“大”定理
The "Runs" Theorem
摘要：
我们给出了一个新的特征的最大重复（或运行）的字符串的基础上林登字。这个特征导致了所谓的“游程”猜想（Kolpakov \& Kucherov（FOCS '99））的证明，该猜想指出长度为$n$的字符串中游程$ρ(n)$的最大数量小于$n$。考虑到过去15年来为解决这个问题所做的大量努力，证明非常简单，并且大大提高了我们对串如何发生的理解。此外，我们还得到了长度为$n$的字符串中游程的最大指数和$σ(n)$的$3n$的上界，改进了Crochemore等人（JDA 2012）的$4.1n$的最佳已知界，以及相关问题的其他改进界。的特征也产生了一个新的，概念上简单的线性时间算法计算所有的运行在一个字符串。我们的算法的一个显着特点是，不同于所有现有的线性时间算法，它不利用字符串的Lempel-Ziv分解。我们还建立了林登树的运行和节点之间的关系，这为Kociumaka等人最近解决的2周期查询问题提供了一个简单的最优解决方案。
英文摘要：
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).
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