The document describes Dijkstra's algorithm, which is used to find the shortest paths between nodes in a graph. It shows the step-by-step workings of the algorithm on a sample graph with 6 nodes. It initializes the graph with distance values and tracks which nodes have been visited as it iteratively explores all possible paths from the source node to determine the shortest distance to each destination node.

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

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Dijkstra's Algorithm

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Dijkstra's Algorithm

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∞ ∞ ∞ ∞ ∞ ∞
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Distance
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Dijkstra's Algorithm

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∞ ∞ ∞ ∞ ∞ ∞
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Distance
∅ ∅ ∅ ∅ ∅ ∅Previous
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Visited
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Dijkstra's Algorithm

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(5, 0)
∞ ∞ ∞ ∞ ∞ ∞
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∅ ∅ ∅ ∅ ∅ ∅Previous
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Dijkstra's Algorithm

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∞ ∞ ∞ ∞ ∞ ∞
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Distance
∅ ∅ ∅ ∅ ∅ ∅Previous
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Dijkstra's Algorithm

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∞ ∞ ∞ ∞ ∞ ∞
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Distance
∅ ∅ ∅ ∅ ∅ ∅Previous
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Visited
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(3, 3)5
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Dijkstra's Algorithm

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∞ ∞ ∞ ∞ ∞ ∞
0 1 2 3 4 5
Distance
∅ ∅ ∅ ∅ ∅ ∅Previous
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Visited
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(3, 3)5
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Dijkstra's Algorithm

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∞ ∞ ∞ ∞ ∞ ∞
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∅ ∅ ∅ ∅ ∅ ∅Previous
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Visited
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(3, 3)5
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(4, 9)
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Dijkstra's Algorithm

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∞ ∞ ∞ ∞ ∞ ∞
0 1 2 3 4 5
Distance
∅ ∅ ∅ ∅ ∅ ∅Previous
Queue
S T
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Visited
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5 3
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(4, 9)
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Dijkstra's Algorithm

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∞ ∞ ∞ ∞ ∞ ∞
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∅ ∅ ∅ ∅ ∅ ∅Previous
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(1, 13)9
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(4, 9)
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Dijkstra's Algorithm

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∞ ∞ ∞ ∞ ∞ ∞
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Distance
∅ ∅ ∅ ∅ ∅ ∅Previous
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Visited
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(1, 13)9
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(4, 9)
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Dijkstra's Algorithm

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∞ ∞ ∞ ∞ ∞ ∞
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∅ ∅ ∅ ∅ ∅ ∅Previous
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Visited
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5 3
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(1, 13)
(4, 7)
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(4, 9)
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Dijkstra's Algorithm

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∞ ∞ ∞ ∞ ∞ ∞
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∅ ∅ ∅ ∅ ∅ ∅Previous
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(1, 13)
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(4, 9)
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Dijkstra's Algorithm

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∞ ∞ ∞ ∞ ∞ ∞
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∅ ∅ ∅ ∅ ∅ ∅Previous
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(1, 13)
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(4, 9)
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(2, 10)
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Dijkstra's Algorithm

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∞ ∞ ∞ ∞ ∞ ∞
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(1, 13)
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Dijkstra's Algorithm

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∞ ∞ ∞ ∞ ∞ ∞
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∅ ∅ ∅ ∅ ∅ ∅Previous
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(1, 13)
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Dijkstra's Algorithm

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∞ ∞ ∞ ∞ ∞ ∞
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(1, 13)
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Dijkstra's Algorithm

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∞ ∞ ∞ ∞ ∞ ∞
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(1, 13)
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Dijkstra's Algorithm

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∞ ∞ ∞ ∞ ∞ ∞
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(1, 13)
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Dijkstra's Algorithm

