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This approach utilizes the Union-Find (or Disjoint Set Union, DSU) data structure to detect cycles. The key operations in Union-Find are 'find', which determines the representative of a set, and 'union', which merges two sets. When processing each edge, we attempt to unite the vertices. If both vertices are already in the same set, a cycle is detected, and this edge is the redundant one.
Time Complexity: O(n) where n is the number of edges because we process each edge once.
Space Complexity: O(n) due to the storage of the parent and rank arrays.
This Java solution implements the Union-Find structure with methods to find and union elements. The findRedundantConnection method iterates through edges, returning the first non-unionable edge as the result.
The DFS approach involves simulating the construction of the graph and utilizing DFS to detect cycles whenever a new edge is being added. If a cycle is detected upon adjacent traversal, that edge is the redundant one.
Time Complexity: O(n^2), potentially examining each pair of nodes connected by added edges in the worst case.
Space Complexity: O(n), dictated by recursive stack depth in the DFS and the graph representation.
1var findRedundantConnection = function(edges) {
2 let graph = {};
3 const dfs = function(source, target, visited) {
4 if (source === target) return true;
5 visited.add(source);
6 for (let neighbor of graph[source] || []) {
7 if (!visited.has(neighbor)) {
8 if (dfs(neighbor, target, visited)) return true;
9 }
10 }
11 return false;
12 };
13 for (let [u, v] of edges) {
14 if (graph[u] && graph[v] && dfs(u, v, new Set())) {
15 return [u, v];
16 }
17 if (!graph[u]) graph[u] = new Set();
18 if (!graph[v]) graph[v] = new Set();
19 graph[u].add(v);
20 graph[v].add(u);
21 }
22 return [];
23};The JavaScript solution uses an adjacency list structure and conducts DFS on a subset of vertices. Pre-checks for edge redundancy locate its provenance if cycles signify extraneous articulation.