1. Topological
Sort
CSE 373
Data Structures
Lecture 19

2. Topological Sort Problem: Find an order in which all these be taken.
142
322
143
321
Problem: Find an order in
which all these
be taken.
courses
can
326
370
341
Example:
370
326
142
321
421
143
341
401
378
322
378
421
In order
to take
a course, you must
401
take all of its prerequisites first
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Topological Sort - Lecture 19 3

3. Topological Sort Given a digraph G = (V, its vertices such that:
E), find a linear ordering of
for
any
edge
(v,
w) in E,
v precedes
w in
the
ordering B C A F D E
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Topological Sort - Lecture 19 4

4. Topo sort - good example B C B C Any linear ordering
all the arrows go to
in which A
the
right F is
a valid solution D E A B F C D E
Note that F can go anywhere in this list because it is not connected.
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Topological Sort - Lecture 19 5

5. Topo sort - bad example B C B C Any linear ordering in which
an arrow goes to the left A F is
not a
valid
solution D E A B F C E D
NO!
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6. Not all can be Sorted • A directed graph with a cycle cannot be
topologically
sorted. B C A F D E
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7. vertices v1,v2, …,vk Cycles • Given a digraph G = (V,E), a cycle is a
sequence of
that
vertices v1,v2,
…,vk
such › G
k < 1
v1 = vk
(vi,vi+1) in E
for 1 < i < k. •
is acyclic if it has no cycles.
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Topological Sort - Lecture 19 8

8. Topo
sort algorithm - 1
Step 1: Identify vertices that have no incoming
edges •
The
“in-degree” of
these
vertices
is zero B C A F D E
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Topological
Sort - Lecture 19 9

9. Topo
sort
algorithm - 1a
Step 1: Identify vertices that have no incoming edges •
If no such vertices, graph has only cycle(s)
(cyclic
graph)
Topological
sort
not
possible – Halt. B C A
Example
of a cyclic graph D
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10. Topo sort algorithm - 1b Step 1: Identify vertices that have no
incoming
edges •
Select
one such
vertex
Select B C A F D E
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11. Topo sort algorithm - 2 Step 2: Delete this
vertex of in-degree 0 and all
its outgoing edges
output. B A
from the
graph.
Place
it in
the C A F D E
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12. Continue until done Repeat Step 1 and Step 2 until graph is empty
Select B C A F D E
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13. B Select B. Copy to sorted list. Delete B and its edges. B C F A B D E
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14. C Select C. Copy to sorted list. Delete C and its edges. C F A B C D E
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15. D Select D. Copy to sorted list. Delete D and its edges. F A B C D D E
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Sort - Lecture 19 16

16. E, F Select E. F. Copy to sorted list. Delete E and its edges. F andB C D E F E
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17. Done B C A F Remove from algorithm and serve. D E A B C D E F 11/27/02
Topological Sort - Lecture 19 18

18. Topological Ordering Algorithm: Examplev2 v3 v6 v5 v4 v7 v1 v1
Topological order:

19. Topological Ordering Algorithm: Examplev2 v2 v3 v6 v5 v4 v7
Topological order: v1

20. Topological Ordering Algorithm: Examplev3 v3 v6 v5 v4 v7
Topological order: v1, v2

21. Topological Ordering Algorithm: Examplev6 v5 v4 v4 v7
Topological order: v1, v2, v3

22. Topological Ordering Algorithm: Examplev6 v5 v5 v7
Topological order: v1, v2, v3, v4

23. Topological Ordering Algorithm: Examplev6 v6 v7
Topological order: v1, v2, v3, v4, v5

24. Topological Ordering Algorithm: Examplev7 v7
Topological order: v1, v2, v3, v4, v5, v6

25. Topological Ordering Algorithm: Examplev2 v3 v6 v5 v4 v1 v2 v3 v4 v5 v6 v7 v7 v1
Topological order: v1, v2, v3, v4, v5, v6, v7.

26. Topological Sort Algorithm 1. 2. 3.
Store each vertex’s In-Degree in an array D
Initialize queue with all “in-degree=0” vertices
While there are vertices remaining in the queue:
(a)
(b) (c)
Dequeue and output a vertex
Reduce In-Degree of all vertices adjacent to it by 1
Enqueue any of these vertices whose In-Degree became zero
all vertices are output then success, 4. If
otherwise there is a cycle.
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Topological Sort - Lecture 19 24

27. Topological Sort Analysis • Initialize In-Degree array: O(|V| + |E|)
Initialize Queue with In-Degree 0 vertices: O(|V|) Dequeue and output vertex: ›
|V| vertices, each takes only O(1) to dequeue and
output: O(|V|) •
Reduce In-Degree of all vertices adjacent
and Enqueue any In-Degree 0 vertices:
to a vertex ›
O(|E|) •
For input graph G=(V,E) run time =
O(|V|
+ |E|) ›
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Linear time!
Topological Sort - Lecture 19 26