365. Water and Jug Problem

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You are given two jugs with capacities x and y litres. There is an infinite amount of water supply available. You need to determine whether it is possible to measure exactly z litres using these two jugs.

If z liters of water is measurable, you must have z liters of water contained within one or both buckets by the end.

Operations allowed:

• Fill any of the jugs completely with water.

• Empty any of the jugs.

• Pour water from one jug into another till the other jug is completely full or the first jug itself is empty.

Example 1: (From the famous "Die Hard" example)

Input: x = 3, y = 5, z = 4
Output: True

Example 2:

Input: x = 2, y = 6, z = 5
Output: False

Solution

這題會需要知道

要滿足條件的(z,x,y) 要有z 為gcd(x,y) 的倍數的關係。

數學證明不是重點。

比較像是考數學定義，而非程式。

by the way GCD 的complexity is O(log(a+b))

    public boolean canMeasureWater(int x, int y, int z) {
        if(z == 0) return true;
        if(z > x + y) return false;
        int gcd = GCD(x,y);
        return z%gcd == 0;
    }
    
    public int GCD(int x, int y){
        if(x == 0) return y;
        else if( x > y) return GCD(x%y, y);
        else return GCD(y%x, x);
    }