Satisfiability of Equality Equations

Problem Description

You are given an array of strings equations that represent relationships between variables where each string equations[i] is of length 4 and takes one of two different forms: x_i==y_i or x_i!=y_i. Here, x_i and y_i are lowercase letters (not necessarily different) that represent one-letter variable names. Return true if it is possible to assign integers to variable names so as to satisfy all the given equations, or false otherwise.

Key Insights

• The problem can be viewed as a graph where each variable is a node.
• The == equations represent connected components, while != equations impose restrictions on those components.
• If two variables are in the same connected component and there is a != equation between them, the solution is impossible.
• We can use the Union-Find (Disjoint Set Union) data structure to efficiently manage and check the connected components.

Space and Time Complexity

Time Complexity: O(N * α(N)), where N is the number of equations and α is the Inverse Ackermann function, which grows very slowly.
Space Complexity: O(1) for the union-find structure if we ignore the input size.

Solution

To solve the problem, we will use the Union-Find algorithm. The approach involves the following steps:

1. Initialize a parent array for each variable (26 lowercase letters).
2. Process all == equations to unite the variables into connected components.
3. After processing == equations, check all != equations to ensure that no two variables in the same component are marked as unequal.
4. If any such contradiction is found, return false. Otherwise, return true.

Code Solutions