Dijkstra’s Algorithm (Shortest Path)

 

Dijkstra’s Algorithm (Shortest Path) 🚀

Dijkstra’s Algorithm is used to find the shortest path from a single source vertex to all other vertices in a graph with non-negative weights.

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🔹 How It Works

✅ Input: A graph represented as an adjacency list or matrix and a starting node.
✅ Output: The shortest path distances from the source to all nodes.
✅ Time Complexity: O((V + E) log V) (using a priority queue).
✅ Uses: Greedy Approach and Min-Heap (Priority Queue) for efficiency.

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🔹 Algorithm Steps

1️⃣ Initialize:

• Set the distance to the source as 0 and all other nodes as ∞.
• Use a min-priority queue (heap) to store vertices by distance.

2️⃣ Process Nodes:

• Extract the node with the minimum distance.
• Update distances of neighboring nodes if a shorter path is found.

3️⃣ Repeat Until All Nodes Processed:

• Continue picking the next closest node and updating paths.

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🔹 Python Implementation

import heapq

def dijkstra(graph, start):
    # Number of vertices
    V = len(graph)
    
    # Distance array, initialized to "infinity"
    distances = {node: float('inf') for node in graph}
    distances[start] = 0  # Distance to source is 0
    
    # Priority queue to get the next minimum distance node
    pq = [(0, start)]  # (distance, node)
    
    while pq:
        current_distance, current_node = heapq.heappop(pq)  # Get the closest node
        
        # If found a shorter path before, skip processing
        if current_distance > distances[current_node]:
            continue
        
        # Explore neighbors
        for neighbor, weight in graph[current_node].items():
            distance = current_distance + weight  # Calculate new distance
            
            if distance < distances[neighbor]:  # Found a shorter path
                distances[neighbor] = distance
                heapq.heappush(pq, (distance, neighbor))  # Push updated distance
    
    return distances

# Example Graph (Adjacency List)
graph = {
    'A': {'B': 1, 'C': 4},
    'B': {'A': 1, 'C': 2, 'D': 5},
    'C': {'A': 4, 'B': 2, 'D': 1},
    'D': {'B': 5, 'C': 1}
}

source = 'A'
shortest_paths = dijkstra(graph, source)

# Print shortest distances from source
print("Shortest distances from source:", source)
for node, distance in shortest_paths.items():
    print(f"{node}: {distance}")

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🔹 Example

Graph Representation:

       A
      / \
     1   4
    /     \
   B --2-- C
    \     /
     5   1
      \ /
       D

Shortest Paths from 'A':

A → A = 0
A → B = 1
A → C = 3  (via B)
A → D = 4  (via C)

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🔹 Applications of Dijkstra’s Algorithm

✅ Navigation & GPS Systems (Google Maps, Waze)
✅ Network Routing (OSPF routing in computer networks)
✅ AI & Game Development (Pathfinding in games using A*)
✅ Robotics & Automation (Finding shortest paths in warehouse robots)

Warshall’s Algorithm

 

🔹 Warshall’s Algorithm (Transitive Closure) 🚀

Warshall’s Algorithm is used to compute the transitive closure of a directed graph. It determines whether a path exists between every pair of nodes.

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🔹 How It Works

Given a graph represented as an adjacency matrix, Warshall’s algorithm updates the matrix to indicate whether a path exists between each pair of vertices.

✅ Input: A boolean adjacency matrix (1 if there is an edge, 0 if there is none).
✅ Output: The transitive closure matrix, where reach[i][j] = 1 if there exists a path from i to j.
✅ Time Complexity: O(V³) (since it uses three nested loops).

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🔹 Algorithm Steps

1️⃣ Initialize the adjacency matrix reach[][] (1 if an edge exists, else 0).
2️⃣ Iterate over all intermediate nodes (k):

• For every pair of vertices (i, j), check if going through k provides a shorter path.
• If reach[i][k] = 1 and reach[k][j] = 1, then set reach[i][j] = 1.
  3️⃣ Final matrix shows if a path exists between any two vertices.

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🔹 Python Implementation

def warshall_algorithm(graph):
    V = len(graph)  # Number of vertices
    reach = [row[:] for row in graph]  # Copy of the input graph

    for k in range(V):  # Intermediate node
        for i in range(V):  # Start node
            for j in range(V):  # End node
                reach[i][j] = reach[i][j] or (reach[i][k] and reach[k][j])

    return reach

# Example graph (Adjacency Matrix)
graph = [
    [1, 1, 0, 1],
    [0, 1, 1, 0],
    [0, 0, 1, 1],
    [0, 0, 0, 1]
]

result = warshall_algorithm(graph)

# Print the transitive closure matrix
for row in result:
    print(row)

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🔹 Example

Input Graph (Adjacency Matrix)

   0  1  2  3
0 [1, 1, 0, 1]
1 [0, 1, 1, 0]
2 [0, 0, 1, 1]
3 [0, 0, 0, 1]

Transitive Closure Output

[1, 1, 1, 1]
[0, 1, 1, 1]
[0, 0, 1, 1]
[0, 0, 0, 1]

This means that from node 0, we can reach all other nodes.

