The Shortest Path Algorithm is a fundamental algorithm used in graph theory and pathfinding applications. It plays a crucial role in various domains, including network routing, navigation systems, logistics planning, and transportation optimization. This algorithm aims to find the shortest path between two nodes in a graph, considering either weighted or unweighted edges. In this presentation, we will explore the principles and techniques behind the Shortest Path Algorithm. We will discuss popular algorithms such as Dijkstra's algorithm, Breadth-First Search (BFS), and Depth-First Search (DFS) that are commonly used to solve the shortest path problem. We will delve into their advantages, limitations, and the types of graphs they are suitable for. Additionally, we will cover essential concepts related to the algorithm, such as graph traversal, distance calculation, edge weights, and data structures used to store and process graph information efficiently. We will also touch upon dynamic programming approaches that can optimize the computation of shortest paths in certain scenarios. Throughout the presentation, we will highlight real-world applications that rely on the Shortest Path Algorithm, including network routing in telecommunications, GPS navigation systems, and supply chain optimization. We will showcase how this algorithm enables efficient decision-making, minimizes travel distances, and optimizes resource allocation in various domains. By the end of this presentation, you will have a comprehensive understanding of the Shortest Path Algorithm and its practical implications. You will recognize its significance in solving complex pathfinding problems and appreciate its role in improving efficiency and reducing costs in diverse industries.

1. Shortest Path Algorithm
CS 250 Data Structures and Algorithms
Prof. Dr. Faisal Shafait
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2. Shortest path
Learning Objectives
In this lecture you will learn
The need to calculate the shortest path and its various
applications
Implementation of Dijkstra’s Algorithm
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3. Shortest path
Shortest path
Goal. find the shortest path and its length
Assumption. weights on all edges is a positive number.
negative vertices may end up in a cycle; each pass decreasing the
total length
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4. Shortest path
Shortest path: applications
Circuit design. time taken for a change in input to affect an output
Internet. Packets are passed from the source, through routers, to
their destination
considers low latency ( to minimize time), or
minimum hop count
Maps. finding shortest route between to points on a map
rush hour
scheduled events (e.g., road construction)
unscheduled events (e.g., accidents)
Courier company. to minimize the cost of different factors:
salary of the truck driver
tolls costs
bonuses for being early
penalties for being late
cost of fuel
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5. Shortest path
Dijkstra’s Algorithm
Input.
a directed weighted graph
Basic idea.
FIND an unvisited vertex vcur with shortest distance to it
MARK it as visited
FOR EACH adjacent unvisited vertex vadj to the vertex vcur:
ADD weight of the connecting edge to distance of vertex vcur, gives
dcur
CONDITION if dcur < dadj , update dadj and set vadj parent to vcurrent
Halting point.
when all vertices are visited
Output.
a shortest path from source vertex to destination vertex
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6. Shortest path
Dijkstra’s Algorithm
Initialize the table.
visited to false.
cost to ∞.
parent to 0.
Vertex Visited Distance Parent
1 F ∞ 0
2 F ∞ 0
3 F ∞ 0
4 F ∞ 0
5 F ∞ 0
6 F ∞ 0
7 F ∞ 0
8 F ∞ 0
9 F ∞ 0
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7. Shortest path
Dijkstra’s Algorithm
Visit vertex 1.
cost to 0.
three paths: (1,2), (1,4) and (1,5) with costs 4, 1 and 8.
Vertex Visited Distance Parent
1 T 0 0
2 F 4 1
3 F ∞ 0
4 F 1 1
5 F 8 1
6 F ∞ 0
7 F ∞ 0
8 F ∞ 0
9 F ∞ 0
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8. Shortest path
Dijkstra’s Algorithm
Visit vertex 4, the minimum-cost unvisited vertex.
three paths: (1,4,5), (1,4,7) and (1,4,8) with costs 12, 10 and 9.
ignore path (1,4,5); there is a shorter path to vertex 5.
