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How to Win Coding Competitions: Secrets of Champions
Week 4: Algorithms on GraphsLecture 5: Topological sort
Maxim BuzdalovSaint Petersburg 2016
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Intuition
Assume you are solving a problem from a competition.What should
you do?
And in which order?
Read the statement
Write the solution template
Get the names of I/O files
Invent the algorithmWrite the solution
Submit the solution
2 / 7
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Intuition
Assume you are solving a problem from a competition.What should
you do?
And in which order?
Read the statement
Write the solution template
Get the names of I/O files
Invent the algorithmWrite the solution
Submit the solution
2 / 7
-
Intuition
Assume you are solving a problem from a competition.What should
you do?And in which order?
Read the statement
Write the solution template
Get the names of I/O files
Invent the algorithmWrite the solution
Submit the solution
2 / 7
-
Intuition
Assume you are solving a problem from a competition.What should
you do?And in which order?
Read the statement
Write the solution template
Get the names of I/O files
Invent the algorithmWrite the solution
Submit the solution
2 / 7
-
Topological sort
For the directed acyclic graph G = 〈V ,E 〉,a topological sort is
an assignment I : V → [1;N] of indices to vertices, such that:
I for every two distinct vertices u, v it holds that I (u) 6= I
(v)I for each edge (u, v) ∈ E it holds that I (u) < I (v)
Sometimes it is convenient to have an array o[i ] such that:
I o[I (v)] = v , or, just the same, I (o[i ]) = i
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Topological sort
For the directed acyclic graph G = 〈V ,E 〉,a topological sort is
an assignment I : V → [1;N] of indices to vertices, such that:
I for every two distinct vertices u, v it holds that I (u) 6= I
(v)I for each edge (u, v) ∈ E it holds that I (u) < I (v)
Sometimes it is convenient to have an array o[i ] such that:
I o[I (v)] = v , or, just the same, I (o[i ]) = i
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Topological sort
For the directed acyclic graph G = 〈V ,E 〉,a topological sort is
an assignment I : V → [1;N] of indices to vertices, such that:
I for every two distinct vertices u, v it holds that I (u) 6= I
(v)I for each edge (u, v) ∈ E it holds that I (u) < I (v)
Sometimes it is convenient to have an array o[i ] such that:
I o[I (v)] = v , or, just the same, I (o[i ]) = i
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Topological sort
For the directed acyclic graph G = 〈V ,E 〉,a topological sort is
an assignment I : V → [1;N] of indices to vertices, such that:
I for every two distinct vertices u, v it holds that I (u) 6= I
(v)I for each edge (u, v) ∈ E it holds that I (u) < I (v)
Sometimes it is convenient to have an array o[i ] such that:
I o[I (v)] = v , or, just the same, I (o[i ]) = i
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Topological sort
For the directed acyclic graph G = 〈V ,E 〉,a topological sort is
an assignment I : V → [1;N] of indices to vertices, such that:
