Graph Algorithms

Graph Representation & Traversal

• Adjacency Matrix: 2D array — graph[i][j] = 1 if edge exists. O(V²) space. O(1) edge check. Best for dense graphs or when V is small
• Adjacency List: Array of linked lists — each vertex stores list of its neighbors. O(V+E) space. Better for sparse graphs (most real-world graphs)
• BFS (Breadth-First Search): Use queue — visit all neighbors before going deeper. Finds shortest path in unweighted graphs. O(V+E) time. Level-order exploration
• DFS (Depth-First Search): Use stack or recursion — go as deep as possible before backtracking. Detects cycles, topological sort, connected components. O(V+E)
• Dijkstra's Algorithm: Shortest path from source to all vertices in weighted graph (non-negative weights). Uses priority queue (min-heap). O((V+E) log V)
• Bellman-Ford: Shortest path with negative weights. Relax all edges V-1 times. Detects negative cycles on Vth iteration. O(V×E). Slower but more versatile than Dijkstra

Try It Yourself: Bubble Sort

#include <stdio.h>

void bubbleSort(int arr[], int n) {
    for (int i = 0; i < n - 1; i++) {
        for (int j = 0; j < n - i - 1; j++) {
            if (arr[j] > arr[j + 1]) {
                int temp = arr[j];
                arr[j] = arr[j + 1];
                arr[j + 1] = temp;
            }
        }
    }
}

void printArray(int arr[], int n) {
    for (int i = 0; i < n; i++)
        printf("%d ", arr[i]);
    printf("
");
}

int main() {
    int arr[] = {64, 34, 25, 12, 22, 11, 90};
    int n = sizeof(arr) / sizeof(arr[0]);

    printf("Unsorted: ");
    printArray(arr, n);

    bubbleSort(arr, n);

    printf("Sorted:   ");
    printArray(arr, n);
    return 0;
}

Quick Quiz — Algorithms