Background The breakpoint median in the set S n of permutations on n terms is known to have some unusual behavior, especially if the input genomes are maximally different to each other. The mathematical study of the set of medians is complicated by the facts that breakpoint distance is not a metric but a pseudo-metric, and that it does not define a geodesic space. Results We introduce the notion of partial geodesic, or geodesic patch between two permutations, and show that if two permutations are medians, then every permutation on a geodesic patch between them is also a median. We also prove the conjecture that the input permutations themselves are medians.
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机译：背景技术已知在N项中的排列中的设置S n 的断点中值具有一些不寻常的行为，特别是如果输入基因组彼此最大地不同。该组中位数的数学研究对断点距离不是度量而是伪度量的事实，并且它没有定义测地空间。结果我们在两个排列之间介绍了部分测地域的概念，或者在两个排列之间介绍了测地贴片，并表明如果两个排列是中位数，则它们之间的测地贴片上的每个排列也是一个中值。我们还证明了猜想输入排列本身是中位数。
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