https://github.com/emory-courses/cs253

1. Shortest Path
Algorithm
Data Structures and Algorithms
Emory University
Jinho D. Choi

2. Emory University Logo Guidelines
-
2
0
1
2
3
4
5
4
5
10
92
3
3
4
Dijkstra's Algorithm

3. Emory University Logo Guidelines
-
2
0
1
2
3
4
5
4
5
10
9
S T
2
3
3
4
Dijkstra's Algorithm

4. Emory University Logo Guidelines
-
2
0
1
2
3
4
5
4
5
10
9
∞ ∞ ∞ ∞ ∞ ∞
0 1 2 3 4 5
Distance
∅ ∅ ∅ ∅ ∅ ∅Previous
S T
2
3
3
4
Dijkstra's Algorithm

5. Emory University Logo Guidelines
-
2
0
1
2
3
4
5
4
5
10
9
∞ ∞ ∞ ∞ ∞ ∞
0 1 2 3 4 5
Distance
∅ ∅ ∅ ∅ ∅ ∅Previous
Queue
S T
Visited
2
3
3
4
Dijkstra's Algorithm

6. Emory University Logo Guidelines
-
2
0
1
2
3
4
5
4
5
10
9
(5, 0)
∞ ∞ ∞ ∞ ∞ ∞
0 1 2 3 4 5
Distance
∅ ∅ ∅ ∅ ∅ ∅Previous
Queue
S T
0
Visited
2
3
3
4
Dijkstra's Algorithm

7. Emory University Logo Guidelines
-
2
0
1
2
3
4
5
4
5
10
9
∞ ∞ ∞ ∞ ∞ ∞
0 1 2 3 4 5
Distance
∅ ∅ ∅ ∅ ∅ ∅Previous
Queue
S T
0
Visited
5
2
3
3
4
Dijkstra's Algorithm

8. Emory University Logo Guidelines
-
2
0
1
2
3
4
5
4
5
10
9
∞ ∞ ∞ ∞ ∞ ∞
0 1 2 3 4 5
Distance
∅ ∅ ∅ ∅ ∅ ∅Previous
Queue
S T
0
Visited
3
(3, 3)5
5
2
3
3
4
Dijkstra's Algorithm

9. Emory University Logo Guidelines
-
2
0
1
2
3
4
5
4
5
10
9
∞ ∞ ∞ ∞ ∞ ∞
0 1 2 3 4 5
Distance
∅ ∅ ∅ ∅ ∅ ∅Previous
Queue
S T
0
Visited
3
(3, 3)5
5
2
3
3
4
Dijkstra's Algorithm

10. Emory University Logo Guidelines
-
2
0
1
2
3
4
5
4
5
10
9
∞ ∞ ∞ ∞ ∞ ∞
0 1 2 3 4 5
Distance
∅ ∅ ∅ ∅ ∅ ∅Previous
Queue
S T
0
Visited
3
(3, 3)5
5
9
5
(4, 9)
2
3
3
4
Dijkstra's Algorithm

11. Emory University Logo Guidelines
-
2
0
1
2
3
4
5
4
5
10
9
∞ ∞ ∞ ∞ ∞ ∞
0 1 2 3 4 5
Distance
∅ ∅ ∅ ∅ ∅ ∅Previous
Queue
S T
0
Visited
3
5 3
5
9
5
(4, 9)
2
3
3
4
Dijkstra's Algorithm

12. Emory University Logo Guidelines
-
2
0
1
2
3
4
5
4
5
10
9
∞ ∞ ∞ ∞ ∞ ∞
0 1 2 3 4 5
Distance
∅ ∅ ∅ ∅ ∅ ∅Previous
Queue
S T
0
Visited
3
5 3
13
53
(1, 13)9
5
(4, 9)
2
3
3
4
Dijkstra's Algorithm

13. Emory University Logo Guidelines
-
2
0
1
2
3
4
5
4
5
10
9
∞ ∞ ∞ ∞ ∞ ∞
0 1 2 3 4 5
Distance
∅ ∅ ∅ ∅ ∅ ∅Previous
Queue
S T
0
Visited
3
5 3
13
53
(1, 13)9
5
(4, 9)
2
3
3
4
Dijkstra's Algorithm

