1. Dijkstra’s Algorithm Shortest Path (Single Source)
Comparison to Floyd’s Algorithm: Solves a simpler problem O(n2) vs. O(n3)

2. Dijkstra’s: A Greedy Algorithm
What does greedy mean?
At the beginning of each step, the algorithm picks the best option.
Q: What other problem used a greedy algorithm?
A: Knapsack problem
Recall that there were 3 greedy strategies
Highest $$$
Lowest weight
Highest $$$/weight ratio

3. Dijkstra’s: A Dynamic Programming Algorithm
Recall that dynamic programming refers to
an algorithm that chooses to store information
rather than re-compute it.
To really be considered Dynamic Programming
Store or update information at each step
Use the collective information to compute the next step

4. Recall Floyd’s Algorithm
for (int k = 0; k < N; k++) // For each vertex
// Update the matrix
for (int i = 0; i < N; i++)
for (int j = 0; j < N; j++)
// Allow vertex k to be an intermediate hop
if (M[i][k]+M[k][j] < M[i][j]) then
M[i][j] = M[i][k]+M[k][j]

5. Recall Floyd’s Algorithm
The beauty of Floyd’s algorithm is that it progressively minimizes the distance between vertices
Each step incorporates more and more possibilities,
i.e., we add another intermediate hop.
Step 0 i j k1
Step 1 k2 k3
Step 2
Step 3

6. Recall Floyd’s Algorithm
However, at the end of each step, the algorithm can rule out inferior possibilities.
if (M[i][k]+M[k][j] < M[i][j]) then
M[i][j] = M[i][k]+M[k][j]
If the i  k  j is a better option than i  j then
we never consider i  j in the future.
We always know that hopping through k is better

7. Recall Floyd’s Algorithm
Implicitly Floyd’s algorithm computes the minimum path between any two vertices given O(n!) possibilities.
But, it actually does NOT compute all the possibilities
hops that are not optimal are quickly identified and never considered in the future. k 4 5 i j 8

8. How is Dijkstra’s algorithm better?
Its really not better.
Its more appropriate when you have a designated starting vertex.
Since you know the source vertex, you can limit your exploration.
If you limit your exploration properly, you can compute the minimum distance to all vertices in O(n2) computations.

9. a b c d e f g h i j Initial Step: Given a graph Adjacency List a null28 13 17 8 14 38 6 7 32 5 20 21 4 26 11 30 10 45 3 46
Initial Step:
Given a graph
Adjacency List V
Prev
Dist a
null b  c d e f g h i j a d b 5 28 c 8 13 f 6 17 g 11 14 32 e i 4 26 38 46 3 7 h j 45 10 20 21 30

10. Pick the V with the smallest dist28 8
Step 1:
Pick the V with the smallest dist V
Prev
Dist a
null b  c d e f g h i j a b c 13 17 5 32 14 38 6 7 46 d e f 26 3 11 20 21 4 26 5 30 a d b 5 28 c 8 13 f 6 17 g 11 14 32 e i 4 26 38 46 3 7 h j 45 10 20 21 30 g h i 11 45 10 5 j

11. Iterate over its adj. list and update the table28 8
Step 1:
Mark it as visited.
Iterate over its adj. list and update the table V
Prev
Dist a
null b 28 c  d 5 e f g h i j a b c 13 17 5 32 14 38 6 7 46 d e f 26 3 11 20 21 4 26 5 30 a 5 d 28 b g h i b 8 c 13 a 11 c 6 f 17 b d 11 g 14 b 32 a 45 10 5 e 4 i 26 d 38 b 46 f f 3 e 7 c j g 5 h 45 j h 10 j 20 d 21 e 30 i i 11 h 26 f j 5 i

12. Pick v with smallest dist.28 8
Step 2:
Pick v with smallest dist. V
Prev
Dist a
null b 28 c  d 5 e f g h i j a b c 13 17 5 32 14 38 6 7 46 d e f 26 3 11 20 21 4 26 5 30 a 5 d 28 b g h i b 8 c 13 a 11 c 6 f 17 b d 11 g 14 b 32 a 45 10 5 e 4 i 26 d 38 b 46 f f 3 e 7 c j g 5 h 45 j h 10 j 20 d 21 e 30 i i 11 h 26 f j 5 i

