Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Nearly Tight Lower Bounds for Succinct Range Minimum Query   Given an array of distinct integers $A[1\ldots n]$, the Range Minimum Query
(RMQ) problem requires us to construct a data structure from $A$, supporting
the RMQ query: given an interval $[a,b]\subseteq[1,n]$, return the index of the
minimum element in subarray $A[a\ldots b]$, i.e. return
$\text{argmin}_{i\in[a,b]}A[i]$. 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 $2n-1.5\log
n+n/\left(\frac{\log n}{t}\right)^t+\tilde{O}(n^{3/4})$ bits of space and
$O(t)$ query time, where the additive $\tilde{O}(n^{3/4})$ is a pre-computed
lookup table used in the RAM model, assuming the word-size is $\Theta(\log n)$
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 $2n-1.5\log n+n/(\log n)^{O(t^2\log^2t)}$ bits of space, assuming
the word-size is $\Theta(\log n)$ 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, $2n-1.5\log
n+n/(\log n)^{O(t\log^2t)}$ bits of space is necessary in the cell-probe model
with word-size $\Theta(\log n)$ bits. We emphasize that, in terms of time
complexity, our lower bound is tight up to a polylogarithmic factor.

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The categories of this paper are: plasm-ph
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The categories of this paper are: q-bio
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The categories of this paper are: funct-an
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The categories of this paper are: nucl-ex
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The categories of this paper are: gr-qc
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The categories of this paper are: hep-ex
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The categories of this paper are: nlin
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The categories of this paper are: atom-ph
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The categories of this paper are: supr-con
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The categories of this paper are: ao-sci
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The categories of this paper are: dg-ga
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The categories of this paper are: comp-gas
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The categories of this paper are: hep-ph
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The categories of this paper are: q-alg
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The categories of this paper are: chao-dyn
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The categories of this paper are: adap-org