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780. Reaching Points

Description

Given four integers sx, sy, tx, and ty, return true if it is possible to convert the point (sx, sy) to the point (tx, ty) through some operations, or false otherwise.

The allowed operation on some point (x, y) is to convert it to either (x, x + y) or (x + y, y).

 

Example 1:

Input: sx = 1, sy = 1, tx = 3, ty = 5
Output: true
Explanation:
One series of moves that transforms the starting point to the target is:
(1, 1) -> (1, 2)
(1, 2) -> (3, 2)
(3, 2) -> (3, 5)

Example 2:

Input: sx = 1, sy = 1, tx = 2, ty = 2
Output: false

Example 3:

Input: sx = 1, sy = 1, tx = 1, ty = 1
Output: true

 

Constraints:

  • 1 <= sx, sy, tx, ty <= 109

Solutions

Solution 1

Thinking

From \((sx,sy)\) we may add one coordinate onto the other. The target can be huge, so searching forward does not terminate well. The inverse subtracts the smaller from the larger—i.e. modulo.

While both coordinates exceed the start and differ, replace the larger by \(a\bmod b\). Once one coordinate matches, the other must drop to the start by a multiple of that value.

Equal to the start succeeds; otherwise fail. Modulo folds many subtractions into one.

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class Solution:
    def reachingPoints(self, sx: int, sy: int, tx: int, ty: int) -> bool:
        while tx > sx and ty > sy and tx != ty:
            if tx > ty:
                tx %= ty
            else:
                ty %= tx
        if tx == sx and ty == sy:
            return True
        if tx == sx:
            return ty > sy and (ty - sy) % tx == 0
        if ty == sy:
            return tx > sx and (tx - sx) % ty == 0
        return False

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class Solution {
    public boolean reachingPoints(int sx, int sy, int tx, int ty) {
        while (tx > sx && ty > sy && tx != ty) {
            if (tx > ty) {
                tx %= ty;
            } else {
                ty %= tx;
            }
        }
        if (tx == sx && ty == sy) {
            return true;
        }
        if (tx == sx) {
            return ty > sy && (ty - sy) % tx == 0;
        }
        if (ty == sy) {
            return tx > sx && (tx - sx) % ty == 0;
        }
        return false;
    }
}

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class Solution {
public:
    bool reachingPoints(int sx, int sy, int tx, int ty) {
        while (tx > sx && ty > sy && tx != ty) {
            if (tx > ty)
                tx %= ty;
            else
                ty %= tx;
        }
        if (tx == sx && ty == sy) return true;
        if (tx == sx) return ty > sy && (ty - sy) % tx == 0;
        if (ty == sy) return tx > sx && (tx - sx) % ty == 0;
        return false;
    }
};

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func reachingPoints(sx int, sy int, tx int, ty int) bool {
    for tx > sx && ty > sy && tx != ty {
        if tx > ty {
            tx %= ty
        } else {
            ty %= tx
        }
    }
    if tx == sx && ty == sy {
        return true
    }
    if tx == sx {
        return ty > sy && (ty-sy)%tx == 0
    }
    if ty == sy {
        return tx > sx && (tx-sx)%ty == 0
    }
    return false
}

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