I for every two distinct vertices u, v it holds that I (u) 6= I
(v)I for each edge (u, v) ∈ E it holds that I (u) < I (v)
Sometimes it is convenient to have an array o[i ] such that:
I o[I (v)] = v , or, just the same, I (o[i ]) = i
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Topological sort using Depth First Search
procedure TopoSort(V , E )U ← ∅, I ← ∅, t ← |V |, o ← {}A(v) =
{u | (v , u) ∈ E}for v ∈ V do
if v /∈ U then DFS(v , A) end ifend for
end procedure
procedure DFS(v , A)U ← U ∪ vfor u ∈ A(v) do
if u /∈ U then DFS(u, A) end ifend forI (v)← t, o[t]← v , t ← t
− 1
end procedure
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Topological sort using Depth First Search
procedure TopoSort(V , E )U ← ∅, I ← ∅, t ← |V |, o ← {}A(v) =
{u | (v , u) ∈ E}for v ∈ V do
if v /∈ U then DFS(v , A) end ifend for
end procedure
procedure DFS(v , A)U ← U ∪ vfor u ∈ A(v) do
if u /∈ U then DFS(u, A) end ifend forI (v)← t, o[t]← v , t ← t
− 1
end procedure
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Topological sort using Depth First Search
procedure TopoSort(V , E )U ← ∅, I ← ∅, t ← |V |, o ← {}A(v) =
{u | (v , u) ∈ E}for v ∈ V do
if v /∈ U then DFS(v , A) end ifend for
end procedure
procedure DFS(v , A)U ← U ∪ vfor u ∈ A(v) do
if u /∈ U then DFS(u, A) end ifend forI (v)← t, o[t]← v , t ← t
− 1
end procedure
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Topological sort using Depth First Search
procedure TopoSort(V , E )U ← ∅, I ← ∅, t ← |V |, o ← {}A(v) =
{u | (v , u) ∈ E}for v ∈ V do
if v /∈ U then DFS(v , A) end ifend for
end procedure
procedure DFS(v , A)U ← U ∪ vfor u ∈ A(v) do
if u /∈ U then DFS(u, A) end ifend forI (v)← t, o[t]← v , t ← t
− 1
end procedure
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Topological sort using Depth First Search
procedure TopoSort(V , E )U ← ∅, I ← ∅, t ← |V |, o ← {}A(v) =
{u | (v , u) ∈ E}for v ∈ V do
if v /∈ U then DFS(v , A) end ifend for
end procedure
procedure DFS(v , A)U ← U ∪ vfor u ∈ A(v) do
if u /∈ U then DFS(u, A) end ifend forI (v)← t, o[t]← v , t ← t
− 1
end procedure
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Topological sort using Depth First Search
procedure TopoSort(V , E )U ← ∅, I ← ∅, t ← |V |, o ← {}A(v) =
{u | (v , u) ∈ E}for v ∈ V do
if v /∈ U then DFS(v , A) end ifend for
end procedure
procedure DFS(v , A)U ← U ∪ vfor u ∈ A(v) do
if u /∈ U then DFS(u, A) end ifend forI (v)← t, o[t]← v , t ← t
− 1
end procedure
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Topological sort using Depth First Search
procedure TopoSort(V , E )U ← ∅, I ← ∅, t ← |V |, o ← {}A(v) =
{u | (v , u) ∈ E}for v ∈ V do
if v /∈ U then DFS(v , A) end ifend for
end procedure
procedure DFS(v , A)U ← U ∪ vfor u ∈ A(v) do
if u /∈ U then DFS(u, A) end ifend forI (v)← t, o[t]← v , t ← t
− 1
end procedure
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Topological sort using Depth First Search
procedure TopoSort(V , E )U ← ∅, I ← ∅, t ← |V |, o ← {}A(v) =
{u | (v , u) ∈ E}for v ∈ V do
if v /∈ U then DFS(v , A) end ifend for
end procedure
procedure DFS(v , A)U ← U ∪ vfor u ∈ A(v) do
if u /∈ U then DFS(u, A) end ifend forI (v)← t, o[t]← v , t ← t
− 1
end procedure