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🔹 Applications of Warshall’s Algorithm

✅ Database Systems: Finding indirect relationships (e.g., foreign key dependencies).
✅ Computer Networks: Identifying reachable nodes in a network.
✅ Social Networks: Finding if a person can be connected via mutual friends.
✅ Compiler Optimization: Analyzing dependencies between code statements.

Shortest Path Algorithms in Graphs

 

🔹 Shortest Path Algorithms in Graphs 🚀

Finding the shortest path between two nodes in a graph is crucial for many real-world applications, such as navigation systems, network routing, and AI pathfinding. Here are the most commonly used algorithms to solve this problem.

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1️⃣ Dijkstra’s Algorithm (Single-Source Shortest Path)

✅ Use Case: Best for graphs with non-negative weights.
✅ Time Complexity: O((V + E) log V) using a priority queue (Heap).
✅ How It Works:

• Starts from a source node and finds the shortest distance to all other nodes.
• Uses a priority queue (min-heap) to always expand the closest node first.
• Fails with negative weights because it assumes once a node’s shortest path is found, it won’t change.

🔹 Best For: Road networks, GPS systems, and real-time traffic routing.

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2️⃣ Bellman-Ford Algorithm (Handles Negative Weights)

✅ Use Case: Works for graphs with negative weights, but no negative cycles.
✅ Time Complexity: O(VE)
✅ How It Works:

• Iterates (V - 1) times, relaxing each edge to update distances.
• Detects negative-weight cycles (if further relaxation is possible after V-1 iterations).

🔹 Best For: Financial systems (arbitrage detection), network routing.

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3️⃣ Floyd-Warshall Algorithm (All-Pairs Shortest Path)

✅ Use Case: Finds the shortest path between all pairs of nodes.
✅ Time Complexity: O(V³)
✅ How It Works:

• Uses dynamic programming to compare paths via intermediate nodes.
• Efficient for small, dense graphs but slow for large graphs.

🔹 Best For: Network topology analysis, finding shortest paths in small graphs.

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4️⃣ A Algorithm (Heuristic-Based Pathfinding)*

✅ Use Case: Optimized for grid-based pathfinding (e.g., games, robotics).
✅ Time Complexity: O(E) (depends on heuristic).
✅ How It Works:

• Uses Dijkstra’s idea but adds a heuristic function (h) that estimates the remaining cost to the goal.
• Expands nodes that seem closest to the destination first.

🔹 Best For: AI, robotics, game development (finding paths in maps).

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🔹 Comparison Table

Algorithm	Handles Negative Weights?	Best For	Time Complexity
Dijkstra’s	❌ No	Non-negative weighted graphs	O((V + E) log V)
Bellman-Ford	✅ Yes	Detecting negative cycles	O(VE)
Floyd-Warshall	✅ Yes	Small all-pairs shortest paths	O(V³)
A*	❌ No	Grid-based AI pathfinding	O(E) (heuristic-based)

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🔹 Real-World Applications

✅ Navigation Apps (Google Maps, Waze): Dijkstra’s or A* Algorithm
✅ Internet Routing Protocols (OSPF, BGP): Bellman-Ford Algorithm
✅ AI & Robotics: A* Algorithm for movement optimization
✅ Logistics & Supply Chain: Shortest route planning in warehouse automation

Transitive Closure in Graph Theory

 

Transitive Closure in Graph Theory 🚀

Transitive Closure is a fundamental concept in graph theory used to determine the reachability of vertices in a directed graph. It tells us whether there is a path between any two vertices, either directly or through intermediate nodes.

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🔹 What is Transitive Closure?

Given a directed graph G(V, E), its transitive closure is another graph G’(V, E') where:
✅ There is an edge (u, v) in E' if there exists a path from u to v in the original graph G.

Simply put, if you can get from node A to node B by following a sequence of edges, then in the transitive closure, we directly add an edge from A to B.

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🔹 How to Compute Transitive Closure?

There are multiple ways to compute the transitive closure of a graph, with two common algorithms:

1️⃣ Floyd-Warshall Algorithm (O(V³) Time Complexity)

• A dynamic programming approach that updates reachability in a matrix form.
• Uses a boolean adjacency matrix, where reach[i][j] = 1 if a path exists from i to j.
• Iterates over all possible intermediate vertices to update reachability.

2️⃣ Warshall’s Algorithm (Modified Floyd-Warshall)

• Specifically optimized for Boolean matrices.
• Uses the logic: $$reach[i][j] = reach[i][j] \text{ OR } (reach[i][k] \text{ AND } reach[k][j])$$ where k is an intermediate node.

3️⃣ DFS (O(V + E) for Each Vertex)

• A depth-first search (DFS) is performed for each vertex to mark all reachable nodes.
• Efficient for sparse graphs with fewer edges.

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🔹 Example of Transitive Closure

Given Graph:

A → B  
B → C  
C → D  

Transitive Closure:

A → B, A → C, A → D  
B → C, B → D  
C → D  

Every vertex is now directly connected to every other reachable vertex.

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🔹 Applications of Transitive Closure

✅ Database Query Optimization: Finding indirect relationships in relational databases.
✅ Social Networks: Checking if one person can be indirectly connected to another.
✅ Compiler Design: Determining dependencies in program flow analysis.
✅ Routing and Navigation: Finding all reachable locations from a given node.