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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 <vector>
2#include <string>
3using namespace std;
4
5class Solution {
6public:
7 int find(vector<int>& parent, int x) {
8 if (parent[x] != x) {
9 parent[x] = find(parent, parent[x]);
10 }
11 return parent[x];
12 }
13
14 void unionSet(vector<int>& parent, int x, int y) {
int rootX = find(parent, x);
int rootY = find(parent, y);
if (rootX != rootY) {
parent[rootY] = rootX;
}
}
bool equationsPossible(vector<string>& equations) {
vector<int> parent(26);
for (int i = 0; i < 26; i++) {
parent[i] = i;
}
for (const string& eq : equations) {
if (eq[1] == '=') {
int index1 = eq[0] - 'a';
int index2 = eq[3] - 'a';
unionSet(parent, index1, index2);
}
}
for (const string& eq : 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;
}
};The C++ solution follows the same logic as the C solution, but uses the vector data structure from the standard library for better type safety and easier management. The use of find and unionSet remains the same, managing connectivity for '==' equations and checking for invalid connectivity in '!=' equations.
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.
In Java, the graph representation is achieved using array lists for adjacency list management. BFS searches validate inequality constraints within connected variable clusters. Traversal encompasses adjacency navigation where each set of resets ensures new traversal roots conform to disjoint segmentations.