Bellman-Ford-Moore Algorithm and Dijkstra’s Algorithm Teaching material for the course of "Tecniche di Programmazione" at Politecnico di Torino in year 2012/2013. More information: http://bit.ly/tecn-progr

1. Bellman-Ford-Moore Algorithm
Single-source shortest path algorithm

2. Bellman-Ford-Moore Algorithm
A.A. 2012/2013Tecniche di programmazione2
 Solution to the single-source shortest path (SSSP)
problem in graph theory
 Based on relaxation (for every vertex, relax all possible
edges)
 Does not work in presence of negative cycles
 but it is able to detect the problem
 O(V∙E)

3. Bellman-Ford-Moore Algorithm
A.A. 2012/2013Tecniche di programmazione3
dist[s] ← 0 (distance to source vertex is zero)
for all v ∈ V–{s}
do dist[v] ←∞ (set all other distances to infinity)
for i← 0 to |V|
for all (u, v) ∈ E
do if dist[v] > dist[u] + w(u, v) (if new shortest path found)
then d[v] ←d[u] + w(u, v) (set new value of shortest path)
(if desired, add traceback code)
for all (u, v) ∈ E (sanity check)
do if dist[v] > dist[u] + w(u, v)
then PANIC!

4. Dijkstra’s Algorithm
Yet another SSSP algorithm

5. Dijkstra’s algorithm
A.A. 2012/2013Tecniche di programmazione5
 Solution to the single-source shortest path (SSSP)
problem in graph theory
 Works on both directed and undirected graphs
 All edges must have nonnegative weights
 the algorithm would miserably fail
 Greedy
… but guarantees the optimum!

6. Dijkstra’s algorithm
A.A. 2012/2013Tecniche di programmazione6
dist[s] ←0 (distance to source vertex is zero)
for all v ∈ V–{s}
do dist[v] ←∞ (set all other distances to infinity)
S←∅ (S, the set of visited vertices is initially empty)
Q←V (Q, the queue initially contains all vertices)
while Q ≠∅ (while the queue is not empty)
do u ← mindistance(Q,dist) (select e  Q with the min. distance)
S←S∪{u} (add u to list of visited vertices)
for all v ∈ neighbors[u]
do if dist[v] > dist[u] + w(u, v) (if new shortest path found)
then d[v] ←d[u] + w(u, v) (set new value of shortest path)
(if desired, add traceback code)

7. Dijkstra Animated Example

8. Dijkstra Animated Example

9. Dijkstra Animated Example

10. Dijkstra Animated Example

11. Dijkstra Animated Example

12. Dijkstra Animated Example

13. Dijkstra Animated Example

14. Dijkstra Animated Example

15. Dijkstra Animated Example

16. Dijkstra Animated Example

17. Why it works
A.A. 2012/2013Tecniche di programmazione17
 A formal proof would take longer than this presentation,
but we can understand how the argument works
intuitively
 Think of Djikstra’s algorithm as a water-filling algorithm
 Remember that all edge’s weights are positive

18. Dijkstra efficiency
A.A. 2012/2013Tecniche di programmazione18
 The simplest implementation is:
O(𝐸 + 𝑉2)
 But it can be implemented more efficently:
O(𝐸 + 𝑉 ∙ log 𝑉)
Floyd–Warshall: O(V3)
Bellman-Ford-Moore : O(V∙E)

19. Applications
A.A. 2012/2013Tecniche di programmazione19
 Dijkstra’s algorithm calculates the shortest path to every
vertex from vertex s (SSSP)
 It is about as computationally expensive to calculate the
shortest path from vertex u to every vertex using
Dijkstra’s as it is to calculate the shortest path to some
particular vertex t
 Therefore, anytime we want to know the optimal path to
some other vertex t from a determined origin s, we can
use Dijkstra’s algorithm (and stop as soon t exit from Q)

20. Applications
A.A. 2012/2013Tecniche di programmazione20
 Traffic Information Systems are most prominent use
 Mapping (Map Quest, Google Maps)
 Routing Systems

