What is Shortest Path Algorithm?

The shortest path algorithms aim to find the path in a graph where the sum of edge weights is minimized.

SSSP(Single Source Shortest Path) - Starting Vertex Exists

Time Complexity: O(VElogV) - V: All nodes, E: Edges connected to each node, logV: For each edge's relaxation
- relax ⇒ insertion and extraction-min of Min-Heap data structure
- Considering all edges in the graph as E: E = O(V^2), so O(V^2logV)
Principle: Find the shortest path by continuously following the minimum weight from the starting vertex.
Features:
- Does not allow negative weights (cannot guarantee the shortest path when negative weights exist).
- Greedy algorithm.
- Performs relaxation by following the minimum weight in each iteration.
Data Structure in Use: Adjacency list {Vertex: [Neighbor Vertex, Neighbor Vertex…]}, Min-Heap, Fibonacci Heap
- Using a Fibonacci Heap achieves a time complexity of O(V^2).

Time Complexity: O(VE) - V: All vertices
- E = O(V^2), so O(V^3)
Principle: Find the shortest path from the starting vertex to all vertices. It allows negative weights and can detect negative weight cycles.
Features:
- Allows negative weights.
- Does not allow negative cycles but can detect them.
- Dynamic Programming (DP) algorithm.
- Performs relaxation by considering all edges in each iteration.
Data Structure in Use: Graph edge list {Edge:(Start, End, Weight), …}

Principle: Update the shortest paths by considering all vertices while traversing all other vertices.
Data Structure in Use: Graph weight matrix
Time Complexity: O(N^3) - triple for loop
Features:
- Allows negative weights.
- Calculates the shortest path for all pairs of vertices in one go.