1. What are the contributions mentioned in the paper "On space efficient two dimensional range minimum data structures" ?
The authors study the trade-off between the space and query time of the problem.. The authors show that every algorithm enabled to access A during the query and using O ( N/c ) bits additional space requires Ω ( c ) query time, for any c where 1 ≤ c ≤ N.. In two dimensions, the authors complement the lower bound with an indexing data structure of size O ( N/c ) bits additional space which can be preprocessed in O ( N ) time and achieves O ( c log c ) query time.
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2. What is the space for a data structure encoding these arrays?
Since the number of different arrays in the set is (m2 !) n′ , the space for a data structure encoding these arrays is Ω(log(m2 !) n′) = Ω(mn log m) bits.
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3. How is the subquery q2 answered in O(1) time?
The subquery q2 is answered in O(1) time by using a constant query time LCA structure [5], querying the Cartesian trees in constant time [16], and performing O(1) probes into the matrix.
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4. How can the authors ensure that each sequence has length exactly d + n/c?
By padding each of these sequences with further 0s and 1s, the authors can ensure that each sequence has length exactly d + n/c and contains exactly n/c zeros.
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![Table 2. Results for the 2D-RMQ problem. The contributions of [13, 9, 4] and Theorem 1 can be generalized to the multidimensional version of the problem.](/figures/table-2-results-for-the-2d-rmq-problem-the-contributions-of-32l97z9b.webp)