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Topological sort using Depth First Search
procedure TopoSort(V , E )U ← ∅, I ← ∅, t ← |V |, o ← {}A(v) =
{u | (v , u) ∈ E}for v ∈ V do
if v /∈ U then DFS(v , A) end ifend for
end procedure
procedure DFS(v , A)U ← U ∪ vfor u ∈ A(v) do
if u /∈ U then DFS(u, A) end ifend forI (v)← t, o[t]← v , t ← t
− 1
end procedure
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Topological sort using Depth First Search
procedure TopoSort(V , E )U ← ∅, I ← ∅, t ← |V |, o ← {}A(v) =
{u | (v , u) ∈ E}for v ∈ V do
if v /∈ U then DFS(v , A) end ifend for
end procedure
procedure DFS(v , A)U ← U ∪ vfor u ∈ A(v) do
if u /∈ U then DFS(u, A) end ifend forI (v)← t, o[t]← v , t ← t
− 1
end procedure
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Topological sort using Depth First Search
procedure TopoSort(V , E )U ← ∅, I ← ∅, t ← |V |, o ← {}A(v) =
{u | (v , u) ∈ E}for v ∈ V do
if v /∈ U then DFS(v , A) end ifend for
end procedure
procedure DFS(v , A)U ← U ∪ vfor u ∈ A(v) do
if u /∈ U then DFS(u, A) end ifend forI (v)← t, o[t]← v , t ← t
− 1
end procedure
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Topological sort using Depth First Search
procedure TopoSort(V , E )U ← ∅, I ← ∅, t ← |V |, o ← {}A(v) =
{u | (v , u) ∈ E}for v ∈ V do
if v /∈ U then DFS(v , A) end ifend for
end procedure
procedure DFS(v , A)U ← U ∪ vfor u ∈ A(v) do
if u /∈ U then DFS(u, A) end ifend forI (v)← t, o[t]← v , t ← t
− 1
end procedure
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Topological sort using Depth First Search
procedure TopoSort(V , E )U ← ∅, I ← ∅, t ← |V |, o ← {}A(v) =
{u | (v , u) ∈ E}for v ∈ V do
if v /∈ U then DFS(v , A) end ifend for
end procedure
procedure DFS(v , A)U ← U ∪ vfor u ∈ A(v) do
if u /∈ U then DFS(u, A) end ifend forI (v)← t, o[t]← v , t ← t
− 1
end procedure
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Topological sort using Depth First Search
procedure TopoSort(V , E )U ← ∅, I ← ∅, t ← |V |, o ← {}A(v) =
{u | (v , u) ∈ E}for v ∈ V do
if v /∈ U then DFS(v , A) end ifend for
end procedure
procedure DFS(v , A)U ← U ∪ vfor u ∈ A(v) do
if u /∈ U then DFS(u, A) end ifend forI (v)← t, o[t]← v , t ← t
− 1
end procedure
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Topological sort using Depth First Search
procedure TopoSort(V , E )U ← ∅, I ← ∅, t ← |V |, o ← {}A(v) =
{u | (v , u) ∈ E}for v ∈ V do
if v /∈ U then DFS(v , A) end ifend for
end procedure
procedure DFS(v , A)U ← U ∪ vfor u ∈ A(v) do
if u /∈ U then DFS(u, A) end ifend forI (v)← t, o[t]← v , t ← t
− 1
end procedure
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Topological sort using Depth First Search
procedure TopoSort(V , E )U ← ∅, I ← ∅, t ← |V |, o ← {}A(v) =
{u | (v , u) ∈ E}for v ∈ V do
if v /∈ U then DFS(v , A) end ifend for
end procedure
procedure DFS(v , A)U ← U ∪ vfor u ∈ A(v) do
if u /∈ U then DFS(u, A) end ifend forI (v)← t, o[t]← v , t ← t
− 1
end procedure
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Topological sort using Depth First Search
procedure TopoSort(V , E )U ← ∅, I ← ∅, t ← |V |, o ← {}A(v) =
{u | (v , u) ∈ E}for v ∈ V do
if v /∈ U then DFS(v , A) end ifend for
