![Figure 7: Pr[p ∈ [xl, xr]] is a bivariate piecewise linear function in xl and xr. It consists of s2 pieces and each piece covers a rectangular region in the xlxr-plane.](/figures/figure-7-pr-p-xl-xr-is-a-bivariate-piecewise-linear-function-28fmgkt3.webp)
Range Searching on Uncertain Data
∗
Pankaj K. Agarwal
Duke University
Durham, NC, USA
pankaj@cs.duke.edu
Siu-Wing Cheng
HKUST
Hong Kong, China
scheng@cse.ust.hk
Yufei Tao
CUHK
Hong Kong, China
taoyf@cse.cuhk.edu.hk
Ke Yi
HKUST
Hong Kong, China
yike@cse.ust.hk
Abstract
Querying uncertain data has emerged as an important problem in data management due to
the imprecise nature of many measurement data. In this paper we study answering range queries
over uncertain data. Specifically, we are given a collection P of n uncertain points in R, each
represented by its one-dimensional probability density function (pdf). The goal is to build a data
structure on P such that given a query interval I and a probability threshold τ , we can quickly
report all points of P that lie in I with probability at least τ . We present various structures with
linear or near-linear space and (poly)logarithmic query time. Our structures support pdf’s that
are either histograms or more complex ones such as Gaussian or piecewise algebraic.
1 Introduction
Range searching, namely preprocessing a set of points into a data structure so that all points within
a given query range can be reported efficiently, is one of the most widely studied topics in computa-
tional geometry and database systems [2], with a wide range of applications. Most of the works to
date deal with certain data, that is, the points are given their precise locations in R
d
. Recent years,
however, have witnessed a dramatically increasing amount of attention devoted to managing un-
certain data because many real-world measurements are inherently accompanied with uncertainty.
Besides the recent efforts in the data management community (see the survey [15]), various issues
related with data uncertainty have also been studied in artificial intelligence [20], machine learning
[5], statistics [18], and many other areas.
A popular approach to model data uncertainty [13, 25] is to consider each uncertain point p
as a probability distribution over space. It is usually assumed that the points are independent but
∗
A preliminary version of this paper appeared as “Indexing uncertain data” in ACM Symposium on Principles of
Database Systems (PODS), 2009. P. K. Agawal is supported by NSF under grants CNS-05-40347, CCF-06 -35000,
IIS-07-13498, and CCF-09-40671, by ARO grants W911NF-07-1-0376 and W911NF-08-1-0452, by an NIH grant 1P50-
GM-08183-01, by a DOE grant OEG-P200A070505, and by a grant from the U.S.–Israel Binational Science Foundation.
S.-W. Cheng is supported by HKRGC under grant GRF 612107; Y. Tao is supported by HKRGC under grants GRF
1202/06, GRF 4161/07, and GRF 4173/08; and K. Yi is supported by Hong Kong Direct Allocation Grant (DAG07/08).
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it is not necessary. The generally agreed semantics for querying uncertain data is the thresholding
approach [13, 16], i.e., for a particular threshold τ, retrieve all the tuples that appear in the query
range with probability at least τ. This problem turns out to be nontrivial even in one dimension. The
na¨ıve approach of examining each point one by one and computing its probability of being inside
the query is obviously very expensive. Note that the independence assumption among the uncertain
points is irrelevant as far as range queries are concerned.
Problem definition. We now define our problem more formally. Let P = {p
1
, . . . , p
n
} be a set of
n uncertain points in R, where each p
i
is specified by its probability density function (pdf) f
i
: R →
R
+
∪{0}. We assume that each f
i
is a piecewise-uniform function, i.e., a histogram, consisting of at
most s pieces for some integer s ≥ 1. In practice, such a histogram can be used to approximate any
pdf with arbitrary precision. In some applications each point p
i
has a discrete pdf, namely, it could
appear at one of a few locations, each with a certain probability. This case can also be represented
by the histogram model using infinitesimal pieces around these locations, so the histogram model
also incorporates the discrete pdf case. We will adopt the histogram model by default throughout
the paper. For simplicity, we assume s to be a constant for most of the discussion. Some of our
structures also support more complicated pdf’s (such as Gaussian or piecewise algebraic), and we
will explicitly say so for these structures.
Given the set P and the associated pdf’s, the goal is to build a data structure on them so that for
a query interval I and a threshold τ , all points p such that Pr[p ∈ I] ≥ τ are reported efficiently.
We also consider the version where τ is fixed in advance. We refer to the former as the variable
threshold version and the latter as the fixed threshold version of the problem. The latter version is
useful since in many applications the threshold is always fixed at, say, 0.5. Moreover, the user can
often tolerate some error ε in the probability. In this case we can build 1/ε fixed-threshold structures
with τ = ε, 2ε, . . . , 1, so that a query with any threshold can be answered with error at most ε.
