1. Chapter 13 Graph Algorithms
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Acknowledgement: These slides are adapted from slides provided with Data Structures and Algorithms in C++, Goodrich, Tamassia and Mount (Wiley 2004) and slides from Nancy M. Amato 1 1

2. Directed Graphs Shortest Path 12/7/2017 7:14 AM BOS ORD JFK SFO DFW
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3. Digraphs E A digraph is a graph whose edges are all directed DB D
A digraph is a graph whose edges are all directed
Short for “directed graph”
Applications
one-way streets
flights
task scheduling

4. Digraph Properties E D A graph 𝐺=(𝑉,𝐸) such that CB D
A graph 𝐺=(𝑉,𝐸) such that
Each edge goes in one direction:
Edge (𝑎,𝑏) goes from 𝑎 to 𝑏, but not 𝑏 to 𝑎
If 𝐺 is simple, 𝑚 < 𝑛(𝑛−1)
If we keep in-edges and out-edges in separate adjacency lists, we can perform listing of incoming edges and outgoing edges in time proportional to their size

5. Digraph Application
Scheduling: edge (𝑎,𝑏) means task 𝑎 must be completed before 𝑏 can be started
The good life
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6. Directed DFS A C E B D
We can specialize the traversal algorithms (DFS and BFS) to digraphs by traversing edges only along their direction
In the directed DFS algorithm, we have four types of edges
discovery edges
back edges
forward edges
cross edges
A directed DFS starting at a vertex 𝑠 determines the vertices reachable from 𝑠

7. Reachability
DFS tree rooted at 𝑣: vertices reachable from 𝑣 via directed paths A C E D A C E B D F A C E B D F

8. Strong Connectivity Each vertex can reach all other vertices a g c d eb e f g

9. Strong Connectivity Algorithm
Pick a vertex 𝑣 in 𝐺
Perform a DFS from 𝑣 in 𝐺
If there’s a 𝑤 not visited, print “no”
Let 𝐺’ be 𝐺 with edges reversed
Perform a DFS from 𝑣 in 𝐺’
Else, print “yes”
Running time: 𝑂(𝑛+𝑚) c d e b f a
G’: g c d e b f

10. Strongly Connected Components
Maximal subgraphs such that each vertex can reach all other vertices in the subgraph
Can also be done in 𝑂(𝑛+𝑚) time using DFS, but is more complicated (similar to biconnectivity). a d c b e f g
{ a , c , g }
{ f , d , e , b }

11. Transitive Closure B A D C E G
Given a digraph 𝐺, the transitive closure of 𝐺 is the digraph 𝐺 ∗ such that
G* has the same vertices as G
if 𝐺 has a directed path from 𝑢 to 𝑣 (𝑢  𝑣), 𝐺 ∗ has a directed edge from 𝑢 to 𝑣
The transitive closure provides reachability information about a digraph B A D C E G*

12. Computing the Transitive Closure
If there's a way to get from A to B and from B to C, then there's a way to get from A to C.
We can perform DFS starting at each vertex
𝑂(𝑛(𝑛+𝑚))
Alternatively ... Use dynamic programming: The Floyd-Warshall Algorithm

13. Floyd-Warshall Transitive Closure
Idea #1: Number the vertices 1, 2, …,𝑛.
Idea #2: Consider paths that use only vertices numbered 1, 2, …,𝑘, as intermediate vertices:
Uses only vertices numbered 𝑖,…,𝑘
(add this edge if it’s not already in) i j
Uses only vertices
numbered 𝑖,…,𝑘−1
Uses only vertices
numbered 𝑘,…,𝑗 k

14. Floyd-Warshall’s Algorithm
Number vertices 𝑣 1 ,…, 𝑣 𝑛
Compute digraphs 𝐺 0 ,…, 𝐺 𝑛
𝐺 0 ←𝐺
𝐺 𝑘 has directed edge 𝑣 𝑖 , 𝑣 𝑗 if 𝐺 has a directed path from 𝑣 𝑖 to 𝑣 𝑗
We have that 𝐺 𝑛 = 𝐺 ∗
In phase 𝑘, digraph 𝐺 𝑘 is computed from 𝐺 𝑘−1
Running time: 𝑂 𝑛 3 , assuming 𝐺.areAdjacent 𝑣 𝑖 , 𝑣 𝑗 is 𝑂 1 (e.g., adjacency matrix)
Algorithm FloydWarshall(𝐺)
Input: Digraph 𝐺
Output: Transitive Closure 𝐺 ∗ of 𝐺
Name each vertex 𝑣∈𝐺.vertices( ) with 𝑖=1…𝑛
𝐺 0 ←𝐺
for 𝑘←1…𝑛 do
𝐺 𝑘 ← 𝐺 𝑘−1
for 𝑖←1…𝑛 | 𝑖≠𝑘 do
for 𝑗←1…𝑛 | 𝑗≠𝑖,𝑘 do
if 𝐺 𝑘−1 .areAdjacent 𝑣 𝑖 , 𝑣 𝑘 ∧ 𝐺 𝑘−1 .areAdjacent 𝑣 𝑘 , 𝑣 𝑗 ∧ ¬ 𝐺 𝑘 .areAdjacent 𝑣 𝑖 , 𝑣 𝑗 then
𝐺 𝑘 .insertDirectedEdge v i , v j
return 𝐺 𝑛

