Shortest Path problem

What is the Shortest Path Problem in Graphs?

The shortest path problem in computer science is all about finding the minimum cost path between two nodes (or vertices) in a graph.

Graphs are used to model road networks, communication networks, maps, and more. So, the shortest path problem helps in finding:

• The shortest route between two cities,

• The fastest way to send data between computers,

• Or even the cheapest travel cost across stations.

Key Concepts of the Shortest Path Problem

Graph Terminology:

• Vertex/Node: A point (e.g., city, router)

• Edge: A connection between two vertices (e.g., road)

• Weight: A number assigned to an edge (e.g., distance, time, cost)

• Path Weight/Cost: Total sum of edge weights in a path

• Shortest Path: Path with the lowest total cost

Example

Imagine this graph of cities:

D --(2)--> E --(4)--> C --(4)--> F

The shortest path from D to F is:
 D → E → C → F, with total cost: 2 + 4 + 4 = 10

There may be other routes, but none with a smaller total weight.

How to Solve the Shortest Path Problem?

There are two most common algorithms used:

1. Dijkstra’s Algorithm

• Works when all edge weights are positive

• Finds shortest path from one source to all other vertices

• Used in GPS apps, network routing

2. Bellman-Ford Algorithm

• Works even if the graph has negative weights

• Can detect negative weight cycles

• Slower but more powerful

What is Relaxation?

In both algorithms, we check each edge to see if we can find a shorter path to a vertex than previously known.
This process is called relaxing an edge.

Every time we find a shorter path, we update the distance.

Positive vs Negative Edge Weights

Positive Weights

Graphs with all positive weights are easier to work with and are perfect for Dijkstra’s Algorithm.

For example:

• A → C with weight 4 means cost is $4

• All travel, fuel, or time costs are positive numbers

Negative Weights

Negative weights mean that you earn or gain something when traveling that edge.

For example:

• C → A with weight -3 means you gain $3

Only the Bellman-Ford Algorithm can handle negative weights safely.

Negative Cycles – The Dangerous Loops

A negative cycle is a loop in a graph where the total sum of weights is negative.

Example:

A → E → B → C → A with weights: 5, 2, -4, -4 = Total: -1

Why is it a problem?

You can keep going around this loop and reduce the path weight infinitely:

• 1st round: cost = -1

• 2nd round: cost = -2

• 3rd round: cost = -3

• And so on...

Result: No shortest path can be found, because it gets shorter forever.

That’s why:

• Dijkstra fails if there's a negative edge.

• Bellman-Ford can detect negative cycles and report them.

Summary Table

Real-Life Applications of Shortest Path

• Google Maps, GPS route finding

• Network Routing (like Internet Protocols)

• Dynamic Programming and AI search

• Cost optimization in logistics and transport

• Telecommunications

Final Thought:

The Shortest Path Problem helps us solve real-world navigation and optimization challenges.
Understanding Dijkstra’s and Bellman-Ford gives you a strong foundation in algorithms used in tech giants like Google, Uber, Amazon, and more.