AbstractWe consider the problem of encoding two-dimensional arrays, whose elements come from a total order, for answering $${\text{Top-}}{k}$$
Top-
k
queries. The aim is to obtain encodings that use space close to the information-theoretic lower bound, which can be constructed efficiently. For an $$m \times n$$
m
×
n
array, with $$m \le n$$
m
≤
n
, we first propose an encoding for answering 1-sided $${\textsf {Top}}{\text {-}}k{}$$
Top
-
k
queries, whose query range is restricted to $$[1 \dots m][1 \dots a]$$
[
1
⋯
m
]
[
1
⋯
a
]
, for $$1 \le a \le n$$
1
≤
a
≤
n
. Next, we propose an encoding for answering for the general (4-sided) $${\textsf {Top}}{\text {-}}k{}$$
Top
-
k
queries that takes $$(m\lg {{(k+1)n \atopwithdelims ()n}}+2nm(m-1)+o(n))$$
(
m
lg
(
k
+
1
)
n
n
+
2
n
m
(
m
-
1
)
+
o
(
n
)
)
bits, which generalizes the joint Cartesian tree of Golin et al. [TCS 2016]. Compared with trivial $$O(nm\lg {n})$$
O
(
n
m
lg
n
)
-bit encoding, our encoding takes less space when $$m = o(\lg {n})$$
m
=
o
(
lg
n
)
. In addition to the upper bound results for the encodings, we also give lower bounds on encodings for answering 1 and 4-sided $${\textsf {Top}}{\text {-}}k{}$$
Top
-
k
queries, which show that our upper bound results are almost optimal.
Information-Theoretic Lower Bounds on the Oracle Complexity of Stochastic Convex Optimization