15. Floyd-Warshall Example
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16. Floyd-Warshall, Iteration 1
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17. Floyd-Warshall, Iteration 2
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18. Floyd-Warshall, Iteration 3
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19. Floyd-Warshall, Iteration 4
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20. Floyd-Warshall, Iteration 5
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21. Floyd-Warshall, Iteration 6
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22. Floyd-Warshall, Conclusion
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23. DAGs and Topological OrderingB A D C E
DAG G
A directed acyclic graph (DAG) is a digraph that has no directed cycles
A topological ordering of a digraph is a numbering
𝑣 1 ,…, 𝑣 𝑛
Of the vertices such that for every edge 𝑣 𝑖 , 𝑣 𝑗 , we have 𝑖<𝑗
Example: in a task scheduling digraph, a topological ordering a task sequence that satisfies the precedence constraints
Theorem - A digraph admits a topological ordering if and only if it is a DAG B A D C E
Topological ordering of G v1 v2 v3 v4 v5

24. Exercise Topological Sorting
write c.s. program
play
wake up
eat
nap
study computer sci.
more c.s.
work out
sleep
dream about graphs
A typical student day
bake cookies
Number vertices, so that 𝑢,𝑣 in 𝐸 implies 𝑢<𝑣

25. Exercise Topological Sorting
write c.s. program
play
wake up
eat
nap
study computer sci.
more c.s.
work out
sleep
dream about graphs
A typical student day 1 2 3 4 5 6 7 8 9 10 11
bake cookies
Number vertices, so that 𝑢,𝑣 in 𝐸 implies 𝑢<𝑣

26. Algorithm for Topological Sorting
Note: This algorithm is different than the one in the book
Algorithm TopologicalSort 𝐺
𝐻←𝐺
𝑛←𝐺.numVertices
while ¬𝐻.empty do
Let 𝑣 be a vertex with no outgoing edges
Label 𝑣←𝑛
𝑛←𝑛−1
𝐻.eraseVertex 𝑣

27. Implementation with DFS
Simulate the algorithm by using depth-first search
𝑂(𝑛+𝑚) time.
Algorithm topologicalDFS 𝐺
Input: DAG 𝐺
Output: Topological ordering of 𝑔
𝑛←𝐺.numVertices
Initialize all vertices as 𝑈𝑁𝐸𝑋𝑃𝐿𝑂𝑅𝐸𝐷
for each vertex 𝑣∈𝐺.vertices do
if 𝑣.getLabel =𝑈𝑁𝐸𝑋𝑃𝐿𝑂𝑅𝐸𝐷 then
topologicalDFS 𝐺, 𝑣
Algorithm topologicalDFS 𝐺,𝑣
Input: DAG 𝐺, start vertex 𝑣
Output: Labeling of the vertices of 𝐺 in the connected component of 𝑣
𝑣.setLabel(𝑉𝐼𝑆𝐼𝑇𝐸𝐷)
for each 𝑒∈𝑣.outEdges do
𝑤←𝑒.dest( )
if 𝑤.getLabel =𝑈𝑁𝐸𝑋𝑃𝐿𝑂𝑅𝐸𝐷 then
//𝑒 is a discovery edge
topologicalDFS 𝐺, 𝑤
else
//𝑒 is a forward, cross, or back edge
Label 𝑣 with topological number 𝑛
𝑛←𝑛−1

28. Topological Sorting Example

29. Topological Sorting Example9

30. Topological Sorting Example8 9

31. Topological Sorting Example7 8 9

32. Topological Sorting Example7 8 6 9

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34. Topological Sorting Example7 4 8 5 6 9

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36. Topological Sorting Example2 7 4 8 5 6 3 9

37. Topological Sorting Example2 7 4 8 5 6 1 3 9