13. Iterate over its adj. list and update the table28 8
Step 2:
Mark it as visited.
Iterate over its adj. list and update the table V
Prev
Dist a
null b d 19 c  5 e f g 16 h i j a b c 13 17 5 32 14 38 6 7 46 d e f 26 3 11 20 21 4 26 5 30 a 5 d 28 b g h i b 8 c 13 a 11 c 6 f 17 b d 11 g 14 b 32 a 45 10 5 e 4 i 26 d 38 b 46 f f 3 e 7 c j g 5 h 45 j h 10 j 20 d 21 e 30 i i 11 h 26 f j 5 i

14. Pick v with smallest dist.28 8
Step 3:
Pick v with smallest dist. V
Prev
Dist a
null b d 19 c  5 e f g 16 h i j a b c 13 17 5 32 14 38 6 7 46 d e f 26 3 11 20 21 4 26 5 30 a 5 d 28 b g h i b 8 c 13 a 11 c 6 f 17 b d 11 g 14 b 32 a 45 10 5 e 4 i 26 d 38 b 46 f f 3 e 7 c j g 5 h 45 j h 10 j 20 d 21 e 30 i i 11 h 26 f j 5 i

15. Iterate over its adj. list and update the table28 8
Step 3:
Mark it as visited.
Iterate over its adj. list and update the table V
Prev
Dist a
null b d 19 c  5 e f g 16 h 21 i j 61 a b c 13 17 5 32 14 38 6 7 46 d e f 26 3 11 20 21 4 26 5 30 a 5 d 28 b g h i b 8 c 13 a 11 c 6 f 17 b d 11 g 14 b 32 a 45 10 5 e 4 i 26 d 38 b 46 f f 3 e 7 c j g 5 h 45 j h 10 j 20 d 21 e 30 i i 11 h 26 f j 5 i

16. Pick v with smallest dist.28 8
Step 4:
Pick v with smallest dist. V
Prev
Dist a
null b d 19 c  5 e f g 16 h 21 i j 61 a b c 13 17 5 32 14 38 6 7 46 d e f 26 3 11 20 21 4 26 5 30 a 5 d 28 b g h i b 8 c 13 a 11 c 6 f 17 b d 11 g 14 b 32 a 45 10 5 e 4 i 26 d 38 b 46 f f 3 e 7 c j g 5 h 45 j h 10 j 20 d 21 e 30 i i 11 h 26 f j 5 i

17. Iterate over its adj. list and update the table28 8
Step 4:
Mark it as visited.
Iterate over its adj. list and update the table V
Prev
Dist a
null b d 19 c 27 5 e  f g 16 h 21 i j 61 a b c 13 17 5 32 14 38 6 7 46 d e f 26 3 11 20 21 4 26 5 30 a 5 d 28 b g h i b 8 c 13 a 11 c 6 f 17 b d 11 g 14 b 32 a 45 10 5 e 4 i 26 d 38 b 46 f f 3 e 7 c j g 5 h 45 j h 10 j 20 d 21 e 30 i i 11 h 26 f j 5 i

18. Pick v with smallest dist.28 8
Step 5:
Pick v with smallest dist. V
Prev
Dist a
null b d 19 c 27 5 e  f g 16 h 21 i j 61 a b c 13 17 5 32 14 38 6 7 46 d e f 26 3 11 20 21 4 26 5 30 a 5 d 28 b g h i b 8 c 13 a 11 c 6 f 17 b d 11 g 14 b 32 a 45 10 5 e 4 i 26 d 38 b 46 f f 3 e 7 c j g 5 h 45 j h 10 j 20 d 21 e 30 i i 11 h 26 f j 5 i

19. Iterate over its adj. list and update the table28 8
Step 5:
Mark it as visited.
Iterate over its adj. list and update the table V
Prev
Dist a
null b d 19 c 27 5 e h 42 f  g 16 21 i 51 j 31 a b c 13 17 5 32 14 38 6 7 46 d e f 26 3 11 20 21 4 26 5 30 a 5 d 28 b g h i b 8 c 13 a 11 c 6 f 17 b d 11 g 14 b 32 a 45 10 5 e 4 i 26 d 38 b 46 f f 3 e 7 c j g 5 h 45 j h 10 j 20 d 21 e 30 i i 11 h 26 f j 5 i

