Is there an easy function from a pair of 32-bit ints to a single 64-bit int that preserves rotational order?

Question

This is a question that came up in the context of sorting points with integer coordinates into clockwise order, but this question is not about how to do that sorting.

This question is about the observation that 2-d vectors have a natural cyclic ordering. Unsigned integers with usual overflow behavior (or signed integers using twos-complement) also have a natural cyclic ordering. Can you easily map from the first ordering to the second?

So, the exact question is whether there is a map from pairs of twos-complement signed 32-bit integers to unsigned (or twos-complement signed) 64-bit integers such that any list of vectors that is in clockwise order maps to integers that are in decreasing (modulo overflow) order?

Some technical cases that people will likely ask about:

• Yes, vectors that are multiples of each other should map to the same thing
• No, I don't care which vector (if any) maps to 0
• No, the images of antipodal vectors don't have to differ by 2^63 (although that is a nice-to-have)

The obvious answer is that since there are only around 0.6*2^64 distinct slopes, the answer is yes, such a map exists, but I'm looking for one that is easily computable. I understand that "easily" is subjective, but I'm really looking for something reasonably efficient and not terrible to implement. So, in particular, no counting every lattice point between the ray and the positive x-axis (unless you know a clever way to do that without enumerating them all).

An important thing to note is that it can be done by mapping to 65-bit integers. Simply project the vector out to where it hits the box bounded by x,y=+/-2^62 and round toward negative infinity. You need 63 bits to represent that integer and two more to encode which side of the box you hit. The implementation needs a little care to make sure you don't overflow, but only has one branch and two divides and is otherwise quite cheap. It doesn't work if you project out to 2^61 because you don't get enough resolution to separate some slopes.

Also, before you suggest "just use atan2", compute atan2(1073741821,2147483643) and atan2(1073741820,2147483641)

EDIT: Expansion on the "atan2" comment:

Given two values x_1 and x_2 that are coprime and just less than 2^31 (I used 2^31-5 and 2^31-7 in my example), we can use the extended Euclidean algorithm to find y_1 and y_2 such that y_1/x_1-y_2/x_2 = 1/(x_1*x_2) ~= 2^-62. Since the derivative of arctan is bounded by 1, the difference of the outputs of atan2 on these values is not going to be bigger than that. So, there are lots of pairs of vectors that won't be distinguishable by atan2 as vanilla IEEE 754 doubles.

If you have 80-bit extended registers and you are sure you can retain residency in those registers throughout the computation (and don't get kicked out by a context switch or just plain running out of extended registers), then you're fine. But, I really don't like the correctness of my code relying on staying resident in extended registers.

Answer 1

Here's one possible approach, inspired by a comment in your question. (For the tl;dr version, skip down to the definition of point_to_line at the bottom of this answer: that gives a mapping for the first quadrant only. Extension to the whole plane is left as a not-too-difficult exercise.)

Your question says:

in particular, no counting every lattice point between the ray and the positive x-axis (unless you know a clever way to do that without enumerating them all).

There is an algorithm to do that counting without enumerating the points; its efficiency is akin to that of the Euclidean algorithm for finding greatest common divisors. I'm not sure to what extent it counts as either "easily computable" or "clever".

Suppose that we're given a point (p, q) with integer coordinates and both p and q positive (so that the point lies in the first quadrant). We might as well also assume that q < p, so that the point (p, q) lies between the x-axis y = 0 and the diagonal line y = x: if we can solve the problem for the half of the first quadrant that lies below the diagonal, we can make use of symmetry to solve it generally.

Write M for the bound on the size of p and q, so that in your example we want M = 2^31.

Then the number of lattice points strictly inside the triangle bounded by:

• the x-axis y = 0
• the ray y = (q/p)x that starts at the origin and passes through (p, q), and
• the vertical line x = M

is the sum as x ranges over integers in (0, M) of ⌈qx/p⌉ - 1.

For convenience, I'll drop the -1 and include 0 in the range of the sum; both those changes are trivial to compensate for. And now the core functionality we need is the ability to evaluate the sum of ⌈qx/p⌉ as x ranges over the integers in an interval [0, M). While we're at it, we might also want to be able to compute a closely-related sum: the sum of ⌊qx/p⌋ over that same range of x (and it'll turn out that it makes sense to evaluate both of these together).

For testing purposes, here are slow, naive-but-obviously-correct versions of the functions we're interested in, here written in Python:

def floor_sum_slow(p, q, M):
    """
    Sum of floor(q * x / p) for 0 <= x < M.

    Assumes p positive, q and M nonnegative.
    """
    return sum(q * x // p for x in range(M))


def ceil_sum_slow(p, q, M):
    """
    Sum of ceil(q * x / p) for 0 <= x < M.

    Assumes p positive, q and M nonnegative.
    """
    return sum((q * x + p - 1) // p for x in range(M))

And an example use:

>>> floor_sum_slow(51, 43, 2**28)  # takes several seconds to complete
30377220771239253
>>> ceil_sum_slow(140552068, 161600507, 2**28)
41424305916577422

These sums can be evaluated much faster. The first key observation is that if q >= p, then we can apply the Euclidean "division algorithm" and write q = ap + r for some integers a and r. The sum then simplifies: the ap part contributes a factor of a * M * (M - 1) // 2, and we're reduced from computing floor_sum(p, q, M) to computing floor_sum(p, r, M). Similarly, the computation of ceil_sum(p, q, M) reduces to the computation of ceil_sum(p, q % p, M).