end procedure
procedure DFS(v , A)U ← U ∪ vfor u ∈ A(v) do
if u /∈ U then DFS(u, A) end ifend forI (v)← t, o[t]← v , t ← t
− 1
end procedure
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Topological sort using Depth First Search
procedure TopoSort(V , E )U ← ∅, I ← ∅, t ← |V |, o ← {}A(v) =
{u | (v , u) ∈ E}for v ∈ V do
if v /∈ U then DFS(v , A) end ifend for
end procedure
procedure DFS(v , A)U ← U ∪ vfor u ∈ A(v) do
if u /∈ U then DFS(u, A) end ifend forI (v)← t, o[t]← v , t ← t
− 1
end procedure
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Topological sort using Depth First Search
procedure TopoSort(V , E )U ← ∅, I ← ∅, t ← |V |, o ← {}A(v) =
{u | (v , u) ∈ E}for v ∈ V do
if v /∈ U then DFS(v , A) end ifend for
end procedure
procedure DFS(v , A)U ← U ∪ vfor u ∈ A(v) do
if u /∈ U then DFS(u, A) end ifend forI (v)← t, o[t]← v , t ← t
− 1
end procedure
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Topological sort using Depth First Search
procedure TopoSort(V , E )U ← ∅, I ← ∅, t ← |V |, o ← {}A(v) =
{u | (v , u) ∈ E}for v ∈ V do
if v /∈ U then DFS(v , A) end ifend for
end procedure
procedure DFS(v , A)U ← U ∪ vfor u ∈ A(v) do
if u /∈ U then DFS(u, A) end ifend forI (v)← t, o[t]← v , t ← t
− 1
end procedure
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Topological sort using Depth First Search
procedure TopoSort(V , E )U ← ∅, I ← ∅, t ← |V |, o ← {}A(v) =
{u | (v , u) ∈ E}for v ∈ V do
if v /∈ U then DFS(v , A) end ifend for
end procedure
procedure DFS(v , A)U ← U ∪ vfor u ∈ A(v) do
if u /∈ U then DFS(u, A) end ifend forI (v)← t, o[t]← v , t ← t
− 1
end procedure
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Topological sort using Depth First Search
procedure TopoSort(V , E )U ← ∅, I ← ∅, t ← |V |, o ← {}A(v) =
{u | (v , u) ∈ E}for v ∈ V do
if v /∈ U then DFS(v , A) end ifend for
end procedure
procedure DFS(v , A)U ← U ∪ vfor u ∈ A(v) do
if u /∈ U then DFS(u, A) end ifend forI (v)← t, o[t]← v , t ← t
− 1
end procedure
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Topological sort using Depth First Search
procedure TopoSort(V , E )U ← ∅, I ← ∅, t ← |V |, o ← {}A(v) =
{u | (v , u) ∈ E}for v ∈ V do
if v /∈ U then DFS(v , A) end ifend for
end procedure
procedure DFS(v , A)U ← U ∪ vfor u ∈ A(v) do
if u /∈ U then DFS(u, A) end ifend forI (v)← t, o[t]← v , t ← t
− 1
end procedure
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Topological sort using Depth First Search
procedure TopoSort(V , E )U ← ∅, I ← ∅, t ← |V |, o ← {}A(v) =
{u | (v , u) ∈ E}for v ∈ V do
if v /∈ U then DFS(v , A) end ifend for
end procedure
procedure DFS(v , A)U ← U ∪ vfor u ∈ A(v) do
if u /∈ U then DFS(u, A) end ifend forI (v)← t, o[t]← v , t ← t
− 1
end procedure
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Topological sort using Depth First Search
procedure TopoSort(V , E )U ← ∅, I ← ∅, t ← |V |, o ← {}A(v) =
{u | (v , u) ∈ E}for v ∈ V do
if v /∈ U then DFS(v , A) end ifend for
end procedure
procedure DFS(v , A)U ← U ∪ vfor u ∈ A(v) do
if u /∈ U then DFS(u, A) end ifend forI (v)← t, o[t]← v , t ← t
− 1
end procedure
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Topological sort using Depth First Search
