Encoding Range Minimum Queries

We consider the problem of encoding range minimum queries (RMQs): given an array A[1..n] of distinct totally ordered values, to pre-process A and create a data structure that can answer the query RMQ(i, j), which returns the index containing the smallest element in A[i..j], without access to the array A at query time. We give a data structure whose space usage is 2n + o(n) bits, which is asymptotically optimal for worst-case data, and answers RMQs in O(1) worstcase time. This matches the previous result of Fischer and Heun, but is obtained ∗An extended abstract of some of the results in Sections 1 and 2 appeared in Proc. 18th Annual International Conference on Computing and Combinatorics (COCOON 2012), Springer LNCS 7434, pp. 396–407. †Research supported by NSF grant CCF-1018370 and BSF grant 2010437. ‡Partially funded by Millennium Nucleus Information and Coordination in Networks ICM/FIC P10-024F, Chile. §Research partly supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (Grant number 2012-0008241). 1 ar X iv :1 31 1. 43 94 v1 [ cs .D S] 1 8 N ov 2 01 3 in a more natural way. Furthermore, our result can encode the RMQs of a random array A in 1.919n + o(n) bits in expectation, which is not known to hold for Fischer and Heun’s result. We then generalize our result to the encoding range top-2 query (RT2Q) problem, which is like the encoding RMQ problem except that the query RT2Q(i, j) returns the indices of both the smallest and second-smallest elements of A[i..j]. We introduce a data structure using 3.272n+o(n) bits that answers RT2Qs in constant time, and also give lower bounds on the effective entropy of RT2Q.