The problem of parameterized range majority asks us to preprocess a string of length n such that, given the endpoints of a range, one can quickly find all the distinct elements whose relative frequencies in that range are more than a threshold \(\tau\). This is a more tractable version of the classical problem of finding the range mode, which is unlikely to be solvable in polylogarithmic time and linear space. In this paper we give the first linear-space solution with optimal \(\mathcal {O}\!\left( {1 / \tau } \right)\) query time, even when \(\tau\) can be specified with the query. We then consider data structures whose space is bounded by the entropy of the distribution of the symbols in the sequence. For the case when the alphabet size \(\sigma\) is polynomial on the computer word size, we retain the optimal time within optimally compressed space (i.e., with sublinear redundancy). Otherwise, either the compressed space is increased by an arbitrarily small constant factor or the time rises to any function in \((1/\tau )\cdot \omega (1)\). We obtain the same results on the complementary problem of parameterized range minority.

参数化范围多数的问题要求我们对长度为n的字符串进行预处理，以便在给定范围的端点的情况下，可以快速找到所有在该范围内的相对频率大于阈值\（\ tau \）的不同元素。。这是找到范围模式的经典问题的一个更易处理的版本，在多对数时间和线性空间中这不太可能解决。在本文中，我们给出了具有最优\（\ mathcal {O} \！\ left（{1 / \ tau} \ right）\）查询时间的第一个线性空间解，即使\（\ tau \）可以与查询一起指定。然后，我们考虑其空间受序列中符号分布的熵限制的数据结构。对于字母大小\（\ sigma \）在计算机单词大小上是多项式的情况，我们将最佳时间保留在最佳压缩空间内（即，具有亚线性冗余）。否则，要么压缩空间增加任意小的常数因数，要么时间增加到\（（1 / tau）\ cdot \ omega（1）\）中的任何函数。我们在参数化范围少数的互补问题上获得了相同的结果。