procedure TopoSort(V , E )U ← ∅, I ← ∅, t ← |V |, o ← {}A(v) =
{u | (v , u) ∈ E}for v ∈ V do
if v /∈ U then DFS(v , A) end ifend for
end procedure
procedure DFS(v , A)U ← U ∪ vfor u ∈ A(v) do
if u /∈ U then DFS(u, A) end ifend forI (v)← t, o[t]← v , t ← t
− 1
end procedure
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Topological sort using Depth First Search
procedure TopoSort(V , E )U ← ∅, I ← ∅, t ← |V |, o ← {}A(v) =
{u | (v , u) ∈ E}for v ∈ V do
if v /∈ U then DFS(v , A) end ifend for
end procedure
procedure DFS(v , A)U ← U ∪ vfor u ∈ A(v) do
if u /∈ U then DFS(u, A) end ifend forI (v)← t, o[t]← v , t ← t
− 1
end procedure
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Topological sort using Depth First Search
procedure TopoSort(V , E )U ← ∅, I ← ∅, t ← |V |, o ← {}A(v) =
{u | (v , u) ∈ E}for v ∈ V do
if v /∈ U then DFS(v , A) end ifend for
end procedure
procedure DFS(v , A)U ← U ∪ vfor u ∈ A(v) do
if u /∈ U then DFS(u, A) end ifend forI (v)← t, o[t]← v , t ← t
− 1
end procedure
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Topological sort using Depth First Search
procedure TopoSort(V , E )U ← ∅, I ← ∅, t ← |V |, o ← {}A(v) =
{u | (v , u) ∈ E}for v ∈ V do
if v /∈ U then DFS(v , A) end ifend for
end procedure
procedure DFS(v , A)U ← U ∪ vfor u ∈ A(v) do
if u /∈ U then DFS(u, A) end ifend forI (v)← t, o[t]← v , t ← t
− 1
end procedure
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Topological sort using Depth First Search
procedure TopoSort(V , E )U ← ∅, I ← ∅, t ← |V |, o ← {}A(v) =
{u | (v , u) ∈ E}for v ∈ V do
if v /∈ U then DFS(v , A) end ifend for
end procedure
procedure DFS(v , A)U ← U ∪ vfor u ∈ A(v) do
if u /∈ U then DFS(u, A) end ifend forI (v)← t, o[t]← v , t ← t
− 1
end procedure
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Topological sort using Depth First Search
procedure TopoSort(V , E )U ← ∅, I ← ∅, t ← |V |, o ← {}A(v) =
{u | (v , u) ∈ E}for v ∈ V do
if v /∈ U then DFS(v , A) end ifend for
end procedure
procedure DFS(v , A)U ← U ∪ vfor u ∈ A(v) do
if u /∈ U then DFS(u, A) end ifend forI (v)← t, o[t]← v , t ← t
− 1
end procedure
7
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Topological sort using Depth First Search
procedure TopoSort(V , E )U ← ∅, I ← ∅, t ← |V |, o ← {}A(v) =
{u | (v , u) ∈ E}for v ∈ V do
if v /∈ U then DFS(v , A) end ifend for
end procedure
procedure DFS(v , A)U ← U ∪ vfor u ∈ A(v) do
if u /∈ U then DFS(u, A) end ifend forI (v)← t, o[t]← v , t ← t
− 1
end procedure
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Topological sort using Depth First Search
procedure TopoSort(V , E )U ← ∅, I ← ∅, t ← |V |, o ← {}A(v) =
{u | (v , u) ∈ E}for v ∈ V do
if v /∈ U then DFS(v , A) end ifend for
end procedure
procedure DFS(v , A)U ← U ∪ vfor u ∈ A(v) do
if u /∈ U then DFS(u, A) end ifend forI (v)← t, o[t]← v , t ← t
− 1
end procedure
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Topological sort and graphs with cycles
What will happen if topological sort is called on a graph with
cycles?
I DFS will eventually find a cycle
I It may return an error which indicates that there is no
topological sortI But what if we ignore it?
