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The DFS approach involves exploring as far as possible along each branch before backing up. Start from the source vertex and explore each adjacent vertex, marking visited vertices. If you reach the destination during your exploration, return true. If no path to destination is found by the time all possible vertices are visited, return false.
Time Complexity: O(V + E), where V is the number of vertices and E is the number of edges, due to the need to traverse the entire graph in the worst case.
Space Complexity: O(V + E) for storing the adjacency list and the visited array.
1#include <stdbool.h>
2#include <stdlib.h>
3
4void dfs(int node, int **adjList, int *adjListSize, bool *visited) {
5 visited[node] = true;
6 for (int i = 0; i < adjListSize[node]; i++) {
7 int neighbor = adjList[node][i];
8 if (!visited[neighbor]) {
9 dfs(neighbor, adjList, adjListSize, visited);
10 }
11 }
12}
13
14bool validPath(int n, int **edges, int edgesSize, int *edgesColSize, int source, int destination) {
15 if (source == destination) return true;
16
17 // Create adjacency list
18 int *adjListSize = calloc(n, sizeof(int));
19 int **adjList = malloc(n * sizeof(int *));
20
21 for (int i = 0; i < n; i++) {
22 adjList[i] = malloc(0);
23 }
24
25 for (int i = 0; i < edgesSize; i++) {
26 int u = edges[i][0];
27 int v = edges[i][1];
28
29 adjListSize[u]++;
30 adjListSize[v]++;
31
32 adjList[u] = realloc(adjList[u], adjListSize[u] * sizeof(int));
33 adjList[v] = realloc(adjList[v], adjListSize[v] * sizeof(int));
34
35 adjList[u][adjListSize[u] - 1] = v;
36 adjList[v][adjListSize[v] - 1] = u;
37 }
38
39 bool *visited = calloc(n, sizeof(bool));
40
41 dfs(source, adjList, adjListSize, visited);
42
43 bool result = visited[destination];
44
45 // Free memory
46 for (int i = 0; i < n; i++) {
47 free(adjList[i]);
48 }
49 free(adjList);
50 free(adjListSize);
51 free(visited);
52
53 return result;
54}This C implementation of the DFS approach creates an adjacency list from the given edge list. It uses a recursive depth-first search to explore the graph, marking nodes as visited. The search begins at the source and continues until it either finds the destination or runs out of vertices to explore. If the destination is found, the function returns true; otherwise, it returns false. The adjacency list is implemented with dynamic memory allocation to store neighboring vertices efficiently.
The Union-Find approach is effective for problems involving connectivity checks among different components. This data structure helps efficiently merge sets of connected components and determine whether two vertices are in the same connected component by checking if they share the same root. Initialize each vertex as its own component, then loop through the edges to perform union operations, connecting the components. Finally, check if the source and destination vertices belong to the same component.
Time Complexity: O(Eα(V)), where α is the Inverse Ackermann function which grows very slowly. Equivalent to O(E) practically.
Space Complexity: O(V) for storing parent and rank arrays.
private int[] parent;
private int[] rank;
public UnionFind(int size) {
parent = new int[size];
rank = new int[size];
for (int i = 0; i < size; i++) {
parent[i] = i;
rank[i] = 0;
}
}
public int Find(int x) {
if (parent[x] != x) {
parent[x] = Find(parent[x]);
}
return parent[x];
}
public void Union(int x, int y) {
int rootX = Find(x);
int rootY = Find(y);
if (rootX != rootY) {
if (rank[rootX] > rank[rootY]) {
parent[rootY] = rootX;
} else if (rank[rootX] < rank[rootY]) {
parent[rootX] = rootY;
} else {
parent[rootY] = rootX;
rank[rootX]++;
}
}
}
}
public class Solution {
public bool ValidPath(int n, int[][] edges, int source, int destination) {
UnionFind uf = new UnionFind(n);
foreach (var edge in edges) {
uf.Union(edge[0], edge[1]);
}
return uf.Find(source) == uf.Find(destination);
}
}This C# program employs the Union-Find pattern using classes to ensure disjoint set fusion and path checks efficiently. Path compression boosts search speed, while union by rank minimizes height. The algorithm ensures source and destination belong to the same root component, forming the basis for determining path existence.