Minimal Distance to Pi

Here is a problem from Week of Code 29 hosted by Hackerrank.

Problem. Given two integers q_1 and q_2 (1\le q_1 \le q_2 \le 10^{15}), find and print a common fraction p/q such that q_1 \le q \le q_2 and \left|p/q-\pi\right| is minimal. If there are several fractions having minimal distance to \pi, choose the one with the smallest denominator.

Note that checking all possible denominators does not work as iterating for 10^{15} times would exceed the time limit (2 seconds for C or 10 seconds for Ruby).

The problem setter suggested the following algorithm in the editorial of the problem:

  1. Given q, it is easy to compute p such that r(q) := p/q is the closest rational to \pi among all rationals with denominator q.
  2. Find the semiconvergents of the continued fraction of \pi with denominators \le 10^{15}.
  3. Start from q = q_1, and at each step increase q by the smallest denominator d of a semiconvergent such that r(q+d) is closer to \pi than r(q). Repeat until q exceeds q_2.

Given q, let d = d(q) be the smallest increment to the denominator q such that r(q+d) is closer to \pi than r(q). To justify the algorithm, one needs to prove the d is the denominator of one of the semiconvergents. The problem setter admits that he does not have a formal proof.

Inspired by the problem setter’s approach, here is a complete solution to the problem. Note that \pi should not be special in this problem, and can be replaced by any other irrational number \theta. Without loss of generality, we may assume that \theta\in(0,1).

We first introduce the Farey intervals of \theta.

  1. Start with the interval (0/1, 1/1).
  2. Suppose the last interval is (a/b, c/d). Cut it by the mediant of a/b and c/d and choose one of the intervals (a/b, (a+c)/(b+d)), ((a+c)/(b+d), c/d) that contains \theta as the next interval.

We call the intervals appeared in the above process Farey intervals of \theta. For example, take \theta = \pi - 3 = 0.1415926.... The Farey intervals are:

\begin{gathered}(0/1, 1/1), (0/1, 1/2), (0/1, 1/3), (0/1, 1/4), (0/1, 1/5), \\ (0/1, 1/6), (0/1, 1/7), (1/8, 1/7), (2/15, 1/7),\cdots\end{gathered}

The Farey sequence of order n, denoted by F_n, is the sequence of completely reduced fractions between 0 and 1 which when in lowest terms have denominators less than or equal to n, arranged in order of increasing size. Fractions which are neighboring terms in any Farey sequence are known as a Farey pair. For example, Farey sequence of order 5 is
F_5 = (0/1, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 1/1).

Using the Stern–Brocot tree, one can prove that

Lemma 1. For every Farey interval (a/b, c/d) of \theta, the pair (a/b, c/d) is a Farey pair. Conversely, for every Farey pair (a/b, c/d), if \theta \in (a/b, c/d), then (a/b, c/d) is a Farey interval.
We say p/q is a good rational approximation of \theta if every rational between p/q and \theta (exclusive) has a denominator greater than q.

By the definition of Farey sequence, incorporating with Lemma 1, we know that

Lemma 2. A rational is an endpoint of a Farey interval of \theta if and only if it is a good rational approximation of \theta.

In fact, one can show that the endpoints of Farey intervals and semiconvergents of continued fraction are the same thing! Thereof, the problem setter’s claim follows immediately from:

Proposition. Given q, let r(q) = p / q be the rational closest to \theta with denominator q. If d = d(q) is the minimal increment to q such that r(q + d) = (p + c) / (q + d) is closer to \theta than r(q), then c/d is a good rational approximation.

Remark. The proposition states that the increments to p/q always come from a good rational approximation. It is stronger than the problem setter’s statement, which only asserts that the increment to the q comes from a good rational approximation.

Proof. In (x, y)-plane, plot the trapezoid defined by

\left| y/x - \theta \right| < \left|p/q - \theta\right|, q < x < q + d.
Geometric interpretation

Also we interpret rational numbers p/q, (p+c)/(q+d) as points A = (q, p), B = (q+d, p+c). Let the line through (q, p) parallel to y=\theta x intersect the vertical line x = q+d at C = (q+d, p+\theta d). By the definition of d, we know that the trapezoid does not contain lattice points. In particular, there is no lattice point in the interior of the triangle ABC. In the coordinate system with origin at A, B has coordinate (d, c) and the line through A, C is y = \theta x. Since triangle ABC contains no lattice points, there is no (b, a) with b < d such that a/b is between \theta and c/d. In other words, c/d is a good rational approximation. [qed]

Here is a fine print of the algorithm. Because floats may not have enough precision for the purpose of computation, we will instead use a convergent of the continuous fraction of \pi instead. All the computations will then happen in \mathbb{Q}. Finally, we present the algorithm.

P = Rational(5706674932067741, 1816491048114374) - 3
# find endpoints of Farey intervals
a, b, c, d = 0, 1, 1, 1
farey = [[a,b],[c,d]]
while (f = b + d) <= max - min
  farey << [e = a + c, f]
  if P < Rational(e, f)
    c, d = e, f
    a, b = e, f
min, max =
p_min = (P * q).round
# increase p_min/min by frations in farey
while min <= max
  c, d = nil, nil
  farey.each do |a, b|
    break if min + b > max
    if (Rational(p_min + a, min + b) - P).abs < (Rational(p_min, min) - P).abs
      c, d = a, b
  break if d == nil
  p_min += c; min += d
puts "#{p_min + 3 * min}/#{min}"

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