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 \(\theta\in(0,1)\).

Let me 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:

$$(0/1, 1/1), (0/1, 1/2), (0/1, 1/3), (0/1, 1/4), (0/1, 1/5), (0/1, 1/6), (0/1, 1/7), (1/8, 1/7), (2/15, 1/7), \cdots$$

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 neighbouring 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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