14. Emory University Logo Guidelines
-
2
0
1
2
3
4
5
4
5
10
9
∞ ∞ ∞ ∞ ∞ ∞
0 1 2 3 4 5
Distance
∅ ∅ ∅ ∅ ∅ ∅Previous
Queue
S T
0
Visited
3
5 3
13
53
(1, 13)
(4, 7)
97
53
(4, 9)
2
3
3
4
Dijkstra's Algorithm

15. Emory University Logo Guidelines
-
2
0
1
2
3
4
5
4
5
10
9
∞ ∞ ∞ ∞ ∞ ∞
0 1 2 3 4 5
Distance
∅ ∅ ∅ ∅ ∅ ∅Previous
Queue
S T
0
Visited
3
5 3
13
53
(1, 13)
4
97
53
(4, 9)
2
3
3
4
Dijkstra's Algorithm

16. Emory University Logo Guidelines
-
2
0
1
2
3
4
5
4
5
10
9
∞ ∞ ∞ ∞ ∞ ∞
0 1 2 3 4 5
Distance
∅ ∅ ∅ ∅ ∅ ∅Previous
Queue
S T
0
Visited
3
5 3
13
53
(1, 13)
4
97
53
(4, 9)
10
4
(2, 10)
2
3
3
4
Dijkstra's Algorithm

17. Emory University Logo Guidelines
-
2
0
1
2
3
4
5
4
5
10
9
∞ ∞ ∞ ∞ ∞ ∞
0 1 2 3 4 5
Distance
∅ ∅ ∅ ∅ ∅ ∅Previous
Queue
S T
0
Visited
3
5 3
13
53
(1, 13)
4
97
53
10
4
(2, 10)
2
3
3
4
Dijkstra's Algorithm

18. Emory University Logo Guidelines
-
2
0
1
2
3
4
5
4
5
10
9
∞ ∞ ∞ ∞ ∞ ∞
0 1 2 3 4 5
Distance
∅ ∅ ∅ ∅ ∅ ∅Previous
Queue
S T
0
Visited
3
5 3
13
53
(1, 13)
4
97
53
10
4
2
2
3
3
4
Dijkstra's Algorithm

19. Emory University Logo Guidelines
-
2
0
1
2
3
4
5
4
5
10
9
∞ ∞ ∞ ∞ ∞ ∞
0 1 2 3 4 5
Distance
∅ ∅ ∅ ∅ ∅ ∅Previous
Queue
S T
0
Visited
3
5 3
13
53
(1, 13)
4
97
53
10
4
2
12
2
(0, 12)
2
3
3
4
Dijkstra's Algorithm

20. Emory University Logo Guidelines
-
2
0
1
2
3
4
5
4
5
10
9
∞ ∞ ∞ ∞ ∞ ∞
0 1 2 3 4 5
Distance
∅ ∅ ∅ ∅ ∅ ∅Previous
Queue
S T
0
Visited
3
5 3
13
53
(1, 13)
4
97
53
10
4
2
12
2
(0, 12)
2
3
3
4
Dijkstra's Algorithm

21. Emory University Logo Guidelines
-
2
0
1
2
3
4
5
4
5
10
9
∞ ∞ ∞ ∞ ∞ ∞
0 1 2 3 4 5
Distance
∅ ∅ ∅ ∅ ∅ ∅Previous
Queue
S T
0
Visited
3
5 3
13
53
(1, 13)
4
97
53
10
4
2
12
2
0
2
3
3
4
Dijkstra's Algorithm

22. Emory University Logo Guidelines
-
2
0
1
2
3
4
5
4
5
10
9
∞ ∞ ∞ ∞ ∞ ∞
0 1 2 3 4 5
Distance
∅ ∅ ∅ ∅ ∅ ∅Previous
Queue
S T
0
Visited
3
5 3
13
53
4
97
53
10
4
2
12
2
0
2
3
3
4
1
Dijkstra's Algorithm

23. Emory University Logo Guidelines
-
2
0
1
2
3
4
5
4
5
10
9
∞ ∞ ∞ ∞ ∞ ∞
0 1 2 3 4 5
Distance
∅ ∅ ∅ ∅ ∅ ∅Previous
Queue
S T
0
Visited
3
5 3
13
53
4
97
53
10
4
2
12
2
0
2
3
3
4
1
Dijkstra's Algorithm