Applications. The problem of range searching over uncertain data was first introduced by Cheng
et al. [13] and has numerous applications in practice. For example, a certain measurement, say
temperature, may be taken by multiple sensors in a sensor network. Due to various imprecision
factors, the readings of these sensors may not be identical, in which case the temperature of a
location can be conveniently modeled as a pdf. In this context, a query in our problem would
retrieve “all the locations whose temperatures are between 100 and 120 degrees with probability
at least 50%”. It is not hard to see that there are many similar scenarios involving uncertain data.
In fact, our problem is also important even in several traditional applications where no uncertainty
seems to exist. For instance, consider a movie rating system (such as the one at Amazon) where
each reviewer can give a rating from 1 to 10. A query of our problem would find “all the movies
such that at least 90% of the ratings it receives are at least 8”.
Previous results. The problem of range searching on uncertain data has received much attention
in the database community over the last few years. The earliest work [13] considered the above
problem in a simpler form, namely, where each f
i
(x) is a uniform distribution — a special case of
our definition in which the histogram consists of only one piece. For the fixed-threshold version
with threshold 0 < τ ≤ 1, they proposed a structure of O(nτ
−1
) size with O(τ
−1
log n + k) query
time, where k is the output size. These bounds depend on τ
−1
, which can be arbitrarily large.
This structure does not extend to histograms consisting of two or more pieces. They presented
heuristics for the variable threshold version without any performance guarantees. Tao et al. [25]
2
considered the problem in two and higher dimensions, and presented some data structures based on
space partitioning heuristics. They prune points whose probability of being inside the query range
is either too low or too high, but the query procedure visits all points of P in the worst case. Finally,
yet another heuristic is presented in [22], but it is still the same as a sequential scan in the worst
case.
Cheng et al. [13] also showed that the fixed-threshold version of the problem is at least as
difficult as 2D halfplane range-reporting (i.e., report all points lying in a query halfplane), and that
it can be reduced to 2D simplex queries (report all points lying in a query triangle). However
the complexities of these two problems differ significantly: With linear space, a halfplane range-
reporting query can be answered in time Θ(log n + k) [11], while the latter takes Ω(
√
n) time [12].
So there is a significant gap between the current upper and lower bounds for range searching over
uncertain data.
Also related is the work by Singh et al. [24], who considered the problem of querying uncertain
data that are categorical, namely, each random object takes a value from a discrete, unordered
domain. The structures presented there are again heuristic solutions.
Our results. In this paper, we make a significant theoretical step towards understanding the com-
plexity of range searching on uncertain data. We present linear or near-linear size data structures for
both the fixed and variable threshold versions of the problem, with logarithmic or polylogarithmic
query times. Specifically, we obtain the following results.
For the fixed-threshold version, we present a linear-size structure that answers a query in O(log n+
k) time (Section 2). These bounds are clearly optimal (in the comparison model of computation).
We first show that this problem can be reduced to a so-called segments-below-point problem: stor-
ing a set of segments in R
2
so that all segments lying below a query point can be reported quickly.
Then we present an optimal structure for the segments-below-point problem — a linear-size struc-
ture with O(log n + k) query time. This result shows that the fixed-threshold version has exactly
the same complexity as the halfplane range-reporting problem, closing the large gap left in [13]. In
Section 3 we present a simpler structure of size O(nα(n) log n) and query time O(log n + k). This
structure extends to more general pdf’s, such as Gaussian distributions or other piecewise algebraic
pdf’s.
For the variable-threshold version, we use a different reduction and show that it can be solved
by carefully storing a number of points in R
3
in a structure for answering halfspace range queries.
Combining with the recent result of Afshani and Chan [1] for 3D halfspace range reporting, we
obtain a structure for the variable-threshold version of our problem with O(n log
2
n) space and
O(log
3
n + k) query time (Section 4). Although the bounds have extra log factors in this case, our
result shows that this problem is still significantly easier than 2D simplex queries.
Finally, we show that our structures can be dynamized, supporting insertions and deletions of
(uncertain) points with a slight increase in the query time.
2 Fixed-Threshold Range Queries
We present an optimal structure for answering range queries on uncertain data where the probability
threshold τ is fixed. Our structure uses linear space and answers a query in the optimal O(log n+k)
time. These bounds do not depend on the particular value of τ. We first describe in Section 2.1
the reduction to the segments-below-point problem. Next we describe a segment-tree based data
3
x x x
f(x) F (x)
g(x)
a b c d e a b c d e a b c d e
b
c
d
e
(i)
x
l
x
r
(ii)
(iii)
(x
l
, x
r
)
Figure 1: Reduction to the segments-below-point problem: (i) pdf, (ii) cdf, and (iii) threshold
function.
structure that uses linear space and answers a query in O(
√
n + k) time or uses O(n log n) space
and answers a query in O(log n + k) time (Section 2.3). We then improve this structure to achieve
linear size and O(log n + k) query time simultaneously (Section 2.4). We conclude this section by
describing how we make the structure dynamic.
2.1 A geometric reduction
Let p be an uncertain point in R, and let f : R → R be its pdf.