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Dijkstra's Algorithm

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Dijkstra's Algorithm

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Dijkstra's Algorithm

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Dijkstra's Algorithm

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Dijkstra's Algorithm

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Dijkstra's Algorithm

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Dijkstra's Algorithm
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∅∅∅∅Previous
∞∞∞∞Distance
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Dijkstra's Algorithm
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∅∅∅∅Previous
∞∞∞∞Distance
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Dijkstra's Algorithm
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∅∅∅∅Previous
∞∞∞∞Distance
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Dijkstra's Algorithm
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∅∅∅∅Previous
∞∞∞∞Distance
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Dijkstra's Algorithm
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∞∞∞∞Distance
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Dijkstra's Algorithm
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∅∅∅∅Previous
∞∞∞∞Distance
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Dijkstra's Algorithm
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∅∅∅∅Previous
∞∞∞∞Distance
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(1, 1)
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(2, 2)
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Dijkstra's Algorithm
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∅∅∅∅Previous
∞∞∞∞Distance
0 1 2 3
QueueVisited
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Dijkstra's Algorithm
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∅∅∅∅Previous
∞∞∞∞Distance
0 1 2 3
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(2, 2)1
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(0, 3)
12
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Dijkstra's Algorithm
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∅∅∅∅Previous
∞∞∞∞Distance
0 1 2 3
QueueVisited
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(0, 3)
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Dijkstra's Algorithm
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∅∅∅∅Previous
∞∞∞∞Distance
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QueueVisited
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2 0 12
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Dijkstra's Algorithm
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∅∅∅∅Previous
∞∞∞∞Distance
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2 0 12
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Dijkstra's Algorithm
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∅∅∅∅Previous
∞∞∞∞Distance
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QueueVisited
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2 0 12
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Dijkstra's Algorithm
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∅∅∅∅Previous
∞∞∞∞Distance
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2 0 12
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Dijkstra's Algorithm
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∅∅∅∅Previous
∞∞∞∞Distance
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Dijkstra's Algorithm
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∞∞∞∞Distance
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Dijkstra's Algorithm
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∅∅∅∅Previous
∞∞∞∞Distance
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QueueVisited
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1 2
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-4

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Dijkstra's Algorithm
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∅∅∅∅Previous
∞∞∞∞Distance
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QueueVisited
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1 2
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-4

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Dijkstra's Algorithm
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∅∅∅∅Previous
∞∞∞∞Distance
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(3, 0)1
1 2
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Dijkstra's Algorithm
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∅∅∅∅Previous
∞∞∞∞Distance
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QueueVisited
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Dijkstra's Algorithm
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∅∅∅∅Previous
∞∞∞∞Distance
0 1 2 3
QueueVisited
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(1, 1)
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-4

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Dijkstra's Algorithm
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0 3
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∅∅∅∅Previous
∞∞∞∞Distance
0 1 2 3
QueueVisited
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(1, 1)
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-4

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Dijkstra's Algorithm
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0 3
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∅∅∅∅Previous
∞∞∞∞Distance
0 1 2 3
QueueVisited
TS
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1
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(1, 1)
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(2, 2)
2
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-4

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Dijkstra's Algorithm
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∅∅∅∅Previous
∞∞∞∞Distance
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QueueVisited
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(2, 2)
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-4

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Dijkstra's Algorithm
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∅∅∅∅Previous
∞∞∞∞Distance
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QueueVisited
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(2, 2)1
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(0, 3)
2
2
-4

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Dijkstra's Algorithm
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∅∅∅∅Previous
∞∞∞∞Distance
0 1 2 3
QueueVisited
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1
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(0, 3)
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2
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-4

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Dijkstra's Algorithm
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∅∅∅∅Previous
∞∞∞∞Distance
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QueueVisited
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1
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(0, 3)
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-2
2
(1, -2)
2
2
-4

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Dijkstra's Algorithm
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∅∅∅∅Previous
∞∞∞∞Distance
0 1 2 3
QueueVisited
TS
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1
1
3
1
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1
3
1
(0, 3)
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-2
2
2
2
-4

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

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Dijkstra's Algorithm
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0 3
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∅∅∅∅Previous
∞∞∞∞Distance
0 1 2 3
QueueVisited
TS
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1
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-4

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

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

60. Emory University Logo Guidelines
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Dijkstra's Algorithm
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∅∅∅∅Previous
∞∞∞∞Distance
0 1 2 3
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TS
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61. Emory University Logo Guidelines
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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
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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
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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
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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
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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
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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;
}
}