24. Emory University Logo Guidelines
-
2
0
1
2
3
4
5
4
5
10
9
∞ ∞ ∞ ∞ ∞ ∞
0 1 2 3 4 5
Distance
∅ ∅ ∅ ∅ ∅ ∅Previous
Queue
S T
0
Visited
3
5 3
13
53
4
97
53
10
4
2
12
2
0
2
3
3
4
1
Dijkstra's Algorithm

25. Emory University Logo Guidelines
-
2
0
1
2
3
4
5
4
5
10
9
∞ ∞ ∞ ∞ ∞ ∞
0 1 2 3 4 5
Distance
∅ ∅ ∅ ∅ ∅ ∅Previous
Queue
S T
0
Visited
3
5 3
13
53
4
97
53
10
4
2
12
2
0
2
3
3
4
1
Dijkstra's Algorithm

26. Emory University Logo Guidelines
-
2
0
1
2
3
4
5
4
5
10
9
∞ ∞ ∞ ∞ ∞ ∞
0 1 2 3 4 5
Distance
∅ ∅ ∅ ∅ ∅ ∅Previous
Queue
S T
0
Visited
3
5 3
13
53
4
97
53
10
4
2
12
2
0
2
3
3
4
1
Dijkstra's Algorithm

27. Emory University Logo Guidelines
-
2
0
1
2
3
4
5
4
5
10
9
∞ ∞ ∞ ∞ ∞ ∞
0 1 2 3 4 5
Distance
∅ ∅ ∅ ∅ ∅ ∅Previous
Queue
S T
0
Visited
3
5 3
13
53
4
97
53
10
4
2
12
2
0
2
3
3
4
1
Dijkstra's Algorithm

28. Emory University Logo Guidelines
-
Dijkstra's Algorithm
3
1
0 3
2
∅∅∅∅Previous
∞∞∞∞Distance
0 1 2 3
QueueVisited
21
12
4

29. Emory University Logo Guidelines
-
Dijkstra's Algorithm
3
1
0 3
2
∅∅∅∅Previous
∞∞∞∞Distance
0 1 2 3
QueueVisited
TS
21
12
4

30. Emory University Logo Guidelines
-
Dijkstra's Algorithm
3
1
0 3
2
∅∅∅∅Previous
∞∞∞∞Distance
0 1 2 3
QueueVisited
TS
0
(3, 0)
21
12
4

31. Emory University Logo Guidelines
-
Dijkstra's Algorithm
3
1
0 3
2
∅∅∅∅Previous
∞∞∞∞Distance
0 1 2 3
QueueVisited
TS
0
21
3 12
4

32. Emory University Logo Guidelines
-
Dijkstra's Algorithm
3
1
0 3
2
∅∅∅∅Previous
∞∞∞∞Distance
0 1 2 3
QueueVisited
TS
0
21
3
1
3
(1, 1)12
4

33. Emory University Logo Guidelines
-
Dijkstra's Algorithm
3
1
0 3
2
∅∅∅∅Previous
∞∞∞∞Distance
0 1 2 3
QueueVisited
TS
0
21
3
1
3
(1, 1)12
4

34. Emory University Logo Guidelines
-
Dijkstra's Algorithm
3
1
0 3
2
∅∅∅∅Previous
∞∞∞∞Distance
0 1 2 3
QueueVisited
TS
0
21
3
1
3
(1, 1)
2
3
(2, 2)
12
4

35. Emory University Logo Guidelines
-
Dijkstra's Algorithm
3
1
0 3
2
∅∅∅∅Previous
∞∞∞∞Distance
0 1 2 3
QueueVisited
TS
0
21
3
1
3
2
3
(2, 2)
1 12
4

36. Emory University Logo Guidelines
-
Dijkstra's Algorithm
3
1
0 3
2
∅∅∅∅Previous
∞∞∞∞Distance
0 1 2 3
QueueVisited
TS
0
21
3
1
3
2
3
(2, 2)1
3
1
(0, 3)
12
4

37. Emory University Logo Guidelines
-
Dijkstra's Algorithm
3
1
0 3
2
∅∅∅∅Previous
∞∞∞∞Distance
0 1 2 3
QueueVisited
TS
0
21
3
1
3
2
3
1
3
1
(0, 3)
2 12
4