1
Suppose the histogram of f consists
of s pieces, and let
f(x) = y
i
, for x
i−1
≤ x < x
i
, i = 1, . . . , s.
We set x
0
= −∞, x
s
= ∞, and y
1
= y
s
= 0; see Figure 1 (i). The cumulative distribution
function (cdf) F (x) =
R
x
−∞
f(t)dt is a monotone piecewise-linear function consisting of s pieces;
see Figure 1 (ii). Let the query range be [x
l
, x
r
]. The probability of p falling inside [x
l
, x
r
] is
F (x
r
) −F (x
l
). We define a function g : R → R, which we refer to as the threshold function. For a
given a ∈ R, let g(a) be the minimum value b such that F (b) − F (a) ≥ τ. If no such b exists, g(a)
is set to ∞; see Figure 1 (iii).
Lemma 2.1 The function g(x) is non-decreasing and piecewise linear consisting of at most 2s
pieces.
Proof : Suppose we continuously vary x from −∞ to ∞. For x = −∞, g(x) = min{y | F (y) =
τ}; g(x) stays the same until x reaches x
1
. As we increase x further, g(x) increases linearly, with
the slope depending on the pieces of the histogram f that contain x and g(x). When either x or
g(x) passes through one of the x
i
’s, the slope changes. There are at most 2(s − 1) such changes;
see Figure 1.
Given the description of the pdf f, the function g can be constructed easily. Once we have
the threshold function g, the condition Pr[p ∈ [x
l
, x
r
]] ≥ τ simply becomes checking whether
x
r
≥ g(x
l
). Geometrically, this is equivalent to testing whether the point (x
l
, x
r
) ∈ R
2
lies above
the polygonal line representing the graph of g (see Figure 1). We construct the threshold function g
p
for each point p in P . Let S be the set of at most 2ns segments in R
2
that form the pieces of these n
functions; S can be constructed in O(n) time. We label each segment of g
p
with p. The problem of
1
Through this paper we do not distinguish between a function and its graph.
4
1
2
3
4
5
6
7
Figure 2: The data structure for a set of lines: thick polygonal chain is the lower envelope of S;
L
1
(S) = {1, 2, 6}, L
2
(S) = {3, 7}, L
3
(S) = {4, 5}.
reporting the points of P that lie in the interval [x
l
, x
r
] with probability at least τ becomes reporting
the segments of S that lie below the point (x
l
, x
r
) ∈ R
2
: If the procedure returns a segment labeled
with p, we return the point p. Each polygonal line being x-monotone, no point is reported more
than once.
We thus have the following problem at hand: Let S be a set of n segments in R
2
. Build a
data structure on S so that for a query point q ∈ R
2
, the set of segments in S lying directly below
q, denoted by S[q], can be reported efficiently. For simplicity, we assume the coordinates of the
endpoints of S to be distinct; this assumption can be removed using standard techniques. We call
this problem the segments-below-point problem.
2.2 Half-plane range reporting
We begin by describing a structure for the special case when all segments in S are full lines and
we want to report the lines of S lying below a query point. This problem is dual to the well-known
half-plane range reporting problem, for which there is an O(n)-size structure with O(log n + k)-
time [11]. We briefly describe a variant of this structure (in the dual setting), denoted by H(S),
which we will use as a building block.
If we view each line ` in S as a linear function ` : R → R, then the lower envelope of S is the
graph of the function E
S
(x) = min
`∈S
`(x), i.e., it is the boundary of the unbounded region in the
planar map induced by S that lies below all the lines of S (see Figure 2). We represent the lower
envelope as a sequence x
0
= −∞, `
1
, x
1
, `
2
, . . . , `
r
, x
r
= +∞, where the x
i
’s are the x-coordinates
of the vertices of the lower envelope, and `
i
is the line that appears on the lower envelope in the
interval [x
i−1
, x
i
]. Note that the lines appear along the envelope in decreasing order of their slopes.
We partition S into a sequence L
1
(S), L
2
(S), . . ., of subsets, called layers. L
1
(S) ⊆ S consists of
the lines that appear on the lower envelope of S. For i > 1, L
i
(S) is the set of lines that appear
on the lower envelope of S \
S
i−1
j=1
L
j
(S); see Figure 2. For each i, we store the aforementioned
representation of layer L
i
(S) in a list. To answer a query q = (q
x
, q
y
), we start from L
1
(S) and
locate the interval [x
i−1
, x
i
] that contains q
x
, using binary search. Next we walk along the envelope
of L
1
(S) in both directions, starting from `
i
, to report the lines lying below q, in time linear to the
output size. Then we query the rest of the layers L
2
(S), L
3
(S), . . . in order until no lines have been
reported at a certain layer. By using fractional cascading [10] on the x-coordinates of the envelopes
of these layers, the total query time can be improved to O(k) plus the initial binary search in L
1
(S).
Fractional cascading augments these lists with copies of elements from other lists, but the size of
the structure remains linear, and it can be constructed in O(n log n) time [10, 11]. The following
5