Random necklaces require fewer cuts
| Autor: | Alon, Noga, Elboim, Dor, Pach, J��nos, Tardos, G��bor |
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| Rok vydání: | 2021 |
| Předmět: | |
| DOI: | 10.48550/arxiv.2112.14488 |
| Popis: | It is known that any open necklace with beads of $t$ types in which the number of beads of each type is divisible by $k$, can be partitioned by at most $(k-1)t$ cuts into intervals that can be distributed into $k$ collections, each containing the same number of beads of each type. This is tight for all values of $k$ and $t$. Here, we consider the case of random necklaces, where the number of beads of each type is $km$. Then the minimum number of cuts required for a ``fair'' partition with the above property is a random variable $X(k,t,m)$. We prove that for fixed $k,t,$ and large $m$, this random variable is at least $(k-1)(t+1)/2$ with high probability. For $k=2$, fixed $t$, and large $m$, we determine the asymptotic behavior of the probability that $X(2,t,m)=s$ for all values of $s\le t $. We show that this probability is polynomially small when $s(t+1)/2$, and decays like $��( 1/\log m)$ when $s=(t+1)/2$. We also show that for large $t$, $X(2,t,1)$ is at most $(0.4+o(1))t$ with high probability and that for large $t$ and large ratio $k/\log t$, $X(k,t,1)$ is $o(kt)$ with high probability. 34 pages |
| Databáze: | OpenAIRE |
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