I Every two vertices u and v , where u is reachable from v , but
not vice versa,will have I (u) > I (v)
I Condensation of a graph G : a graph whose vertices are strong
connectivitycomponents of G and whose edges are the edges between
vertices from differentcomponents
I Condensation is a treeI . . . and this tree will be traversed
in a correct order by topological sort!I So we may use topological
sort to find the strong connectivity components
5 / 7
-
Topological sort and graphs with cycles
What will happen if topological sort is called on a graph with
cycles?
I DFS will eventually find a cycle
I It may return an error which indicates that there is no
topological sortI But what if we ignore it?
I Every two vertices u and v , where u is reachable from v , but
not vice versa,will have I (u) > I (v)
I Condensation of a graph G : a graph whose vertices are strong
connectivitycomponents of G and whose edges are the edges between
vertices from differentcomponents
I Condensation is a treeI . . . and this tree will be traversed
in a correct order by topological sort!I So we may use topological
sort to find the strong connectivity components
5 / 7
-
Topological sort and graphs with cycles
What will happen if topological sort is called on a graph with
cycles?
I DFS will eventually find a cycle
I It may return an error which indicates that there is no
topological sort
I But what if we ignore it?
I Every two vertices u and v , where u is reachable from v , but
not vice versa,will have I (u) > I (v)
I Condensation of a graph G : a graph whose vertices are strong
connectivitycomponents of G and whose edges are the edges between
vertices from differentcomponents
I Condensation is a treeI . . . and this tree will be traversed
in a correct order by topological sort!I So we may use topological
sort to find the strong connectivity components
5 / 7
-
Topological sort and graphs with cycles
What will happen if topological sort is called on a graph with
cycles?
I DFS will eventually find a cycle
I It may return an error which indicates that there is no
topological sortI But what if we ignore it?
I Every two vertices u and v , where u is reachable from v , but
not vice versa,will have I (u) > I (v)
I Condensation of a graph G : a graph whose vertices are strong
connectivitycomponents of G and whose edges are the edges between
vertices from differentcomponents
I Condensation is a treeI . . . and this tree will be traversed
in a correct order by topological sort!I So we may use topological
sort to find the strong connectivity components
5 / 7
-
Topological sort and graphs with cycles
What will happen if topological sort is called on a graph with
cycles?
I DFS will eventually find a cycle
I It may return an error which indicates that there is no
topological sortI But what if we ignore it?
I Every two vertices u and v , where u is reachable from v , but
not vice versa,will have I (u) > I (v)
I Condensation of a graph G : a graph whose vertices are strong
connectivitycomponents of G and whose edges are the edges between
vertices from differentcomponents
I Condensation is a treeI . . . and this tree will be traversed
in a correct order by topological sort!I So we may use topological
sort to find the strong connectivity components
5 / 7
-
Topological sort and graphs with cycles
What will happen if topological sort is called on a graph with
cycles?
I DFS will eventually find a cycle
I It may return an error which indicates that there is no
topological sortI But what if we ignore it?
I Every two vertices u and v , where u is reachable from v , but
not vice versa,will have I (u) > I (v)
I Condensation of a graph G : a graph whose vertices are strong
connectivitycomponents of G and whose edges are the edges between
vertices from differentcomponents
I Condensation is a treeI . . . and this tree will be traversed
in a correct order by topological sort!I So we may use topological
sort to find the strong connectivity components
5 / 7
-
Topological sort and graphs with cycles
What will happen if topological sort is called on a graph with
cycles?
I DFS will eventually find a cycle
I It may return an error which indicates that there is no
topological sortI But what if we ignore it?
I Every two vertices u and v , where u is reachable from v , but
not vice versa,will have I (u) > I (v)
I Condensation of a graph G : a graph whose vertices are strong
connectivitycomponents of G and whose edges are the edges between
vertices from differentcomponents
I Condensation is a tree
I . . . and this tree will be traversed in a correct order by
topological sort!I So we may use topological sort to find the
strong connectivity components
5 / 7
-
Topological sort and graphs with cycles
What will happen if topological sort is called on a graph with
cycles?