Vertex Visited Distance Parent
1 T 0 0
2 F 4 1
3 F ∞ 0
4 T 1 1
5 F 8 1
6 F ∞ 0
7 F 10 4
8 F 9 4
9 F ∞ 0
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9. Shortest path
Dijkstra’s Algorithm
Visit vertex 2, the minimum-cost unvisited vertex.
three paths: (1,2,3), (1,2,5) and (1,2,6) with costs 5, 10 and 5.
ignore path (1,2,5); there is a shorter path to vertex 5.
Vertex Visited Distance Parent
1 T 0 0
2 T 4 1
3 F 5 2
4 T 1 1
5 F 8 1
6 F 5 2
7 F 10 4
8 F 9 4
9 F ∞ 0
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10. Shortest path
Dijkstra’s Algorithm
Visit vertex 3, the minimum-cost unvisited vertex.
three paths: (1,2,3,5) and (1,2,3,6) with costs 7 and 12.
ignore path (1,2,3,6); there is a shorter path to vertex 6.
Vertex Visited Distance Parent
1 T 0 0
2 T 4 1
3 T 5 2
4 T 1 1
5 F 7 3
6 F 5 2
7 F 10 4
8 F 9 4
9 F ∞ 0
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11. Shortest path
Dijkstra’s Algorithm
Visit vertex 6, the minimum-cost unvisited vertex.
paths: (1,2,6,8) and (1,2,6,9) with costs 12 and 13.
ignore path (1,2,3,8); there is a shorter path to vertex 8.
Vertex Visited Distance Parent
1 T 0 0
2 T 4 1
3 T 5 2
4 T 1 1
5 F 7 3
6 T 5 2
7 F 10 4
8 F 9 4
9 F 13 6
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12. Shortest path
Dijkstra’s Algorithm
Visit vertex 5, the minimum-cost unvisited vertex.
paths: (1,2,3,5,7), (1,2,3,5,8) and (1,2,3,5,9) with costs 8, 8 and 15.
ignore path (1,2,3,5,9); there is a shorter path to vertex 9.
Vertex Visited Distance Parent
1 T 0 0
2 T 4 1
3 T 5 2
4 T 1 1
5 T 7 3
6 T 5 2
7 F 8 5
8 F 8 5
9 F 13 6
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13. Shortest path
Dijkstra’s Algorithm
Visit vertex 7, the minimum-cost unvisited vertex.
one paths: (1,2,3,5,7,8) with cost 10.
ignore the path; there is a shorter path to vertex 8.
Vertex Visited Distance Parent
1 T 0 0
2 T 4 1
3 T 5 2
4 T 1 1
5 T 7 3
6 T 5 2
7 T 8 5
8 F 8 5
9 F 13 6
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14. Shortest path
Dijkstra’s Algorithm
Visit vertex 8, the minimum-cost unvisited vertex.
one path: (1,2,3,5,8,9) with cost 11.
update the cost to reach 11.
Vertex Visited Distance Parent
1 T 0 0
2 T 4 1
3 T 5 2
4 T 1 1
5 T 7 3
6 T 5 2
7 T 8 5
8 T 8 5
9 F 11 8
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15. Shortest path
Dijkstra’s Algorithm
Visit vertex 9, the minimum-cost unvisited vertex.
one path: (1,2,3,5,8,9) with cost 11.
this is the shortest path from vertex 1 to vertex 9.
Vertex Visited Distance Parent
1 T 0 0
2 T 4 1
3 T 5 2
4 T 1 1
5 T 7 3
6 T 5 2
7 T 8 5
8 T 8 5
9 T 11 8
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16. Shortest path
Reading Material
For further reading, please refer to
Data Structures by Weiss: Chapter 15, Sections 15.3
https://www.programiz.com/dsa/dijkstra-algorithm
https://www.cs.usfca.edu/ galles/visualization/Dijkstra.html
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