20. Pick v with smallest dist.28 8
Step 6:
Pick v with smallest dist. V
Prev
Dist a
null b d 19 c 27 5 e h 42 f  g 16 21 i 51 j 31 a b c 13 17 5 32 14 38 6 7 46 d e f 26 3 11 20 21 4 26 5 30 a 5 d 28 b g h i b 8 c 13 a 11 c 6 f 17 b d 11 g 14 b 32 a 45 10 5 e 4 i 26 d 38 b 46 f f 3 e 7 c j g 5 h 45 j h 10 j 20 d 21 e 30 i i 11 h 26 f j 5 i

21. Iterate over its adj. list and update the table28 8
Step 6:
Mark it as visited.
Iterate over its adj. list and update the table V
Prev
Dist a
null b d 19 c 27 5 e h 42 f 33 g 16 21 i 51 j 31 a b c 13 17 5 32 14 38 6 7 46 d e f 26 3 11 20 21 4 26 5 30 a 5 d 28 b g h i b 8 c 13 a 11 c 6 f 17 b d 11 g 14 b 32 a 45 10 5 e 4 i 26 d 38 b 46 f f 3 e 7 c j g 5 h 45 j h 10 j 20 d 21 e 30 i i 11 h 26 f j 5 i

22. Pick v with smallest dist.28 8
Step 7:
Pick v with smallest dist. V
Prev
Dist a
null b d 19 c 27 5 e h 42 f 33 g 16 21 i 51 j 31 a b c 13 17 5 32 14 38 6 7 46 d e f 26 3 11 20 21 4 26 5 30 a 5 d 28 b g h i b 8 c 13 a 11 c 6 f 17 b d 11 g 14 b 32 a 45 10 5 e 4 i 26 d 38 b 46 f f 3 e 7 c j g 5 h 45 j h 10 j 20 d 21 e 30 i i 11 h 26 f j 5 i

23. Iterate over its adj. list and update the table28 8
Step 7:
Mark it as visited.
Iterate over its adj. list and update the table V
Prev
Dist a
null b d 19 c 27 5 e h 42 f 33 g 16 21 i j 36 31 a b c 13 17 5 32 14 38 6 7 46 d e f 26 3 11 20 21 4 26 5 30 a 5 d 28 b g h i b 8 c 13 a 11 c 6 f 17 b d 11 g 14 b 32 a 45 10 5 e 4 i 26 d 38 b 46 f f 3 e 7 c j g 5 h 45 j h 10 j 20 d 21 e 30 i i 11 h 26 f j 5 i

24. Pick v with smallest dist.28 8
Step 8:
Pick v with smallest dist. V
Prev
Dist a
null b d 19 c 27 5 e h 42 f 33 g 16 21 i j 36 31 a b c 13 17 5 32 14 38 6 7 46 d e f 26 3 11 20 21 4 26 5 30 a 5 d 28 b g h i b 8 c 13 a 11 c 6 f 17 b d 11 g 14 b 32 a 45 10 5 e 4 i 26 d 38 b 46 f f 3 e 7 c j g 5 h 45 j h 10 j 20 d 21 e 30 i i 11 h 26 f j 5 i

25. Iterate over its adj. list and update the table28 8
Step 8:
Mark it as visited.
Iterate over its adj. list and update the table V
Prev
Dist a
null b d 19 c 27 5 e f 36 33 g 16 h 21 i j 31 a b c 13 17 5 32 14 38 6 7 46 d e f 26 3 11 20 21 4 26 5 30 a 5 d 28 b g h i b 8 c 13 a 11 c 6 f 17 b d 11 g 14 b 32 a 45 10 5 e 4 i 26 d 38 b 46 f f 3 e 7 c j g 5 h 45 j h 10 j 20 d 21 e 30 i i 11 h 26 f j 5 i

26. Pick v with smallest dist.28 8
Step 9:
Pick v with smallest dist. V
Prev
Dist a
null b d 19 c 27 5 e f 36 33 g 16 h 21 i j 31 a b c 13 17 5 32 14 38 6 7 46 d e f 26 3 11 20 21 4 26 5 30 a 5 d 28 b g h i b 8 c 13 a 11 c 6 f 17 b d 11 g 14 b 32 a 45 10 5 e 4 i 26 d 38 b 46 f f 3 e 7 c j g 5 h 45 j h 10 j 20 d 21 e 30 i i 11 h 26 f j 5 i