38. Emory University Logo Guidelines
-
Dijkstra's Algorithm
3
1
0 3
2
∅∅∅∅Previous
∞∞∞∞Distance
0 1 2 3
QueueVisited
TS
0
21
3
1
3
2
3
1
3
1
2 0 12
4

39. Emory University Logo Guidelines
-
Dijkstra's Algorithm
3
1
0 3
2
∅∅∅∅Previous
∞∞∞∞Distance
0 1 2 3
QueueVisited
TS
0
21
3
1
3
2
3
1
3
1
2 0 12
4

40. Emory University Logo Guidelines
-
Dijkstra's Algorithm
3
1
0 3
2
∅∅∅∅Previous
∞∞∞∞Distance
0 1 2 3
QueueVisited
TS
0
21
3
1
3
2
3
1
3
1
2 0 12
4

41. Emory University Logo Guidelines
-
Dijkstra's Algorithm
3
1
0 3
2
∅∅∅∅Previous
∞∞∞∞Distance
0 1 2 3
QueueVisited
TS
0
21
3
1
3
2
3
1
3
1
2 0 12
4

42. Emory University Logo Guidelines
-
Dijkstra's Algorithm
3
1
0 3
2
∅∅∅∅Previous
∞∞∞∞Distance
0 1 2 3
QueueVisited
TS
0
21
3
1
3
2
3
1
3
1
2 0 12
4

43. Emory University Logo Guidelines
-
Dijkstra's Algorithm
3
1
0 3
2
∅∅∅∅Previous
∞∞∞∞Distance
0 1 2 3
QueueVisited
TS
0
21
3
1
3
2
3
1
3
1
2 0 12
4

44. Emory University Logo Guidelines
-
Dijkstra's Algorithm
4
1
0 3
2
∅∅∅∅Previous
∞∞∞∞Distance
0 1 2 3
QueueVisited
1
1 2
2
-4

45. Emory University Logo Guidelines
-
Dijkstra's Algorithm
4
1
0 3
2
∅∅∅∅Previous
∞∞∞∞Distance
0 1 2 3
QueueVisited
TS
1
1 2
2
-4

46. Emory University Logo Guidelines
-
Dijkstra's Algorithm
4
1
0 3
2
∅∅∅∅Previous
∞∞∞∞Distance
0 1 2 3
QueueVisited
TS
0
(3, 0)1
1 2
2
-4

47. Emory University Logo Guidelines
-
Dijkstra's Algorithm
4
1
0 3
2
∅∅∅∅Previous
∞∞∞∞Distance
0 1 2 3
QueueVisited
TS
0
1
1
3
2
2
-4

48. Emory University Logo Guidelines
-
Dijkstra's Algorithm
4
1
0 3
2
∅∅∅∅Previous
∞∞∞∞Distance
0 1 2 3
QueueVisited
TS
0
1
1
3
1
3
(1, 1)
2
2
-4

49. Emory University Logo Guidelines
-
Dijkstra's Algorithm
4
1
0 3
2
∅∅∅∅Previous
∞∞∞∞Distance
0 1 2 3
QueueVisited
TS
0
1
1
3
1
3
(1, 1)
2
2
-4

50. Emory University Logo Guidelines
-
Dijkstra's Algorithm
4
1
0 3
2
∅∅∅∅Previous
∞∞∞∞Distance
0 1 2 3
QueueVisited
TS
0
1
1
3
1
3
(1, 1)
2
3
(2, 2)
2
2
-4

51. Emory University Logo Guidelines
-
Dijkstra's Algorithm
4
1
0 3
2
∅∅∅∅Previous
∞∞∞∞Distance
0 1 2 3
QueueVisited
TS
0
1
1
3
1
3
2
3
(2, 2)
1
2
2
-4

52. Emory University Logo Guidelines
-
Dijkstra's Algorithm
4
1
0 3
2
∅∅∅∅Previous
∞∞∞∞Distance
0 1 2 3
QueueVisited
TS
0
1
1
3
1
3
2
3
(2, 2)1
3
1
(0, 3)
2
2
-4

