Affiliation:
1. Dalhousie University, Nova Scotia, Canada
2. University of Waterloo, Ontario, Canada
Abstract
Consider a tree
T
on
n
nodes, each having a weight drawn from [1‥σ]. In this article, we study the problem of supporting various path queries over the tree
T
. The path counting query asks for the number of the nodes on a query path whose weights are in a query range, while the path reporting query requires to report these nodes. The path median query asks for the median weight on a path between two given nodes, and the path selection query returns the
k
-th smallest weight. We design succinct data structures to encode
T
using n
nH
(
W
T
) + 2
n
+
o
(
n
lg σ) bits of space, such that we can support path counting queries in
O
(lg σ/lg lg
n
+ 1)) time, path reporting queries in
O
((
occ
+1)(lg σ / lg lg
n
+ 1)) time, and path median and path selection queries in
O
(lg σ / lg lg σ) time, where
H
(
W
T
) is the entropy of the multiset of the weights of the nodes in
T
and
occ
is the size of the output. Our results not only greatly improve the best known data structures [Chazelle 1987; Krizanc et al. 2005], but also match the lower bounds for path counting, median, and selection queries [Pătraşcu 2007, 2011; Jørgensen and Larsen 2011] when σ = Ω(
n
/polylog(
n
)).
Funder
NSERC
Canada Research Chairs Program
Publisher
Association for Computing Machinery (ACM)
Subject
Mathematics (miscellaneous)
Cited by
5 articles.
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