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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.
The Python graph implementation uses lists of lists for creating adjustable connectivity maps, engaging queues to seek connections and enforce separations via BFS processing. Each inequality discovered in the same component steps up incongruence resolutions, returning false when conflicts emerge.