The second key observation is that we can express floor_sum(p, q, M) in terms of ceil_sum(q, p, N), where N is the ceiling of (q/p)M. To do this, we consider the rectangle [0, M) x (0, (q/p)M), and divide that rectangle into two triangles using the line y = (q/p)x. The number of lattice points within the rectangle that lie on or below the line is floor_sum(p, q, M), while the number of lattice points within the rectangle that lie above the line is ceil_sum(q, p, N). Since the total number of lattice points in the rectangle is (N - 1)M, we can deduce the value of floor_sum(p, q, M) from that of ceil_sum(q, p, N), and vice versa.

Combining those two ideas, and working through the details, we end up with a pair of mutually recursive functions that look like this:

def floor_sum(p, q, M):
    """
    Sum of floor(q * x / p) for 0 <= x < M.

    Assumes p positive, q and M nonnegative.
    """
    a = q // p
    r = q % p
    if r == 0:
        return a * M * (M - 1) // 2
    else:
        N = (M * r + p - 1) // p
        return a * M * (M - 1) // 2 + (N - 1) * M - ceil_sum(r, p, N)


def ceil_sum(p, q, M):
    """
    Sum of ceil(q * x / p) for 0 <= x < M.

    Assumes p positive, q and M nonnegative.
    """
    a = q // p
    r = q % p
    if r == 0:
        return a * M * (M - 1) // 2
    else:
        N = (M * r + p - 1) // p
        return a * M * (M - 1) // 2 + N * (M - 1) - floor_sum(r, p, N)

Performing the same calculation as before, we get exactly the same results, but this time the result is instant:

>>> floor_sum(51, 43, 2**28)
30377220771239253
>>> ceil_sum(140552068, 161600507, 2**28)
41424305916577422

A bit of experimentation should convince you that the floor_sum and floor_sum_slow functions give the same result in all cases, and similarly for ceil_sum and ceil_sum_slow.

Here's a function that uses floor_sum and ceil_sum to give an appropriate mapping for the first quadrant. I failed to resist the temptation to make it a full bijection, enumerating points in the order that they appear on each ray, but you can fix that by simply replacing the + gcd(p, q) term with + 1 in both branches.

from math import gcd

def point_to_line(p, q, M):
    """
    Bijection from [0, M) x [0, M) to [0, M^2), preserving
    the 'angle' ordering.
    """
    if p == q == 0:
        return 0
    elif q <= p:
        return ceil_sum(p, q, M) + gcd(p, q)
    else:
        return M * (M - 1) - floor_sum(q, p, M) + gcd(p, q)

Extending to the whole plane should be straightforward, though just a little bit messy due to the asymmetry between the negative range and the positive range in the two's complement representation.

Here's a visual demonstration for the case M = 7, printed using this code:

M = 7
for q in reversed(range(M)):
    for p in range(M):
        print(" {:02d}".format(point_to_line(p, q, M)), end="")
    print()

Results:

 48 42 39 36 32 28 27
 47 41 37 33 29 26 21
 46 40 35 30 25 20 18
 45 38 31 24 19 16 15
 44 34 23 17 14 12 11
 43 22 13 10 09 08 07
 00 01 02 03 04 05 06

Answer 2

This doesn't meet your requirement for an "easy" function, nor for a "reasonably efficient" one. But in principle it would work, and it might give some idea of how difficult the problem is. To keep things simple, let's consider just the case where 0 < y ≤ x, because the full problem can be solved by splitting the full 2D plane into eight octants and mapping each to its own range of integers in essentially the same way.

A point (x₁, y₁) is "anticlockwise" of (x₂, y₂) if and only if the slope y₁/x₁ is greater than the slope y₂/x₂. To map the slopes to integers in an order-preserving way, we can consider the sequence of all distinct fractions whose numerators and denominators are within range (i.e. up to 2³¹), in ascending numerical order. Note that each fraction's numerical value is between 0 and 1 since we are just considering one octant of the plane.

This sequence of fractions is finite, so each fraction has an index at which it occurs in the sequence; so to map a point (x, y) to an integer, first reduce the fraction y/x to its simplest form (e.g. using Euclid's algorithm to find the GCD to divide by), then compute that fraction's index in the sequence.

It turns out this sequence is called a Farey sequence; specifically, it's the Farey sequence of order 2³¹. Unfortunately, computing the index of a given fraction in this sequence turns out to be neither easy nor reasonably efficient. According to the paper Computing Order Statistics in the Farey Sequence by Corina E. Pǎtraşcu and Mihai Pǎtraşcu, there is a somewhat complicated algorithm to compute the rank (i.e. index) of a fraction in O(n) time, where n in your case is 2³¹, and there is unlikely to be an algorithm in time polynomial in log n because the algorithm can be used to factorise integers.

All of that said, there might be a much easier solution to your problem, because I've started from the assumption of wanting to map these fractions to integers as densely as possible (i.e. no "unused" integers in the target range), whereas in your question you wrote that the number of distinct fractions is about 60% of the available range of size 2⁶⁴. Intuitively, that amount of leeway doesn't seem like a lot to me, so I think the problem is probably quite difficult and you may need to settle for a solution that uses a larger output range, or a smaller input range. At the very least, by writing this answer I might save somebody else the effort of investigating whether this approach is feasible.

Answer 3

Just some random ideas / observations:

(edit: added two more and marked the first one as wrong as pointed out in the comments)

• Divide into 16 22.5° segments instead of 8 45° segments

  If I understand the problem correctly, the lines spread out "more" towards 45°, "wasting" resolution that you need for smaller angles. (Incorrect, see below)

• In the mapping to 62 bit integers, there must be gaps. Identify enough low density areas to map down to 61 bits? Perhaps plot for a smaller problem to potentially see a pattern?

• As the range for x and y is limited, for a given x0, all (legal) x < x0 with y > q must have a smaller angle. Could this help to break down the problem in some way? Perhaps cutting a triangle where points can easily be enumerated out of the problem for each quadrant?