What's the most efficient way to test two inte

Question

Given two inclusive integer ranges [x1:x2] and [y1:y2], where x1 ≤ x2 and y1 ≤ y2, what is the most efficient way to test whether there is any overlap of the two ranges?

A simple implementation is as follows:

bool testOverlap(int x1, int x2, int y1, int y2) {
  return (x1 >= y1 && x1 <= y2) ||
         (x2 >= y1 && x2 <= y2) ||
         (y1 >= x1 && y1 <= x2) ||
         (y2 >= x1 && y2 <= x2);
}

But I expect there are more efficient ways to compute this.

What method would be the most efficient in terms of fewest operations.

Answer 1

You have the most efficient representation already - it's the bare minimum that needs to be checked unless you know for sure that x1 < x2 etc, then use the solutions others have provided.

You should probably note that some compilers will actually optimise this for you - by returning as soon as any of those 4 expressions return true. If one returns true, so will the end result - so the other checks can just be skipped.

Answer 2

If you were dealing with, given two ranges [x1:x2] and [y1:y2], natural / anti-natural order ranges at the same time where:

• natural order: x1 <= x2 && y1 <= y2 or
• anti-natural order: x1 >= x2 && y1 >= y2

then you may want to use this to check:

they are overlapped <=> (y2 - x1) * (x2 - y1) >= 0

where only four operations are involved:

• two subtractions
• one multiplication
• one comparison

Answer 3

If someone is looking for a one-liner which calculates the actual overlap:

int overlap = ( x2 > y1 || y2 < x1 ) ? 0 : (y2 >= y1 && x2 <= y1 ? y1 : y2) - ( x2 <= x1 && y2 >= x1 ? x1 : x2) + 1; //max 11 operations

If you want a couple fewer operations, but a couple more variables:

bool b1 = x2 <= y1;
bool b2 = y2 >= x1;
int overlap = ( !b1 || !b2 ) ? 0 : (y2 >= y1 && b1 ? y1 : y2) - ( x2 <= x1 && b2 ? x1 : x2) + 1; // max 9 operations

Answer 4

Here's my version:

int xmin = min(x1,x2)
  , xmax = max(x1,x2)
  , ymin = min(y1,y2)
  , ymax = max(y1,y2);

for (int i = xmin; i < xmax; ++i)
    if (ymin <= i && i <= ymax)
        return true;

return false;

Unless you're running some high-performance range-checker on billions of widely-spaced integers, our versions should perform similarly. My point is, this is micro-optimization.

Answer 5

What does it mean for the ranges to overlap? It means there exists some number C which is in both ranges, i.e.

x1 <= C <= x2

and

y1 <= C <= y2

Now, if we are allowed to assume that the ranges are well-formed (so that x1 <= x2 and y1 <= y2) then it is sufficient to test

x1 <= y2 && y1 <= x2

Answer 6

return x2 >= y1 && x1 <= y2;