#990

Medium

LeetCode Problem Workspace

Satisfiability of Equality Equations

Determine if it's possible to assign values to variables such that all equations are satisfied based on equality and inequality relationships.

array string union find graph

Problem Statement

You are given an array of strings equations where each element represents a relationship between two variables. The equations are in the form 'xi==yi' or 'xi!=yi', where 'xi' and 'yi' are lowercase letters representing variables. Each equation defines either an equality or inequality relationship between the variables.

Return true if it is possible to assign values to these variables so that all the equations are satisfied, otherwise return false. You must consider the equality relationships first and ensure that no contradictions arise when verifying the inequalities between the variables.

Examples

Example 1

Input: equations = ["a==b","b!=a"]

Output: false

If we assign say, a = 1 and b = 1, then the first equation is satisfied, but not the second. There is no way to assign the variables to satisfy both equations.

Example 2

Input: equations = ["b==a","a==b"]

Output: true

We could assign a = 1 and b = 1 to satisfy both equations.

Constraints

1 <= equations.length <= 500
equations[i].length == 4
equations[i][0] is a lowercase letter.
equations[i][1] is either '=' or '!'.
equations[i][2] is '='.
equations[i][3] is a lowercase letter.

Solution Approach

Union Find Approach

Use a Union Find data structure to group variables that must be equal. Start by processing all the equality equations ('xi==yi'). For each equality, union the variables xi and yi into the same group. Then, check the inequality equations ('xi!=yi'). If two variables are in the same group but are required to be unequal, return false.

Path Compression Optimization

When performing union and find operations, use path compression to flatten the structure of the tree, ensuring that future find operations are faster. This improves the efficiency of the Union Find operations, especially when there are many variables and equations.

Time Complexity Considerations

The time complexity of the Union Find approach with path compression and union by rank is near constant, O(α(n)), where α is the inverse Ackermann function, and n is the number of variables. This makes it very efficient for large inputs, given the problem's constraints.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity is O(α(n)) per union or find operation due to path compression, where n is the number of distinct variables, and α(n) is very small in practice. Space complexity is O(n) due to the storage of parent pointers and ranks for each variable.

What Interviewers Usually Probe

The candidate understands the Union Find technique and its use in grouping variables based on equality.
The candidate optimizes the Union Find operations using path compression, improving the solution's efficiency.
The candidate successfully handles edge cases, such as conflicts between equality and inequality constraints, demonstrating problem-solving skills.

Common Pitfalls or Variants

Common pitfalls

Failing to consider edge cases where inequality conflicts with existing equality relationships.
Not optimizing the Union Find operations with path compression, leading to slower performance for larger test cases.
Confusing the order of processing equality and inequality equations, which may lead to incorrect results.

Follow-up variants

Use a different data structure for union-find, such as a disjoint set with additional optimizations.
Extend the problem to handle more complex relationships, such as multiple inequalities between the same variables.
Apply the solution to solve similar problems involving graph connectivity or equivalence of nodes.

FAQ

How do Union Find operations help solve the Satisfiability of Equality Equations problem?

Union Find helps by grouping variables that must be equal based on the '==' equations and ensuring that no conflicts arise with '!=' equations, making the solution efficient.

What are the key optimizations used in the solution for this problem?

Key optimizations include path compression during find operations and union by rank to ensure efficient Union Find operations, even for large datasets.

What should I do if there is a conflict between equality and inequality in the equations?

If a conflict occurs where two variables must be both equal and unequal, the answer is false. The problem requires identifying such conflicts using Union Find.

What is the time complexity of the Union Find solution for the Satisfiability of Equality Equations?

The time complexity is nearly constant, O(α(n)), where α is the inverse Ackermann function, making the solution efficient even for large inputs.

How does GhostInterview assist in solving the Satisfiability of Equality Equations problem?

GhostInterview provides insights into Union Find techniques, optimizations, and helps ensure you don't miss any important steps in the algorithm, such as handling edge cases.

terminal

Solution

Solution 1

#### Python3

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class Solution:
    def equationsPossible(self, equations: List[str]) -> bool:
        def find(x):
            if p[x] != x:
                p[x] = find(p[x])
            return p[x]

        p = list(range(26))
        for e in equations:
            a, b = ord(e[0]) - ord('a'), ord(e[-1]) - ord('a')
            if e[1] == '=':
                p[find(a)] = find(b)
        for e in equations:
            a, b = ord(e[0]) - ord('a'), ord(e[-1]) - ord('a')
            if e[1] == '!' and find(a) == find(b):
                return False
        return True

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