One (More) Line on the Most Ancient Algorithm in History

Home , Euclidean algorithm

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In this section we review the algorithm and give the new, simpliﬁed runtime analysis. The description of the algorithm is given below in Algorithm 1. The algorithm is given non-negative integers, x and y as an input. We assume w.l.o.g that x>y.

Input: Two non-negative integers, x>y ∈ N Output: gcd(x,y)

1 if y = 0 then return x; 2 if y = 1 then return 1; 3 return gcd(y,x mod y) Algorithm 1: Euclidean Algorithm For the sake of analysis, let us ﬁx x>y and let m denote the number of the (recursive) iterations of algorithm.

Deﬁnition 2.1. For 1 ≤ i ≤ m, let xi and yi denote the values of x and y in the i-th iteration of the algorithm, respectively. In particular, x1 = x,y1 = y.

Observation 2.2. For 1 ≤ i ≤ m: xi >yi ≥ 0.

Proof. Follows from the fact that yi = xi 1 mod yi 1 = xi 1 mod xi, which attains values between − − − 0 and xi − 1. ∆ Deﬁnition 2.3. For 1 ≤ i ≤ m, we deﬁne si = xi + yi.

Next, we will show that si is a potential function for this algorithm. In particular, the function loses a constant fraction of its mass in every step, yet it is bounded away from zero. The following is immediate from the deﬁnition, given the above observation.

Corollary 2.4. For 1 ≤ i ≤ m: si ≥ 1. The following lemma is the heart of the argument.

Lemma 2.5. For 1 ≤ i ≤ m: si ≤ 2/3 · si 1. − Proof. Recall that si 1 = xi 1 + yi 1. Let us divide xi 1 by yi 1 with a reminder. In particular, − − − − −

xi 1 = qi 1 · yi 1 + ri 1, where 0 ≤ ri 1 ≤ yi 1 − 1. − − − − − −

Observe that si = yi 1 + ri 1 and in addition, qi 1 ≥ 1, as xi 1 >yi 1. We obtain the following: − − − − − si 1 = (qi 1+1)yi 1 +ri 1 ≥ 2yi 1+ri 1 ≥ 2yi 1+ri 1−(yi 1−ri 1)/2 = 1.5yi 1 +1.5ri 1 = 1.5si. − − − − − − − − − − − −

The second inequality holds since ri 1

i 1 Corollary 2.6. For 1 ≤ i ≤ m: si ≤ s1 · (2/3) − . Theorem 1 follows immediately from the above.

Proof of Theorem 1. By Corollaries 2.4 and 2.6:

m 1 m 1 1 ≤ sm ≤ s1 · (2/3) − = (x + y) · (2/3) − .

Therefore, m ≤ log1.5(x + y) + 1.

2 2.1 Improved Analysis In this section we show that using a slightly more sophisticated potential function yields an even tighter result. As before, for the sake of analysis, let us ﬁx x>y and let m denote the number 1+√5 ≅ of the (recursive) iterations of algorithm. In addition, let φ = 2 1.61803398875 denote the golden ratio.

∆ 1 Deﬁnition 2.7. For 1 ≤ i ≤ m, we deﬁne si = xi + φ · yi.

As before, for 1 ≤ i ≤ m: si ≥ 1. The following lemma mirrors Lemma 2.5. 1 Lemma 2.8. For 1 ≤ i ≤ m: si ≤ · si 1. φ − 1 Proof. Recall that si 1 = xi 1 + · yi 1. Let us divide xi 1 by yi 1 with a reminder. In particular, − − φ − − − xi 1 = qi 1 · yi 1 + ri 1, where 0 ≤ ri 1 ≤ yi 1 − 1. − − − − − − 1 Observe that si = yi 1 + ·ri 1 and in addition, qi 1 ≥ 1, as xi 1 >yi 1. We obtain the following: − φ − − − − 1 1 1 si 1 = qi 1 + yi 1 +ri 1 ≥ 1+ yi 1 +ri 1 = φ·yi 1 +ri 1 = φ yi 1 + · ri 1 = φ·si. − − φ − − φ − − − − − φ − 1 Recall that 1 + φ = φ. The proof of Theorem 2 follows by repeating the previous argument.

3 Conclusion & Open Questions

In this short note we add one (more) line of analysis to the well-known Euclidean Algorithm. The new analysis does not require any background and can be taught even in an introductory-level undergraduate class. It would be nice to see if we could replace other kind of analyses that rely on Fibonacci numbers by a potential argument. One such example is the analysis of the height of an AVL-Tree [AVL62, Sed83].

References

[AVL62] G. Adelson-Velsky and E. Landis. An algorithm for the organization of information. In Proceedings of the USSR Academy of Sciences (in Russian), volume 146, pages 263–266, 1962. [CLRS09] T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein. Introduction to Algorithms, 3rd Edition. MIT Press, 2009. [Knu97] D. E. Knuth. The art of computer programming, Volume I: Fundamental Algorithms, 3rd Edition. Addison-Wesley, 1997. [Knu98] D. E. Knuth. The art of computer programming, Volume II: Seminumerical Algorithms, 3rd Edition. Addison-Wesley, 1998. [Lam44] G. Lam´e. Note sur la limite du nombre des divisions dans la recherche du plus grand commun diviseur entre deux nombres entiers. Comptes Rendus Acad. Sci., (19):867–870, 1844. [Sed83] R. Sedgewick. Algorithms. Addison-Wesley, 1983.