The "Runs" Theorem

We give a new characterization of maximal repetitions (or runs) in strings based on Lyndon words. The characterization leads to a proof of the so-called runs conjecture (Kolpakov & Kucherov (FOCS ’99)), which states that the maximum number of runs ρ(n) in a string of length n is less than n. The proof is remarkably simple, considering the numerous endeavors to tackle this problem in the last 15 years, and significantly improves our understanding of how runs can occur in strings. Based on the same observation, we also obtain an upper bound of 3n for the maximum sum of exponents σ(n) of runs in a string of length n, improving on the current best bound of 4.1n by Crochemore et al. (JDA 2012). Furthermore, the proof also gives rise to a new, conceptually simple linear-time algorithm for computing all the runs in a string. A notable characteristic of our algorithm is that, unlike all existing linear-time algorithms, it does not utilize the Lempel-Ziv factorization of the string.