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This approach uses the Union-Find data structure to manage connectivity between variables. The core idea is to group variables that are known to be equal using '=='. After processing all such equations, we then check the '!=' equations to ensure that connected variables (i.e., variables that are in the same component) are not incorrectly marked as not equal.
Steps:
Time Complexity: O(n * α(n)) where n is the number of equations, and α is the inverse Ackermann function.
Space Complexity: O(1) as we are using a fixed-size array of 26 characters.
1public class Solution {
2 public int Find(int[] parent, int x) {
3 if (parent[x] != x) {
4 parent[x] = Find(parent, parent[x]);
5 }
6 return parent[x];
7 }
8
9 public void Union(int[] parent, int x, int y) {
10 int rootX = Find(parent, x);
11 int rootY = Find(parent, y);
12 if (rootX != rootY) {
13 parent[rootY] = rootX;
14 }
}
public bool EquationsPossible(string[] equations) {
int[] parent = new int[26];
for (int i = 0; i < 26; i++) {
parent[i] = i;
}
foreach (var eq in equations) {
if (eq[1] == '=') {
int index1 = eq[0] - 'a';
int index2 = eq[3] - 'a';
Union(parent, index1, index2);
}
}
foreach (var eq in equations) {
if (eq[1] == '!') {
int index1 = eq[0] - 'a';
int index2 = eq[3] - 'a';
if (Find(parent, index1) == Find(parent, index2)) {
return false;
}
}
}
return true;
}
}This C# solution utilizes the union-find algorithm similarly to previous implementations. The solution handles each equation, leveraging union operations for '==' and raises failure scenarios using find methodology for '!=' equations if any connected variables conflict.
Another method is to visualize the problem as a graph and determine if the '==' relations create valid partitions without violating '!=' conditions. This is managed using a BFS approach.
Steps:
Time Complexity: O(n) where n is the number of equations as each one is processed once during graph building and inequality checks.
Space Complexity: O(1) as it uses a fixed-size 26x26 adjacency matrix.
using System.Collections.Generic;
public class Solution {
private bool bfsCheck(List<int>[] graph, bool[] visited, int start, int target) {
var queue = new Queue<int>();
queue.Enqueue(start);
visited[start] = true;
while (queue.Count > 0) {
int u = queue.Dequeue();
foreach (int v in graph[u]) {
if (!visited[v]) {
if (v == target) return true;
visited[v] = true;
queue.Enqueue(v);
}
}
}
return false;
}
public bool EquationsPossible(string[] equations) {
var graph = new List<int>[26];
for (int i = 0; i < 26; i++) {
graph[i] = new List<int>();
}
bool[] visited = new bool[26];
foreach (var eq in equations) {
if (eq[1] == '=') {
int u = eq[0] - 'a';
int v = eq[3] - 'a';
graph[u].Add(v);
graph[v].Add(u);
}
}
foreach (var eq in equations) {
if (eq[1] == '!') {
int u = eq[0] - 'a';
int v = eq[3] - 'a';
if (bfsCheck(graph, visited, u, v)) {
return false;
}
Array.Fill(visited, false);
}
}
return true;
}
}The C# implementation writes adjacency information into lists per node for dynamic overlap handling. BFS traversals navigate components mapped by equational verifications, thereby conflicting '!=' suggestions are mitigated. Accessing internals by queue persists through boundary resets for exclusive sections.