The “Runs” Theorem

Abstract

We give a new characterization of maximal repetitions (or runs) in strings based on Lyndon words. The characterization leads to a proof of what was known as the "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. In addition, we 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 best known bound of 4.1n by Crochemore et al. (JDA 2012), as well as other improved bounds on related problems. The characterization 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. We also establish a relationship between runs and nodes of the Lyndon tree, which gives a simple optimal solution to the 2-Period Query problem that was recently solved by Kociumaka et al. (SODA 2015). * A preliminary version of this paper has appeared in [1].

Tighter Bounds and Optimal Algorithms for All Maximal α-gapped Repeats and Palindromes

“…Proof We estimate the maximal number of points that can be placed in [1, n] 2 ⊂ N 2 such that their covered points are disjoint. First, the number of points (·, y) ∈ [1, n] 2 with y < 1/γ is less than n/γ .…”

“…As a starter, we can find ( The main ingredient to our algorithms dealing with (2) and (3) is a data structure for finding maximal equal subwords of a word that start or end at some particular positions. We call the data structure of Lemma 16 an LCE ↔ data structure.…”

“…It is easy to use Lemma 22 as a precomputation step in order to find the occurrences of basic factors in small subwords on the precomputed word efficiently (see also Proof We assume that our RAM model supports retrieving the location of the most-significant set bit in the binary representation of an integer in constant time. 2 Otherwise, as a preliminary step, we store the mapping i → lgi + 1 for every integer i with 1 < i < n in a lookup table, in O(n) time.…”

“…In the rest of this proof, we search for maximal α-gapped repeats whose arms' length is upper bounded by 2 lglgn+1 lgn = 2(lgn) 2 . Setting := α · 2(lgn) 2 + 2(lgn) 2 = 2(α + 1)(lgn) 2 , the lengths of those gapped repeats is at most .…”

“…Equation (12) with k ≥ lglg (2 ) gives O(α + occ m ) running time for the algorithm running on a subword of the cover (each is of length 2 ), where occ m is the number of occurrences of all maximal α-gapped repeats with the above described arm length in the m-th subword of the cover. Summing over all subwords of the cover, we get O(αn) time in total.…”