Range Selection and Median: Tight Cell Probe Lower Bounds and Adaptive Data Structures

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• Department of Computer Science
Range selection is the problem of preprocessing an input
array A of n unique integers, such that given a query (i; j; k),
one can report the k'th smallest integer in the subarray
A[i];A[i+1]; : : : ;A[j]. In this paper we consider static data
structures in the word-RAM for range selection and several
natural special cases thereof.
The rst special case is known as range median, which
arises when k is xed to b(j &#x100000; i + 1)=2c. The second
case, denoted prex selection, arises when i is xed to 0.
Finally, we also consider the bounded rank prex selection
problem and the xed rank range selection problem. In
the former, data structures must support prex selection
queries under the assumption that k for some value
n given at construction time, while in the latter, data
structures must support range selection queries where k
is xed beforehand for all queries. We prove cell probe
lower bounds for range selection, prex selection and range
median, stating that any data structure that uses S words
of space needs
(log n= log(Sw=n)) time to answer a query.
In particular, any data structure that uses n logO(1) n space
needs
(log n= log log n) time to answer a query, and any
data structure that supports queries in constant time, needs
n1+
(1) space. For data structures that uses n logO(1) n
space this matches the best known upper bound.
Additionally, we present a linear space data structure
that supports range selection queries in O(log k= log log n +
log log n) time. Finally, we prove that any data struc-
ture that uses S space, needs
(log = log(Sw=n)) time
to answer a bounded rank prex selection query and

(log k= log(Sw=n)) time to answer a xed rank range se-
lection query. This shows that our data structure is optimal
except for small values of k.
Original language English Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms. SODA 2011 9 Society for Industrial and Applied Mathematics 2011 805-813 Published - 2011 Symposium on Discrete Algorithms. SODA 2011 - Duration: 17 Dec 2010 → …Conference number: 22

Conference

Conference Symposium on Discrete Algorithms. SODA 2011 22 17/12/2010 → …

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