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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.
1var equationsPossible = function(equations) {
2 let parent = new Array(26).fill(0).map((_, index) =>
The JavaScript implementation uses a similar approach with an array initialized with character indices to act as parents. We apply the union-find operations to resolve connectivity driven by '==', verify discrepancies noted in inequalities, and return falsity if conflicts are found.
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.
#include <string>
#include <queue>
#include <cstring>
using namespace std;
class Solution {
public:
bool bfsCheck(vector<vector<int>>& graph, vector<bool>& visited, int start, int target) {
queue<int> q;
q.push(start);
visited[start] = true;
while (!q.empty()) {
int u = q.front(); q.pop();
for (int v = 0; v < 26; ++v) {
if (graph[u][v] && !visited[v]) {
if (v == target) return true;
visited[v] = true;
q.push(v);
}
}
}
return false;
}
bool equationsPossible(vector<string>& equations) {
vector<vector<int>> graph(26, vector<int>(26, 0));
vector<bool> visited(26, false);
for (const auto& eq : equations) {
if (eq[1] == '=') {
int u = eq[0] - 'a';
int v = eq[3] - 'a';
graph[u][v] = graph[v][u] = 1;
}
}
for (const auto& eq : equations) {
if (eq[1] == '!') {
int u = eq[0] - 'a';
int v = eq[3] - 'a';
if (bfsCheck(graph, visited, u, v)) {
return false;
}
fill(visited.begin(), visited.end(), false);
}
}
return true;
}
};The BFS setup is similar to the C implementation, where each inequality check re-initializes visitation states. The solution systematically manages graph composition and per-equation checks to navigate potential conflicts using interlinked components.