21. 21
Dijkstra's Shortest Path Algorithm
 Find shortest path from s to t
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Dijkstra's Shortest Path Algorithm
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
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distance label
S = { }
Q = { s, 2, 3, 4, 5, 6, 7, t }
s
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23. 23
Dijkstra's Shortest Path Algorithm
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distance label
S = { }
Q = { s, 2, 3, 4, 5, 6, 7, t }
delmin

24. 24
Dijkstra's Shortest Path Algorithm
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distance label
S = { s }
Q = { 2, 3, 4, 5, 6, 7, t }
decrease key
X

X
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s
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25. 25
Dijkstra's Shortest Path Algorithm
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distance label
S = { s }
Q = { 2, 3, 4, 5, 6, 7, t }
X

X
X
delmin
s
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Dijkstra's Shortest Path Algorithm
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S = { s, 2 }
Q = { 3, 4, 5, 6, 7, t }
X
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X
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Dijkstra's Shortest Path Algorithm
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S = { s, 2 }
Q = { 3, 4, 5, 6, 7, t }
X
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X
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decrease key
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Dijkstra's Shortest Path Algorithm
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S = { s, 2 }
Q = { 3, 4, 5, 6, 7, t }
X
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X
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delmin
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Dijkstra's Shortest Path Algorithm
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S = { s, 2, 6 }
Q = { 3, 4, 5, 7, t }
X
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44
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Dijkstra's Shortest Path Algorithm
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S = { s, 2, 6 }
Q = { 3, 4, 5, 7, t }
X
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delmin
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Dijkstra's Shortest Path Algorithm
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S = { s, 2, 6, 7 }
Q = { 3, 4, 5, t }
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Dijkstra's Shortest Path Algorithm
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S = { s, 2, 6, 7 }
Q = { 3, 4, 5, t }
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59 X
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X 33X
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Dijkstra's Shortest Path Algorithm
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S = { s, 2, 3, 6, 7 }
Q = { 4, 5, t }
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59 XX51
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X 33X
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Dijkstra's Shortest Path Algorithm
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S = { s, 2, 3, 6, 7 }
Q = { 4, 5, t }
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delmin
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Dijkstra's Shortest Path Algorithm
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S = { s, 2, 3, 5, 6, 7 }
Q = { 4, t }
X
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Dijkstra's Shortest Path Algorithm
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S = { s, 2, 3, 5, 6, 7 }
Q = { 4, t }
X
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X50
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X 33X
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Dijkstra's Shortest Path Algorithm
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S = { s, 2, 3, 4, 5, 6, 7 }
Q = { t }
X
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59 XX51
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X50
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Dijkstra's Shortest Path Algorithm
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S = { s, 2, 3, 4, 5, 6, 7 }
Q = { t }
X
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Dijkstra's Shortest Path Algorithm
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S = { s, 2, 3, 4, 5, 6, 7, t }
Q = { }
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Dijkstra's Shortest Path Algorithm
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S = { s, 2, 3, 4, 5, 6, 7, t }
Q = { }
X
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X
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35X
59 XX51
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X50
X45
X 33X
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41. Shortest Paths wrap-up
A.A. 2012/2013Tecniche di programmazione41
Algorithm Problem Efficiency Limitation
Floyd-Warshall AP O(𝑉3
) No negative cycles
Bellman–Ford SS O(𝑉 ∙ 𝐸) No negative cycles
Repeated- Bellman-Ford AP O(𝑉2
∙ 𝐸) No negative cycles
Dijkstra SS O(𝐸 + 𝑉 ∙ log 𝑉) No negative edges
Repeated Dijkstra AP O(𝑉 ∙ 𝐸 + 𝑉2 ∙ log 𝑉) No negative edges

42. JGraphT
A.A. 2012/2013Tecniche di programmazione42
public class FloydWarshallShortestPaths<V,E>
public class BellmanFordShortestPath<V,E>
public class DijkstraShortestPath<V,E>
// APSP
List<GraphPath<V,E>> getShortestPaths(V v)
GraphPath<V,E> getShortestPath(V a, V b)
// SSSPs
GraphPath<V,E> getPath()

43. Licenza d’uso
A.A. 2012/2013Tecniche di programmazione43
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