53. Emory University Logo Guidelines
-
Dijkstra's Algorithm
4
1
0 3
2
∅∅∅∅Previous
∞∞∞∞Distance
0 1 2 3
QueueVisited
TS
0
1
1
3
1
3
2
3
1
3
1
(0, 3)
2
2
2
-4

54. Emory University Logo Guidelines
-
Dijkstra's Algorithm
4
1
0 3
2
∅∅∅∅Previous
∞∞∞∞Distance
0 1 2 3
QueueVisited
TS
0
1
1
3
1
3
2
3
1
3
1
(0, 3)
2
-2
2
(1, -2)
2
2
-4

55. Emory University Logo Guidelines
-
Dijkstra's Algorithm
4
1
0 3
2
∅∅∅∅Previous
∞∞∞∞Distance
0 1 2 3
QueueVisited
TS
0
1
1
3
1
3
2
3
1
3
1
(0, 3)
2
-2
2
2
2
-4

56. Emory University Logo Guidelines
-
Dijkstra's Algorithm
4
1
0 3
2
∅∅∅∅Previous
∞∞∞∞Distance
0 1 2 3
QueueVisited
TS
0
1
1
3
1
3
2
3
1
3
1
2 0
-2
2
2
2
-4

57. Emory University Logo Guidelines
-
Dijkstra's Algorithm
4
1
0 3
2
∅∅∅∅Previous
∞∞∞∞Distance
0 1 2 3
QueueVisited
TS
0
1
1
3
1
3
2
3
1
3
1
2 0
-2
2
2
2
-4

58. Emory University Logo Guidelines
-
Dijkstra's Algorithm
4
1
0 3
2
∅∅∅∅Previous
∞∞∞∞Distance
0 1 2 3
QueueVisited
TS
0
1
1
3
1
3
2
3
1
3
1
2 0
-2
2
2
2
-4

59. Emory University Logo Guidelines
-
Dijkstra's Algorithm
4
1
0 3
2
∅∅∅∅Previous
∞∞∞∞Distance
0 1 2 3
QueueVisited
TS
0
1
1
3
1
3
2
3
1
3
1
2 0
-2
2
2
2
-4

60. Emory University Logo Guidelines
-
Dijkstra's Algorithm
4
1
0 3
2
∅∅∅∅Previous
∞∞∞∞Distance
0 1 2 3
QueueVisited
TS
0
1
1
3
1
3
2
3
1
3
1
2 0
-2
2
2
2
-4

61. Emory University Logo Guidelines
-
Vertex Distance Pair
5
private class VertexDistancePair implements Comparable<VertexDistancePair>
{
public int vertex;
public double distance;
public VertexDistancePair(int vertex, double distance)
{
this.vertex = vertex;
this.distance = distance;
}
@Override
public int compareTo(VertexDistancePair p)
{
double diff = this.distance - p.distance;
if (diff > 0) return 1;
if (diff < 0) return -1;
return 0;
}
}

62. Emory University Logo Guidelines
-
Dijkstra's Algorithm
6
public Integer[] getShortestPath(Graph graph, int source, int target)
{
PriorityQueue<VertexDistancePair> queue = new PriorityQueue<>();
Integer[] previous = new Integer[graph.size()];
double[] distances = new double[graph.size()];
Set<Integer> visited = new HashSet<>();
init(distances, previous, target);
queue.add(new VertexDistancePair(target, distances[target]));
while (!queue.isEmpty())
{
// To be filled.
}
return previous;
}

63. Emory University Logo Guidelines
-
Dijkstra's Algorithm
6
public Integer[] getShortestPath(Graph graph, int source, int target)
{
PriorityQueue<VertexDistancePair> queue = new PriorityQueue<>();
Integer[] previous = new Integer[graph.size()];
double[] distances = new double[graph.size()];
Set<Integer> visited = new HashSet<>();
init(distances, previous, target);
queue.add(new VertexDistancePair(target, distances[target]));
while (!queue.isEmpty())
{
// To be filled.
}
return previous;
}

64. Emory University Logo Guidelines
-
Dijkstra's Algorithm
6
public Integer[] getShortestPath(Graph graph, int source, int target)
{
PriorityQueue<VertexDistancePair> queue = new PriorityQueue<>();
Integer[] previous = new Integer[graph.size()];
double[] distances = new double[graph.size()];
Set<Integer> visited = new HashSet<>();
init(distances, previous, target);
queue.add(new VertexDistancePair(target, distances[target]));
while (!queue.isEmpty())
{
// To be filled.
}
return previous;
}