27. Iterate over its adj. list and update the table28 8
Step 9:
Mark it as visited.
Iterate over its adj. list and update the table V
Prev
Dist a
null b d 19 c 27 5 e f 36 33 g 16 h 21 i j 31 a b c 13 17 5 32 14 38 6 7 46 d e f 26 3 11 20 21 4 26 5 30 a 5 d 28 b g h i b 8 c 13 a 11 c 6 f 17 b d 11 g 14 b 32 a 45 10 5 e 4 i 26 d 38 b 46 f f 3 e 7 c j g 5 h 45 j h 10 j 20 d 21 e 30 i i 11 h 26 f j 5 i

28. Pick v with smallest dist.28 8
Step 10:
Pick v with smallest dist. V
Prev
Dist a
null b d 19 c 27 5 e f 39 33 g 16 h 21 i j 36 31 a b c 13 17 5 32 14 38 6 7 46 d e f 26 3 11 20 21 4 26 5 30 a 5 d 28 b g h i b 8 c 13 a 11 c 6 f 17 b d 11 g 14 b 32 a 45 10 5 e 4 i 26 d 38 b 46 f f 3 e 7 c j g 5 h 45 j h 10 j 20 d 21 e 30 i i 11 h 26 f j 5 i

29. Iterate over its adj. list and update the table28 8
Step 10:
Mark it as visited.
Iterate over its adj. list and update the table V
Prev
Dist a
null b d 19 c 27 5 e f 39 33 g 16 h 21 i j 36 31 a b c 13 17 5 32 14 38 6 7 46 d e f 26 3 11 20 21 4 26 5 30 a 5 d 28 b g h i b 8 c 13 a 11 c 6 f 17 b d 11 g 14 b 32 a 45 10 5 e 4 i 26 d 38 b 46 f f 3 e 7 c j g 5 h 45 j h 10 j 20 d 21 e 30 i i 11 h 26 f j 5 i

30. 28 8
Traceback
The table can be used to recursively trace all the shortest paths from vertex a V
Prev
Dist a
null b d 19 c 27 5 e f 39 33 g 16 h 21 i j 36 31 a b c 13 17 5 32 14 38 6 7 46 d e f 26 3 11 20 21 4 26 5 30 g h i 11 45 10 5 j

31. What is the shortest path to vertex e?28 8
Traceback
What is the shortest path to vertex e? V
Prev
Dist a
null b d 19 c 27 5 e f 36 33 g 16 h 21 i j 31 a b c 13 17 5 32 14 38 6 7 46 d e f 26 3 11 20 21 4 26 5 30 g h i 11 45 10 5 e j

32. What is the shortest path to vertex e?28 8
Traceback
What is the shortest path to vertex e? V
Prev
Dist a
null b d 19 c 27 5 e f 39 33 g 16 h 21 i j 36 31 a b c 13 17 5 32 14 38 6 7 46 d e f 26 3 11 20 21 4 26 5 30 g h i 11 45 10 5 f e j

33. What is the shortest path to vertex e?28 8
Traceback
What is the shortest path to vertex e? V
Prev
Dist a
null b d 19 c 27 5 e f 39 33 g 16 h 21 i j 36 31 a b c 13 17 5 32 14 38 6 7 46 d e f 26 3 11 20 21 4 26 5 30 g h i 11 c 45 10 5 f e j

34. What is the shortest path to vertex e?28 8
Traceback
What is the shortest path to vertex e? V
Prev
Dist a
null b d 19 c 27 5 e f 39 33 g 16 h 21 i j 36 31 a b c 13 17 5 32 14 38 6 7 46 d e f 26 3 11 20 21 4 26 5 30 b g h i 11 c 45 10 5 f e j

35. What is the shortest path to vertex e?28 8
Traceback
What is the shortest path to vertex e? V
Prev
Dist a
null b d 19 c 27 5 e f 39 33 g 16 h 21 i j 36 31 a b c 13 17 5 32 14 38 6 7 46 d e f 26 3 11 20 21 4 26 d 5 30 b g h i 11 c 45 10 5 f e j

