Stern-Brocot Tree

Stern-Brocot tree is a tree data structure whose vertices correspond to the set of non-negative rational numbers. Thus, this tree provides a very elegant way for constructing the set of fractions m/n, where m and n are relatively prime. To construct the tree, the basic idea is to start with two fractions (0/1, 1/0) and then repeat the following operation:

Insert (m+m’)/(n+n’) between two adjacent fractions m/n and m’/n’

The first step gives us the entry 1/1 between 0/1 and 1/0. Similarly, the 2nd step gives us two more: 0/1, 1/2, 1/1, 2/1, 1/0.

Continuing on like this results in an infinite binary search tree which preserves the usual ordering of rational numbers.

The figure below shows the 1st 4 levels of the Stern-Brocot tree.

The first 4 levels of Stern-Brocot Tree

The first 4 levels of Stern-Brocot Tree

Finding the Path to k in Stern-Brocot Tree

The path from the root of the tree to a number k in the Stern-Brocot tree can be found using binary search. At each node, k will either be in the left half of the tree, or the right half. We continue down the left or right subtree until we finally find k.

Implementation

There’s a couple of things to tackle in our implementation. First, I need an easy way to represent fractions, so I create my own SternBrocotFraction class. I deliberately chose to make it very specific to this algorithm because I needed a special way to handle the fraction 1/0 (which by definition is greater than all other rationals).

Secondly, I needed a good way to represent the path from the root of the tree to k. I do this by using a StringBuilder, and at each step I append either the letter L or R depending on which sub-tree we take. When the search is finished, this gives us a string representation of the path from the root of the tree to the number k. This approach is similar to the approach advocated by ACM Programming Competitions for the “Stern-Brocot Number System” problem.

Here’s the code to find path to a number k:

package com.umairsaeed.algorithm;

public class SternBrocotPath {
  private static final char LEFT_SUB = 'L';
  private static final char RIGHT_SUB = 'R';

  public String findPathTo(SternBrocotFraction f) {
    SternBrocotFraction L = new SternBrocotFraction(0, 1);
    SternBrocotFraction R = new SternBrocotFraction(1, 0);

    StringBuilder results = new StringBuilder();
    SternBrocotPath.find(f, L, R, results);
    return results.toString();
  }

  public static void find(SternBrocotFraction f,
      SternBrocotFraction L,
      SternBrocotFraction R,
      StringBuilder results)
  {
    SternBrocotFraction M = L.add(R);

    if (M.compareTo(f) < 0) {
      L = M;
      results.append(RIGHT_SUB);
      SternBrocotPath.find(f, L, R, results);
    } else if (M.compareTo(f) > 0) {
      R = M;
      results.append(LEFT_SUB);
      SternBrocotPath.find(f, L, R, results);
    }
    return;
  }
}

The special SternBrocotFraction class is:

package com.umairsaeed.algorithm;

public class SternBrocotFraction implements
        Comparable<SternBrocotFraction> {
  private int numerator;
  private int denominator;

  public SternBrocotFraction(int numerator, int denominator) {
    if (denominator < 0) {
      numerator *= -1;
      denominator *= -1;
    }

    this.numerator = numerator;
    this.denominator = denominator;
  }

  public double doubleValue() {
    if (this.denominator == 0) {
      return Double.MAX_VALUE;
    } else {
      return (double) this.numerator /
            (double) this.denominator;
    }
  }

  public SternBrocotFraction add(SternBrocotFraction other) {
    return new SternBrocotFraction(
        this.numerator + other.numerator,
        this.denominator + other.denominator);
  }

  public int compareTo(SternBrocotFraction other) {
    if (this.doubleValue() < other.doubleValue()) {
      return -1;
    } else if (this.doubleValue() > other.doubleValue()) {
      return 1;
    }
    return 0;
  }
}

Finally, some test code to exercise my class:

package com.umairsaeed.algorithm;

public class SternBrocotTester {
  public static void main(String[] args) {
    testSternBrocotPath();
  }

  public static void testSternBrocotPath() {
    SternBrocotPath t = new SternBrocotPath();

    SternBrocotFraction f = new SternBrocotFraction(5, 7);
    System.out.println(t.findPathTo(f));

    f = new SternBrocotFraction(19, 101);
    System.out.println(t.findPathTo(f));

    f = new SternBrocotFraction(977, 331);
    System.out.println(t.findPathTo(f));

    f = new SternBrocotFraction(1049, 7901);
    System.out.println(t.findPathTo(f));
  }
}