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 2
n
+
o
(
n
) bits, which is asymptotically optimal for worst-case data, and answers RMQs in
O
(1) worst-case time. This matches the previous result of Fischer and Heun, but is obtained in a more natural way. Furthermore, our result can encode the RMQs of a random array
A
in 1.919
n
+
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 3.272
n
+
o
(
n
) bits that answers RT2Qs in constant time, and also give lower bounds on the
effective entropy
of the RT2Q problem.
Bounding Worst-Case Data Cache Behavior by Analytically Deriving Cache Reference Patterns