36. What is the shortest path to vertex e?28 8
Traceback
What is the shortest path to vertex e? V
Prev
Dist a
null b d 19 c 27 5 e f 39 33 g 16 h 21 i j 36 31 a b c 13 17 5 32 14 38 6 7 46 d e f 26 3 a 11 20 21 4 26 d 5 30 b g h i 11 c 45 10 5 f e j

37. How to code Dijkstra’s Algorithm
Data Structures:
AdjacencyList (adjList)
adjList.find(label)
Finds a vertex in constant time
Uses the vertex label as the hash index
Sets an iterator to point to label’s adjacency list

38. Vertex v = new Vertex(‘h’); adjList.find(v.getLabel())5 28 c 8 13 f 6 17 g 11 14 32 e i 4 26 38 46 3 7 h j 45 10 20 21 30

39. Vertex v = new Vertex(‘h’); adjList.find(v.getLabel()) // Finds h5 28 a d b 8 13 b c a 6 17 c f b 11 14 32 d g b a 4 26 38 46 e i d b f 3 7 f e c 5 45 g h j 10 20 21 30 h j d e i 11 26 i h f 5 j i

40. Vertex v = new Vertex(‘h’); adjList.find(v.getLabel())
// Moves iterator to front of h’s adjacency list 5 28 a d b 8 13 b c a 6 17 c f b 11 14 32 d g b a 4 26 38 46 e i d b f 3 7 f e c 5 45 g h j 10 20 21 30 h j d e i 11 26 i h f 5 j i

41. How to code Dijkstra’s Algorithm
Data Structures:
AdjacencyList (adjList)
adjList.getNext();
Returns the edge
Moves the iterator to the next edge
Returns null if it reaches the end of label’s adjacency list

42. while (e = adjList.getNext()) print(e.getLabel());5 28 a d b 8 13 b c a 6 17 c f b 11 14 32 d g b a 4 26 38 46 e i d b f 3 7 f e c 5 45 g h j 10 20 21 30 h j d e i 11 26 i h f 5 j i

43. How to code Dijkstra’s Algorithm
Data Structures:
Edge (e)
e.getLabel()
Returns the label of the vertex that the edge points to
e.getDist()
Returns the edge’s distance (weight of the edge)

44. j while (e = adjList.getNext()) print(e.getLabel()); a d b b c a c f b5 28 a d b 8 13 j b c a 6 17 c f b 11 14 32 d g b a 4 26 38 46 e i d b f 3 7 f e c 5 45 g h j 10 20 21 30 h j d e i 11 26 i h f 5 j i

45. j d while (e = adjList.getNext()) println(e.getLabel()); a d b b c a c5 28 a d b 8 13 j d b c a 6 17 c f b 11 14 32 d g b a 4 26 38 46 e i d b f 3 7 f e c 5 45 g h j 10 20 21 30 h j d e i 11 26 i h f 5 j i

46. j d e while (e = adjList.getNext()) println(e.getLabel()); a d b b c a5 28 a d b 8 13 j d e b c a 6 17 c f b 11 14 32 d g b a 4 26 38 46 e i d b f 3 7 f e c 5 45 g h j 10 20 21 30 h j d e i 11 26 i h f 5 j i

47. j d e i while (e = adjList.getNext()) println(e.getLabel()); a d b b c5 28 a d b 8 13 j d e i b c a 6 17 c f b 11 14 32 d g b a 4 26 38 46 e i d b f 3 7 f e c 5 45 g h j 10 20 21 30 h j d e i 11 26 i h f 5 j i

48. j d e i while (e = adjList.getNext()) println(e.getLabel()); a d b b c5 28 a d b 8 13 j d e i
loop terminates b c a 6 17 c f b 11 14 32 d g b a 4 26 38 46 e i d b f 3 7 f e c 5 45 g h j 10 20 21 30 h j d e i
null 11 26 i h f 5 j i

49. How to code Dijkstra’s Algorithm
Data Structures:
PathTable (pTable)
pTable.getMin()
Finds the minimum distance vertex and sets it as visited.
Returns the vertex
Returns null if all vertices are marked visited.