65. Emory University Logo Guidelines
-
Dijkstra's Algorithm
7
private void init(double[] distances, Integer[] previous, int target)
{
for (int i=0; i<distances.length; i++)
{
if (i == target)
distances[i] = 0;
else
{
distances[i] = Double.MAX_VALUE;
previous[i] = null;
}
}
}

66. Emory University Logo Guidelines
-
Dijkstra's Algorithm
7
private void init(double[] distances, Integer[] previous, int target)
{
for (int i=0; i<distances.length; i++)
{
if (i == target)
distances[i] = 0;
else
{
distances[i] = Double.MAX_VALUE;
previous[i] = null;
}
}
}

67. Emory University Logo Guidelines
-
8
while (!queue.isEmpty())
{
VertexDistancePair u = queue.poll();
visited.add(u.vertex);
for (Edge edge : graph.getIncomingEdges(u.vertex))
{
int v = edge.getSource();
if (!visited.contains(v))
{
double dist = distances[u.vertex] + edge.getWeight();
if (dist < distances[v])
{
distances[v] = dist;
previous [v] = u.vertex;
queue.add(new VertexDistancePair(v, dist));
}
}
}
}

68. Emory University Logo Guidelines
-
8
while (!queue.isEmpty())
{
VertexDistancePair u = queue.poll();
visited.add(u.vertex);
for (Edge edge : graph.getIncomingEdges(u.vertex))
{
int v = edge.getSource();
if (!visited.contains(v))
{
double dist = distances[u.vertex] + edge.getWeight();
if (dist < distances[v])
{
distances[v] = dist;
previous [v] = u.vertex;
queue.add(new VertexDistancePair(v, dist));
}
}
}
}
Complexity?

69. Emory University Logo Guidelines
-
Dijkstra's vs A*
9
private void init(double[] distances, Integer[] previous, int target)
{
for (int i=0; i<distances.length; i++)
{
if (i == target)
distances[i] = 0;
else
{
distances[i] = Double.MAX_VALUE;
previous[i] = null;
}
}
}

70. Emory University Logo Guidelines
-
Dijkstra's vs A*
9
private void init(double[] distances, Integer[] previous, int target)
{
for (int i=0; i<distances.length; i++)
{
if (i == target)
distances[i] = 0;
else
{
distances[i] = Double.MAX_VALUE;
previous[i] = null;
}
}
}
+ heuristic(i, target);

71. Emory University Logo Guidelines
-
10
while (!queue.isEmpty())
{
VertexDistancePair u = queue.poll();
visited.add(u.vertex);
for (Edge edge : graph.getIncomingEdges(u.vertex))
{
int v = edge.getSource();
if (!visited.contains(v))
{
double dist = distances[u.vertex] + edge.getWeight();
if (dist < distances[v])
{
distances[v] = dist;
previous [v] = u.vertex;
queue.add(new VertexDistancePair(v, dist));
}
}
}
}

72. Emory University Logo Guidelines
-
10
while (!queue.isEmpty())
{
VertexDistancePair u = queue.poll();
visited.add(u.vertex);
for (Edge edge : graph.getIncomingEdges(u.vertex))
{
int v = edge.getSource();
if (!visited.contains(v))
{
double dist = distances[u.vertex] + edge.getWeight();
if (dist < distances[v])
{
distances[v] = dist;
previous [v] = u.vertex;
queue.add(new VertexDistancePair(v, dist));
}
}
}
}
+ heuristic(v, target);

73. Emory University Logo Guidelines
-
A* Abstract Class
11
public abstract class AStar
{
public Integer[] getShortestPath(Graph graph, int source, int target)
{
PriorityQueue<VertexDistancePair> queue = new PriorityQueue<>();
Integer[] previous = new Integer[graph.size()];
double[] distances = new double[graph.size()];
Set<Integer> visited = new HashSet<>();
...
return previous;
}
}
protected abstract double heuristic(int source, int target);

74. Emory University Logo Guidelines
-
Dijkstra’s Algorithm
12
public class Dijkstra extends AStar
{
@Override
protected double heuristic(int source, int target)
{
return 0;
}
}