Shortest Path (Dijkstra) - Problem

Given a weighted graph represented as an adjacency list and a source vertex, implement Dijkstra's algorithm to find the shortest path from the source to all other vertices.

The graph is represented as a list of edges where each edge is [from, to, weight]. Return an array where result[i] is the shortest distance from the source to vertex i. If a vertex is unreachable, use -1.

Note: All edge weights are non-negative integers. Vertices are numbered from 0 to n-1.

Input & Output

Example 1 — Basic Graph

$ Input: n = 4, edges = [[0,1,4],[0,2,2],[1,3,3],[2,3,1]], source = 0

› Output: [0,4,2,3]

💡 Note: From vertex 0: distance to 0 is 0, to 1 is 4 (direct), to 2 is 2 (direct), to 3 is 3 (via 2→3 path costs 2+1=3, better than 1→3 path costs 4+3=7)

Example 2 — Disconnected Vertex

$ Input: n = 3, edges = [[0,1,5]], source = 0

› Output: [0,5,-1]

💡 Note: From vertex 0: distance to 0 is 0, to 1 is 5 (direct edge), to 2 is -1 (unreachable since no path exists)

Example 3 — Single Vertex

$ Input: n = 1, edges = [], source = 0

› Output: [0]

💡 Note: Only one vertex exists, distance from vertex 0 to itself is 0

Constraints

1 ≤ n ≤ 100
0 ≤ edges.length ≤ n × (n-1)
0 ≤ source < n
0 ≤ weight ≤ 100

Visualization

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Asked in

G Google 45 a Amazon 38 f Meta 32 M Microsoft 28

The key insight is using a priority queue to always process the vertex with minimum distance first, guaranteeing optimal subpaths. Best approach is Dijkstra's algorithm with priority queue. Time: O((V + E) log V), Space: O(V).

Common Approaches

✓ Brute Force with All Paths

⏱️ Time: O(V! × E) Space: O(V)

Generate all possible paths from source to each vertex and compare their total weights to find the minimum distance.

Classic Dijkstra with Priority Queue

⏱️ Time: O((V + E) log V) Space: O(V)

Maintain a priority queue of vertices ordered by distance. Always process the vertex with minimum distance first, guaranteeing optimal substructure.

Brute Force with All Paths — Algorithm Steps

Generate all possible paths from source to each vertex using DFS
Calculate total weight for each path
Select minimum weight path for each destination

Visualization

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Step-by-Step Walkthrough

Start DFS

Begin path exploration from source vertex

Try All Paths

Recursively explore every possible route to target

Compare Costs

Track minimum cost among all valid paths

Code -

solution.c — C

#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <limits.h>

#define MAX_VERTICES 100
#define MAX_EDGES 1000

int graph[MAX_VERTICES][MAX_VERTICES];
int n_global;

int dfs(int node, int target, int currentDist, int* visited) {
    if (node == target) {
        return currentDist;
    }
    
    int minDist = INT_MAX;
    for (int i = 0; i < n_global; i++) {
        if (graph[node][i] != -1 && !visited[i]) {
            visited[i] = 1;
            int dist = dfs(i, target, currentDist + graph[node][i], visited);
            if (dist < minDist) {
                minDist = dist;
            }
            visited[i] = 0;
        }
    }
    
    return minDist;
}

int* solution(int n, int edges[][3], int edgeCount, int source) {
    n_global = n;
    
    // Initialize adjacency matrix
    for (int i = 0; i < n; i++) {
        for (int j = 0; j < n; j++) {
            graph[i][j] = -1;
        }
    }
    
    // Build graph
    for (int i = 0; i < edgeCount; i++) {
        graph[edges[i][0]][edges[i][1]] = edges[i][2];
    }
    
    int* result = (int*)malloc(n * sizeof(int));
    for (int i = 0; i < n; i++) {
        result[i] = -1;
    }
    result[source] = 0;
    
    // Find shortest path to each vertex
    for (int target = 0; target < n; target++) {
        if (target != source) {
            int visited[MAX_VERTICES] = {0};
            visited[source] = 1;
            int dist = dfs(source, target, 0, visited);
            result[target] = (dist == INT_MAX) ? -1 : dist;
        }
    }
    
    return result;
}

int main() {
    int n;
    scanf("%d", &n);
    
    // Simple parsing - read edges
    int edges[MAX_EDGES][3];
    int edgeCount = 0;
    
    char line[1000];
    fgets(line, sizeof(line), stdin); // consume newline
    fgets(line, sizeof(line), stdin);
    
    // Parse edges (simplified)
    char* token = strtok(line, "[],");
    while (token != NULL && edgeCount < MAX_EDGES) {
        if (strlen(token) > 0 && token[0] >= '0' && token[0] <= '9') {
            edges[edgeCount / 3][edgeCount % 3] = atoi(token);
            if (edgeCount % 3 == 2) edgeCount = (edgeCount / 3) + 1;
            else edgeCount++;
        }
        token = strtok(NULL, "[],");
    }
    edgeCount = edgeCount / 3 + (edgeCount % 3 > 0 ? 1 : 0);
    
    int source;
    scanf("%d", &source);
    
    int* result = solution(n, edges, edgeCount, source);
    
    printf("[");
    for (int i = 0; i < n; i++) {
        printf("%d", result[i]);
        if (i < n - 1) printf(",");
    }
    printf("]\n");
    
    free(result);
    return 0;
}

Time & Space Complexity

Time Complexity

⏱️

O(V! × E)

In worst case, we explore V! possible paths, checking E edges for each path

✓ Linear Growth

Space Complexity

O(V)

Recursion stack depth up to V vertices for path exploration

✓ Linear Space

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Shortest Path (Dijkstra) - Problem

Input & Output

Constraints

Visualization

Related Problems

Common Approaches

Brute Force with All Paths — Algorithm Steps

Visualization

Code -

Time & Space Complexity

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