🌐 Major Graph Algorithms Every Programmer Should Know 🚀

Graphs are everywhere — from social networks and Google Maps to recommendation engines and network routing. If you know how to work with them, you can solve some of the most complex real-world problems!

In this blog, we’ll explore major graph algorithms 📊, their logic, examples, Python code, and best use cases — plus a problem-to-algorithm cheat sheet at the end! 💡

1️⃣ Breadth-First Search (BFS) 🔍

📖 Concept:
 BFS explores a graph level by level. Perfect for finding the shortest path in an unweighted graph.

🛠 Example:
 Find the shortest distance between two people in a social network.

💻 Python Code:

from collections import deque

def bfs(graph, start):
    visited = set()
    queue = deque([start])
    while queue:
        node = queue.popleft()
        if node not in visited:
            print(node, end=" ")
            visited.add(node)
            queue.extend(graph[node] - visited)

graph = {
    'A': {'B', 'C'},
    'B': {'A', 'D', 'E'},
    'C': {'A', 'F'},
    'D': {'B'},
    'E': {'B', 'F'},
    'F': {'C', 'E'}
}

bfs(graph, 'A')

✅ Best Use Cases:

• Shortest path in unweighted graphs
• Social network friend recommendations
• Crawling websites

2️⃣ Depth-First Search (DFS) 🕵️‍♂️

📖 Concept:
 DFS dives deep into the graph before backtracking — like exploring a maze fully before trying another path.

🛠 Example:
 Check if there is a route between two cities in a road map.

💻 Python Code:

def dfs(graph, start, visited=None):
    if visited is None:
        visited = set()
    visited.add(start)
    print(start, end=" ")
    for neighbor in graph[start] - visited:
        dfs(graph, neighbor, visited)


graph = {
    'A': {'B', 'C'},
    'B': {'A', 'D', 'E'},
    'C': {'A', 'F'},
    'D': {'B'},
    'E': {'B', 'F'},
    'F': {'C', 'E'}
}

dfs(graph, 'A')

✅ Best Use Cases:

• Detecting cycles in graphs
• Topological sorting (with a modified version)
• Solving puzzles like mazes or Sudoku

3️⃣ Dijkstra’s Algorithm 🛣

📖 Concept:
 Finds the shortest path in a graph with non-negative weights.

🛠 Example:
 Finding the fastest driving route between cities considering road distances.

💻 Python Code:

import heapq

def dijkstra(graph, start):
    distances = {node: float('inf') for node in graph}
    distances[start] = 0
    pq = [(0, start)]

    while pq:
        current_distance, current_node = heapq.heappop(pq)
        if current_distance > distances[current_node]:
            continue
        for neighbor, weight in graph[current_node].items():
            distance = current_distance + weight
            if distance < distances[neighbor]:
                distances[neighbor] = distance
                heapq.heappush(pq, (distance, neighbor))
    
    return distances

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}
}
print(dijkstra(graph, 'A'))

✅ Best Use Cases:

• GPS navigation systems
• Network routing algorithms
• Game AI pathfinding

4️⃣ Bellman-Ford Algorithm ⚡

📖 Concept:
 Finds shortest paths even with negative edge weights (unlike Dijkstra).

🛠 Example:
 Calculating currency conversion rates when some rates might cause a loss.

💻 Python Code:

def bellman_ford(graph, vertices, start):
    distance = {v: float('inf') for v in vertices}
    distance[start] = 0

    for _ in range(len(vertices) - 1):
        for u, v, w in graph:
            if distance[u] + w < distance[v]:
                distance[v] = distance[u] + w
    return distance

edges = [
    ('A', 'B', 4), ('A', 'C', 5),
    ('B', 'C', -3), ('C', 'D', 4)
]

vertices = {'A', 'B', 'C', 'D'}
print(bellman_ford(edges, vertices, 'A'))

✅ Best Use Cases:

• Financial arbitrage detection
• Negative cost network routing

5️⃣ Floyd-Warshall Algorithm 🌍

📖 Concept:
 Finds shortest paths between all pairs of vertices.

🛠 Example:
 Finding the shortest travel time between all cities in a country.

💻 Python Code:

def floyd_warshall(graph):
    dist = dict(graph)
    for k in graph:
        for i in graph:
            for j in graph:
                dist[i][j] = min(dist[i][j], dist[i][k] + dist[k][j])
    return dist

graph = {
    'A': {'A': 0, 'B': 3, 'C': float('inf')},
    'B': {'A': 2, 'B': 0, 'C': 1},
    'C': {'A': float('inf'), 'B': float('inf'), 'C': 0}
}

print(floyd_warshall(graph))

✅ Best Use Cases:

• Precomputing shortest distances in dense graphs
• Flight connection planning

6️⃣ Kruskal’s Algorithm 🌳

📖 Concept:
 Builds a Minimum Spanning Tree (MST) by adding the smallest edges without forming cycles.

🛠 Example:
 Connecting cities with the least road-building cost.

💻 Python Code:

def kruskal(vertices, edges):
    parent = {v: v for v in vertices}

def find(v):
        while parent[v] != v:
            v = parent[v]
        return v
    mst = []
    edges.sort(key=lambda x: x[2])
    for u, v, w in edges:
        if find(u) != find(v):
            mst.append((u, v, w))
            parent[find(u)] = find(v)
    return mst

vertices = ['A', 'B', 'C', 'D']

edges = [
    ('A', 'B', 1),
    ('B', 'C', 4),
    ('A', 'C', 3),
    ('C', 'D', 2)
]

print(kruskal(vertices, edges))

✅ Best Use Cases:

• Designing efficient road networks
• Network cabling with minimum cost

7️⃣ Prim’s Algorithm 🌲

📖 Concept:
 Another MST algorithm — grows the tree from a starting vertex, adding the smallest edge each time.

🛠 Example:
 Laying out electrical wiring in the cheapest way possible.

💻 Python Code:

import heapq

def prim(graph, start):
    visited = set([start])
    edges = [
        (cost, start, to)
        for to, cost in graph[start].items()
    ]
    heapq.heapify(edges)
    mst = []
    while edges:
        cost, frm, to = heapq.heappop(edges)
        if to not in visited:
            visited.add(to)
            mst.append((frm, to, cost))
            for to_next, cost_next in graph[to].items():
                if to_next not in visited:
                    heapq.heappush(edges, (cost_next, to, to_next))
    return mst

graph = {
    'A': {'B': 1, 'C': 3},
    'B': {'A': 1, 'C': 4, 'D': 2},
    'C': {'A': 3, 'B': 4, 'D': 5},
    'D': {'B': 2, 'C': 5}
}

print(prim(graph, 'A'))

✅ Best Use Cases:

• Network design
• Electrical grid layout

🗂 Problem → Algorithm Cheat Sheet 📌

🎯 Final Thoughts

Mastering these algorithms will make you a graph wizard 🧙‍♂️ capable of tackling anything from network optimization to social media analytics.

Next time you face a graph problem, check the cheat sheet — it’s your quick key to choosing the right tool 🔑.