Satisfiability of Equality Equations

update Feb 11 2019, 22:25

LeetCode

Given an array equations of strings that represent relationships between variables, each string equations[i] has length 4 and takes one of two different forms: "a==b" or "a!=b". Here, a and b are lowercase letters (not necessarily different) that represent one-letter variable names.

Return true if and only if it is possible to assign integers to variable names so as to satisfy all the given equations.

Example 1:

Input: ["a==b","b!=a"]
Output: false
Explanation: 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: ["b==a","a==b"]
Output: true
Explanation: We could assign a = 1 and b = 1 to satisfy both equations.

Example 3:

Input: ["a==b","b==c","a==c"]
Output: true

Example 4:

Input: ["a==b","b!=c","c==a"]
Output: false

Example 5:

Input: ["c==c","b==d","x!=z"]
Output: true

Note:

1. 1 <= equations.length <= 500
2. equations[i].length == 4
3. equations[i][0] and equations[i][3] are lowercase letters
4. equations[i][1] is either '=' or '!'
5. equations[i][2] is '='

Basic Idea:

基本思路就是把每个相等的部分放到一起，然后判断是否所有的 a!=b 的equation成立，如果成立，需满足a和b不在同一个部分。

这就是典型的并查集的思路，我们先对所有的 a==b 做union，然后查看不等关系中左右值是否在同一个block里，如果在，则说明不成立。

Java Code:

class Solution {
    private int[] idxs;

    private int find(int x) {
        if (idxs[x] == x) return x;
        else {
            idxs[x] = find(idxs[x]);
        }
        return idxs[x];
    }

    private void union(int x, int y) {
        int root_x = find(x), root_y = find(y);
        if (root_x == root_y) return;
        idxs[root_x] = root_y;
    }

    public boolean equationsPossible(String[] equations) {
        // initialize 
        idxs = new int[256];
        for (int i = 0; i < idxs.length; ++i) idxs[i] = i;
        // process "==" first
        for (String equation : equations) {
            if (equation.charAt(1) == '=') {
                union(equation.charAt(0), equation.charAt(3));
            }
        }
        // check all "!=" 
        for (String equation : equations) {
            if (equation.charAt(1) == '!') {
                if (find(equation.charAt(0)) == find(equation.charAt(3))) {
                    return false;
                }
            }
        }
        return true;
    }
}