Nearly Tight Lower Bounds for Succinct Range Minimum Query
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Given an array of distinct integers A[1… n], the Range Minimum Query (RMQ) problem requires us to construct a data structure from A, supporting the RMQ query: given an interval [a,b]⊆[1,n], return the index of the minimum element in subarray A[a… b], i.e. return argmin_i∈[a,b]A[i]. The fundamental problem has a long history. The textbook solution which uses O(n) words of space and O(1) time by Gabow, Bentley, Tarjan (STOC 1984) and Harel, Tarjan (SICOMP 1984) dates back to 1980s. The state-of-the-art solution is presented by Fischer, Heun (SICOMP 2011) and Navarro, Sadakane (TALG 2014). The solution uses 2n-1.5log n+n/(log n/t)^t+Õ(n^3/4) bits of space and O(t) query time, where the additive Õ(n^3/4) is a pre-computed lookup table used in the RAM model, assuming the word-size is Θ(log n) bits. On the other hand, the only known lower bound is proved by Liu and Yu (STOC 2020). They show that any data structure which solves RMQ in t query time must use 2n-1.5log n+n/(log n)^O(t^2log^2t) bits of space, assuming the word-size is Θ(log n) bits. In this paper, we prove nearly tight lower bound for this problem. We show that, for any data structure which solves RMQ in t query time, 2n-1.5log n+n/(log n)^O(tlog^2t) bits of space is necessary in the cell-probe model with word-size Θ(log n) bits. We emphasize that, in terms of time complexity, our lower bound is tight up to a polylogarithmic factor.
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