On continued fractions. Part 3: In competitive programming

Revision en17, by adamant, 2022-04-03 19:07:16

Hi everyone!

It's been quite some time since I wrote two previous articles in the cycle:

Part 1: Introduction
Part 2: Properties and interpretation
Part 3: In competitive programming

This time I finally decided to publish something on how one can actually use continued fractions in competitive programming problems.

Few months ago, I joined CP-Algorithms as a collaborator. The website also underwent a major design update recently, so I decided it would be great to use this opportunity and publish my new article there, so here it is:

CP-Algorithms — Continued fractions

It took me quite a while to write and I made sure to not only describe common competitive programming challenges related to continued fractions, but also to describe the whole concept from scratch. That being said, article is supposed to be self-contained.

Main covered topics:

  1. Notion of continued fractions, convergents, semiconvergents, complete quotients.
  2. Recurrence to compute convergents, notion of continuant.
  3. Connection of continued fractions with the Stern-Brocot tree and the Calkin-Wilf tree.
  4. Convergence rate with continued fractions.
  5. Linear fractional transformations, quadratic irrationalities.
  6. Geometric interpretation of continued fractions and convergents.

I really hope that I managed to simplify the general story-telling compared to previous 2 articles.

Here are the major problems that are dealt with in the article:

  • Given $$$a_1, \dots, a_n$$$, quickly compute $$$[a_l; a_{l+1}, \dots, a_r]$$$ in queries.
  • Which number of $$$A=[a_0; a_1, \dots, a_n]$$$ and $$$B=[b_0; b_1, \dots, b_m]$$$ is smaller? How to emulate $$$A-\varepsilon$$$ and $$$A+\varepsilon$$$?
  • Given $$$A=[a_0; a_1, \dots, a_n]$$$ and $$$B=[b_0; b_1, \dots, b_m]$$$, compute the continued fraction representations of $$$A+B$$$ and $$$A \cdot B$$$.
  • Given $$$\frac{0}{1} \leq \frac{p_0}{q_0} < \frac{p_1}{q_1} \leq \frac{1}{0}$$$, find $$$\frac{p}{q}$$$ such that $$$(q,p)$$$ is lexicographically smallest and $$$\frac{p_0}{q_0} < \frac{p}{q} < \frac{p_1}{q_1}$$$.
  • Given $$$x$$$ and $$$k$$$, $$$x$$$ is not a perfect square. Let $$$\sqrt x = [a_0; a_1, \dots]$$$, find $$$\frac{p_k}{q_k}=[a_0; a_1, \dots, a_k]$$$ for $$$0 \leq k \leq 10^9$$$.
  • Given $$$r$$$ and $$$m$$$, find the minimum value of $$$q r \pmod m$$$ on $$$1 \leq q \leq n$$$.
  • Given $$$r$$$ and $$$m$$$, find $$$\frac{p}{q}$$$ such that $$$p, q \leq \sqrt{m}$$$ and $$$p q^{-1} \equiv r \pmod m$$$.
  • Given $$$p$$$, $$$q$$$ and $$$b$$$, construct the convex hull of lattice points below the line $$$y = \frac{px+b}{q}$$$ on $$$0 \leq x \leq n$$$.
  • Given $$$A$$$, $$$B$$$ and $$$C$$$, find the maximum value of $$$Ax+By$$$ on $$$x, y \geq 0$$$ and $$$Ax + By \leq C$$$.
  • Given $$$p$$$, $$$q$$$ and $$$b$$$, compute the following sum:
$$$\sum\limits_{x=1}^n \lfloor \frac{px+b}{q} \rfloor.$$$

So far, here is the list of problems that are explained in the article:

And an additional list of practice problems where continued fractions could be useful:

There are likely much more problems where continued fractions are used, please mention them in the comments if you know any!

Finally, since CP-Algorithms is supposed to be a wiki-like project (that is, to grow and get better as time goes by), please feel free to comment on any issues that you might find while reading the article, ask questions or suggest any improvements. You can do so in the comments below or in the issues section of the CP-Algorithms GitHub repo. You can also suggest changes via pull request functionality.

Tags tutorial, cp-algorithms, continued fraction, #continued fractions, euclidean algorithm, number theory

History

 
 
 
 
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  Rev. Lang. By When Δ Comment
en17 English adamant 2022-04-03 19:07:16 2 Tiny change: '[problem:356E]\n* [pr' -> '[problem:346E]\n* [pr'
en16 English adamant 2022-04-02 18:59:41 141
en15 English adamant 2022-04-02 06:27:52 126
en14 English adamant 2022-04-02 06:23:37 33 Tiny change: 'rmations, continued fractions on segments, quadrat' -> 'rmations, quadrat'
en13 English adamant 2022-04-02 06:22:06 614
en12 English adamant 2022-04-02 06:16:15 141 Tiny change: 's smaller?\n- How to em' -> 's smaller? How to em'
en11 English adamant 2022-04-02 05:18:30 303
en10 English adamant 2022-04-01 16:35:49 109
en9 English adamant 2022-04-01 16:09:33 374
en8 English adamant 2022-04-01 03:06:15 2 Tiny change: ' Fractions\n](https://' -> ' Fractions](https://'
en7 English adamant 2022-04-01 02:59:55 103
en6 English adamant 2022-04-01 02:44:14 537
en5 English adamant 2022-03-31 21:14:30 108
en4 English adamant 2022-03-31 21:09:57 137
en3 English adamant 2022-03-31 21:05:35 256
en2 English adamant 2022-03-31 21:05:11 1650 Tiny change: 'article:\n* [Timus' -> 'article:\n\n* [Timus'
en1 English adamant 2022-03-31 19:54:49 3507 Initial revision (published)