I DFS will eventually find a cycle
I It may return an error which indicates that there is no
topological sortI But what if we ignore it?
I Every two vertices u and v , where u is reachable from v , but
not vice versa,will have I (u) > I (v)
I Condensation of a graph G : a graph whose vertices are strong
connectivitycomponents of G and whose edges are the edges between
vertices from differentcomponents
I Condensation is a treeI . . . and this tree will be traversed
in a correct order by topological sort!
I So we may use topological sort to find the strong connectivity
components
5 / 7
-
Topological sort and graphs with cycles
What will happen if topological sort is called on a graph with
cycles?
I DFS will eventually find a cycle
I It may return an error which indicates that there is no
topological sortI But what if we ignore it?
I Every two vertices u and v , where u is reachable from v , but
not vice versa,will have I (u) > I (v)
I Condensation of a graph G : a graph whose vertices are strong
connectivitycomponents of G and whose edges are the edges between
vertices from differentcomponents
I Condensation is a treeI . . . and this tree will be traversed
in a correct order by topological sort!I So we may use topological
sort to find the strong connectivity components
5 / 7
-
Condensation
procedure Condensation(V , E )TopoSort(V , E )AR(v) = {u | (v ,
u) ∈ E}C ← {−1}K ← 1for i from 1 to |V | do
if C [o[i ]] = −1 thenDFS(o[i ], AR , K )K ← K + 1
end ifend for
end procedure
procedure DFS(v , A, K )C [v ]← Kfor u ∈ A(v) do
if C [u] 6= −1 thenDFS(u, A, K )
end ifend for
end procedure
I Second DFS traverses vertices in theorder of topological
sort
I . . . and does it using reversed edges
6 / 7
-
Condensation
procedure Condensation(V , E )TopoSort(V , E )AR(v) = {u | (v ,
u) ∈ E}C ← {−1}K ← 1for i from 1 to |V | do
if C [o[i ]] = −1 thenDFS(o[i ], AR , K )K ← K + 1
end ifend for
end procedure
procedure DFS(v , A, K )C [v ]← Kfor u ∈ A(v) do
if C [u] 6= −1 thenDFS(u, A, K )
end ifend for
end procedure
I Second DFS traverses vertices in theorder of topological
sort
I . . . and does it using reversed edges
6 / 7
-
Condensation
procedure Condensation(V , E )TopoSort(V , E )AR(v) = {u | (v ,
u) ∈ E}C ← {−1}K ← 1for i from 1 to |V | do
if C [o[i ]] = −1 thenDFS(o[i ], AR , K )K ← K + 1
end ifend for
end procedure
procedure DFS(v , A, K )C [v ]← Kfor u ∈ A(v) do
if C [u] 6= −1 thenDFS(u, A, K )
end ifend for
end procedure
I Second DFS traverses vertices in theorder of topological
sort
I . . . and does it using reversed edges
6 / 7
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Example of condensation
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Example of condensation
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Example of condensation
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Example of condensation
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Example of condensation
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Example of condensation
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Example of condensation
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Example of condensation
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Example of condensation
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Example of condensation
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Example of condensation
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Example of condensation
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Example of condensation
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Example of condensation
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Example of condensation
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Example of condensation
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Example of condensation
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Example of condensation
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Example of condensation
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Example of condensation
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Example of condensation
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Example of condensation
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Example of condensation
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Example of condensation
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Example of condensation
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Example of condensation
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Example of condensation
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Example of condensation
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Example of condensation
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Example of condensation
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Example of condensation
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Example of condensation
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Example of condensation
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Example of condensation
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Example of condensation
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Example of condensation
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Example of condensation
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Example of condensation
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Example of condensation
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Example of condensation
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Example of condensation
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Example of condensation
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Example of condensation
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Example of condensation
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Example of condensation
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7 / 7
