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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.
1#include <stdbool.h>
2#include <string.h>
3
4int find(int parent[], int x) {
5 if
The solution uses two functions find and unionSet to implement the Union-Find operations. Each variable is initially its own parent. For each '==' equation, the unionSet function connects the two variables. For each '!=' equation, the find function checks if both variables are connected, returning false if they are, because they shouldn't be connected. The program finally returns true if no such pairs exist.
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.