Dijkstra's Algorithm
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Dijkstra's algorithm allows finding the shortest path between any two vertices in a graph. It works by overestimating the distance of each vertex from the starting point and then visiting neighbors to find shorter paths. The algorithm uses a greedy approach, finding the next best solution at each step. It maintains path distances in an array and maps each vertex to its predecessor in the shortest path. A priority queue is used to efficiently retrieve the closest vertex. The time complexity is O(E Log V) and space is O(V). Applications include social networks, maps, and telephone networks.
![1 function Dijkstra(Graph, source):
2
3 create vertex set Q
4
5 for each vertex v in Graph:
6 dist[v] ← INFINITY
7 prev[v] ← UNDEFINED
8 add v to Q
9 dist[source] ← 0
10
11 while Q is not empty:
12 u ← vertex in Q with min dist[u]
13
14 remove u from Q
15
16 for each neighbor v of u still in Q:
17 alt ← dist[u] + length(u, v)
18 if alt < dist[v]:
19 dist[v] ← alt
20 prev[v] ← u
21
22 return dist[], prev[]
PSEUDOCODE](https://image.slidesharecdn.com/organized4-211211134112/85/Dijkstra-s-Algorithm-8-320.jpg)
Dijkstra's Algorithm
- 1. DIJKSTRA’S ALGORITHM Represented by Group: 3 • Arijit Dhali • Ipsita Raha Department : Electronic & Communication Engineering Year : 2nd Session : 2021-22 Data Structure and Algorithm ES – CSS 301
- 2. TABLE OF CONTENTS 01 Understanding Dijkstra 02 Algorithm and Pseudocode 03 Decoding 04 Complexity
- 3. What is Dijkstra’s Algorithm ? Dijkstra's algorithm is an algorithm that allows us to find the shortest path between any two vertices of a graph. DIJKSTRA’S ALGORITHM
- 4. Dijkstra's Algorithm works on the basis that any subpath B -> D of the shortest path A -> D between vertices A and D is also the shortest path between vertices B and D. Dijkstra used this property in the opposite direction i.e we overestimate the distance of each vertex from the starting vertex. Then we visit each node and its neighbors to find the shortest subpath to those neighbors. The algorithm uses a greedy approach in the sense that we find the next best solution hoping that the end result is the best solution for the whole problem. HOW DIJKSTRA’S ALGORITHM WORK?
- 7. We need to maintain the path distance of every vertex. We can store that in an array of size v, where v is the number of vertices. We also want to be able to get the shortest path, not only know the length of the shortest path. For this, we map each vertex to the vertex that last updated its path length. Once the algorithm is over, we can backtrack from the destination vertex to the source vertex to find the path. A minimum priority queue can be used to efficiently receive the vertex with least path distance. PSEUDOCODE OF DIJKSTRA
- 8. 1 function Dijkstra(Graph, source): 2 3 create vertex set Q 4 5 for each vertex v in Graph: 6 dist[v] ← INFINITY 7 prev[v] ← UNDEFINED 8 add v to Q 9 dist[source] ← 0 10 11 while Q is not empty: 12 u ← vertex in Q with min dist[u] 13 14 remove u from Q 15 16 for each neighbor v of u still in Q: 17 alt ← dist[u] + length(u, v) 18 if alt < dist[v]: 19 dist[v] ← alt 20 prev[v] ← u 21 22 return dist[], prev[] PSEUDOCODE
- 9. CODE FOR DIJKSTRA’S ALGORITHM The implementation of Dijkstra's Algorithm in C is given. The complexity of the code can be improved, but the abstractions are convenient to relate the code with the algorithm. Scan this QR Code using Google Lens to get access to the drive containing necessary coding files Log in using College ID
- 10. Time Complexity: O(E Log V) where, E is the number of edges and V is the number of vertices. Space Complexity: O(V) DIJKSTRA’S ALGORITHM COMPLEXITY
- 11. APPLICATIONS OF DIJKSTRA’S ALGORITHM • To find the shortest path • In social networking applications • In a telephone network • To find the locations in the map
- 12. CONCLUSION When the algorithm finishes, each vertex reachable from the source has v. distance set to the length of the shortest path from the source, and v. parent set to its parent in the tree of shortest paths.
- 13. • https://www.programiz.com/dsa/dijkstra-algorithm • https://en.wikipedia.org/wiki/Dijkstra%27s_algorithm • https://www.quora.com/What-is-the-conclusion-of-Dijkstras-algorithm BIBLIOGRAPHY
- 14. For your precious time towards our presentation THANK YOU!