012

Alternate Topological Sort

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GFE

-1

0

1

-1

-1

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-1

-1 -1

012

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Alternate Topological Sort

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GFE

-1

0

1

-1

-1

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-1

-1 -1

012

Alternate Topological Sort

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A D

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GFE

-1

0

1

-1

-1

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-1

-1 -1

012

Alternate Topological Sort

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GFE

-1

0

1

-1

-1

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-1

-1 -1

012

Alternate Topological Sort

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GFE

-1

0

1

-1

-1

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-1

-1 -1

012

Alternate Topological Sort

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GFE

-1

0

1

-1

-1

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-1

-1 -1

012

Alternate Topological Sort

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GFE

-1

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-1

-1

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-1

-1 -1

012

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Alternate Topological Sort

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GFEI

-1

0

1

-1

-1

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-1

-1 3

0123

Alternate Topological Sort

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GFEID

-1

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-1

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-1

-1 3

01234

Alternate Topological Sort

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GFEIDC

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-1

-1 3

012345

Alternate Topological Sort

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GFEIDC

-1

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-1

-1 3

012345

Alternate Topological Sort

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GFEIDCB

6

0

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-1

-1 3

0123456

Alternate Topological Sort

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GFEIDCBA

6

0

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-1

7 3

01234567

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Alternate Topological Sort

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GFEIDCBA

6

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-1

7 3

01234567

Alternate Topological Sort

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GFEIDCBAH

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7 3

012345678

Numbering

• The numbers list the vertices in reversetopological order

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Topological Sort Complexity

• Same as DFS: O(|V| + |E|)

– For a dense graph?

– For a sparse graph?

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