50. Vertex v = pTable.getMin(); a null b d 19 c  5 e f g 16 h 21 i j 61
Prev
Dist a
null b d 19 c  5 e f g 16 h 21 i j 61

51. Vertex v = pTable.getMin(); a null b d 19 c  5 e f g 16 h 21 i j 61
Prev
Dist a
null b d 19 c  5 e f g 16 h 21 i j 61

52. Vertex v = pTable.getMin(); a null b d 19 c  5 e f g 16 h 21 i j 61 v
Prev
Dist a
null b d 19 c  5 e f g 16 h 21 i j 61 v
label = ‘b’
dist = 19

53. How to code Dijkstra’s Algorithm
Data Structures:
PathTable (pTable)
pTable.update(vertex, prev, dist)
Updates previous vertex and total distance
Label used as the hash index
Constant time update

54. pTable.update(‘c’, v.getLabel(), v.getDist() + 10);
Prev
Dist a
null b d 19 c  5 e f g 16 h 21 i j 61 v
label = ‘b’
dist = 19
pTable.update(‘c’, v.getLabel(), v.getDist() + 10);

55. pTable.update(‘c’, v.getLabel(), v.getDist() + 10);
Prev
Dist a
null b d 19 c  5 e f g 16 h 21 i j 61 v
label = ‘b’
dist = 19
pTable.update(‘c’, v.getLabel(), v.getDist() + 10);

56. pTable.update(‘c’, v.getLabel(), v.getDist() + 10);
Prev
Dist a
null b d 19 c 29 5 e  f g 16 h 21 i j 61 v
label = ‘b’
dist = 19
pTable.update(‘c’, v.getLabel(), v.getDist() + 10);

57. How to code Dijkstra’s Algorithm
Weight w = 0;
Edge e = null;
Label v = new Label(‘a’);
pTable.update(v, null, 0);
while(v = pTable.getMin()) {
adjList.find(v.getLabel());
while (e = adjList.getNext()) {
w = e.getDist() + v.getDist();
pTable.update(e.getLabel(), v.getLabel(), w); }

58. How to code Dijkstra’s Algorithm
Weight w = 0;
Edge e = null;
Label v = new Label(‘a’); // variables
pTable.update(v, null, 0);
while(v = pTable.getMin()) {
adjList.find(v.getLabel());
while (e = adjList.getNext()) {
w = e.getDist() + v.getDist();
pTable.update(e.getLabel(), v.getLabel(), w); }

59. How to code Dijkstra’s Algorithm
Weight w = 0;
Edge e = null;
Label v = new Label(‘a’);
pTable.update(v, null, 0); // Sets A as the start vertex
while(v = pTable.getMin()) {
adjList.find(v.getLabel());
while (e = adjList.getNext()) {
w = e.getDist() + v.getDist();
pTable.update(e.getLabel(), v.getLabel(), w); }

60. How to code Dijkstra’s AlgorithmV
Prev
Dist a
null b  c d e f g h i j
Label v = new Label(‘a’);
pTable.update(v, null, 0);
while(v = pTable.getMin()) {
adjList.find(v.getLabel());
while (e = adjList.getNext()) {
w = e.getDist() + v.getDist();
pTable.update(e.getLabel(), v.getLabel(), w); }

61. How to code Dijkstra’s Algorithm5 d 28 b
Label v = new Label(‘a’);
pTable.update(v, null, 0);
while(v = pTable.getMin()) {
adjList.find(v.getLabel());
while (e = adjList.getNext()) {
w = e.getDist() + v.getDist();
pTable.update(e.getLabel(), v.getLabel(), w); } b 8 c 13 a c 6 f 17 b d 11 g 14 b 32 a e 4 i 26 d 38 b 46 f f 3 e 7 c g 5 h 45 j h 10 j 20 d 21 e 30 i i 11 h 26 f j 5 i

62. How to code Dijkstra’s AlgorithmV
Prev
Dist a
null b 28 c  d 5 e f g h i j a 5 d 28 b
Label v = new Label(‘a’);
pTable.update(v, null, 0);
while(v = pTable.getMin()) {
adjList.find(v.getLabel());
while (e = adjList.getNext()) {
w = e.getDist() + v.getDist();
pTable.update(e.getLabel(), v.getLabel(), w); }