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This approach uses a Union-Find (Disjoint Set Union, DSU) structure to detect cycles and check for nodes with two parents. The goal is to handle two situations: a node having two parents, and a cycle existing in the graph. We iterate through the edges to identify a node with two parents and mark the offending edge. Then, we use the Union-Find structure to track cycles and find the redundant connection based on the identified edges.
Time Complexity: O(n), where n is the number of edges.
Space Complexity: O(n), for storing the parent and rank arrays.
1#include <stdio.h>
2#include <stdlib.h>
3
4typedef struct UnionFind {
5 int* parent;
6 int* rank;
7} UnionFind;
8
9UnionFind* createUnionFind(int size) {
10 UnionFind* uf = (UnionFind*)malloc(sizeof(UnionFind));
11 uf->parent = (int*)malloc(sizeof(int) * size);
12 uf->rank = (int*)malloc(sizeof(int) * size);
13 for (int i = 0; i < size; i++) {
14 uf->parent[i] = i;
15 uf->rank[i] = 1;
16 }
17 return uf;
18}
19
20int find(UnionFind* uf, int x) {
21 if (uf->parent[x] != x) {
22 uf->parent[x] = find(uf, uf->parent[x]);
23 }
24 return uf->parent[x];
25}
26
27void unionSets(UnionFind* uf, int x, int y) {
28 int rootX = find(uf, x);
29 int rootY = find(uf, y);
30 if (rootX != rootY) {
31 if (uf->rank[rootX] > uf->rank[rootY]) {
32 uf->parent[rootY] = rootX;
33 } else if (uf->rank[rootX] < uf->rank[rootY]) {
34 uf->parent[rootX] = rootY;
35 } else {
36 uf->parent[rootY] = rootX;
37 uf->rank[rootX]++;
38 }
39 }
40}
41
42int* findRedundantDirectedConnection(int** edges, int edgesSize, int* edgesColSize, int* returnSize) {
43 int n = edgesSize;
44 int* parent = (int*)malloc(sizeof(int) * (n + 1));
45 for (int i = 0; i <= n; i++) {
46 parent[i] = i;
47 }
48 int* can1 = NULL;
49 int* can2 = NULL;
50 for (int i = 0; i < n; i++) {
51 int u = edges[i][0], v = edges[i][1];
52 if (parent[v] != v) {
53 can1 = (int*)malloc(sizeof(int) * 2);
54 can2 = (int*)malloc(sizeof(int) * 2);
55 can1[0] = parent[v];
56 can1[1] = v;
57 can2[0] = u;
58 can2[1] = v;
59 edges[i][1] = 0;
60 } else {
61 parent[v] = u;
62 }
63 }
64
65 UnionFind* uf = createUnionFind(n + 1);
66 for (int i = 0; i < n; i++) {
67 if (edges[i][1] == 0) continue;
68 int u = edges[i][0], v = edges[i][1];
69 if (find(uf, u) == find(uf, v)) {
70 if (can1) return can1;
71 int* result = (int*)malloc(sizeof(int) * 2);
72 result[0] = u;
73 result[1] = v;
74 return result;
75 }
76 unionSets(uf, u, v);
77 }
78 return can2;
79}
80The implementation uses a Union-Find data structure to manage connectivity among nodes and track potential redundant connections by checking parent-child relationships and detecting cycles. First, it checks whether any node has two parents. If found, it temporarily removes that edge and continues to apply Union-Find to check for any cycles. If a cycle exists without finding a node with two parents, the wrong edge is part of the cycle. Otherwise, it'll be the edge removed previously.
In this method, we focus on identifying two scenarios: an edge creating a cycle in the graph and a node with two parents. With graph traversal, primarily cross-check with parent pointers and DFS for cycle confirmation, fine-tuning insights to pinpoint a last array occurrence redundant connection.
Time Complexity: O(n^2) with recursive verification.
Space Complexity: O(n), memoizing visited nodes.
1import
Java smoothly navigates through the DFS calling structure, ensuring it captures cyclic constructs effectively. With ordered parent review and composite result tones through graph narrative construction, cycles are masterfully exposed.