Indexing Uncertain Spatio-Temporal Data
Tobias Emrich
1
, Hans-Peter Kriegel
1
, Nikos Mamoulis
2
,
Matthias Renz
1
, Andreas Züfle
1
1
Institute for Informatics, Ludwig-Maximilians-Universität-München
2
Department of Computer Science, University of Hong Kong
{emrich, kriegel, renz, zuefle}@dbs.ifi.lmu.de, nikos@cs.hku.hk
ABSTRACT
The advances in sensing and telecommunication technologies allow
the collection and management of vast amounts of spatio-temporal
data combining location and time information. Due to physical and
resource limitations of data collection devices (e.g., RFID readers,
GPS receivers and other sensors) data are typically collected only
at discrete points of time. In-between these discrete time instances,
the positions of tracked moving objects are uncertain. In this work,
we propose novel approximation techniques in order to probabilis-
tically bound the uncertain movement of objects; these techniques
allow for efficient and effective filtering during query evaluation
using an hierarchical index structure. To the best of our knowl-
edge, this is the first approach that supports query evaluation on
very large uncertain spatio-temporal databases, adhering to possi-
ble worlds semantics. We experimentally show that it accelerates
the existing, scan-based approach by orders of magnitude.
Categories and Subject Descriptors
H.3.3 [INFORMATION STORAGE AND RETRIEVAL]: Infor-
mation Search and Retrieval
Keywords
Uncertain Spatio-Temporal Data, Uncertain Trajectory, Indexing
1. INTRODUCTION
Efficient management of large collections of spatio-temporal data
pertaining to mobile entities whose locations change over time is
paramount in a large variety of application domains: military ap-
plications, structural and environmental monitoring, disaster/rescue
management and remediation, Geographic Information Systems (GIS),
Location-Based Services (LBS). The technological enabling fac-
tors for such applications are advances in sensing and communica-
tion/networking, along with the miniaturizations of the computing
devices and development of embedded systems. In almost every
application domain, the location data at different (discrete) time-
instants is obtained via some positioning devices, like GPS-enabled
mobile devices, RFID or road-side sensors. In addition, to reduce
the communication cost, improve the bandwidth utilization, and
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Indexing Uncertain Spatiotemporal Data
r
epresenting uncertain spatiotemporal data
r
epresenting
uncertain
spatiotemporal
data
ace
Q
o
cation spl
o
t
t
t
t
temporalspace
movements between observations are usually modeled by beads
t
s
t
e
t
i
t
j
movements
between
observations
are
usually
modeled
by
beads
theprobabilityofanobjecttobelocatedoutsideabeadateachpointoftime=0
Figure 1: Spatio-Temporal Data.
cope with storage constraints, often the recorded object trajectories
undergo simplification, eliminating some recorded values. Hav-
ing object trajectories sampled only at discrete time-instants and/or
simplified, renders the movement in-between samples uncertain
and query evaluation challenging.
Consider an object o, moving in a one-dimensional space, as il-
lustrated in Figure 1. Having complete information about this tra-
jectory enables answering a query asking whether the object inter-
sects a spatio-temporal window Q. However, this task becomes
difficult if only a few positions (at times {t
s
, t
i
, t
j
, t
e
} in the ex-
ample) of the exact trajectory are recorded. A simple interpolation
approach, which connects temporally consecutive observations by
line segments and assumes a movement with constant direction and
speed between these points, is unacceptable for applications where
probabilistic analysis of the uncertain movement is required. When
taking uncertainty under consideration, the main challenge is that
the space of possible (location, time) positions between two ob-
servations can grow very large. More importantly, the number of
possible trajectories between two observed locations explodes.
A common method to approximate possible locations between
two observations is the beads (or necklace) model ([11, 22]). This
model is based on some constraints about the motion of an ob-
ject. In particular, assuming a maximum speed in each direction of
each dimension, the possible locations that an object can visit be-
tween two exact observations is bounded. Recent work [8] follows
the pragmatic assumption that the uncertain movement of an object
between consecutive observations can be described by a Markov-
Chain model, which captures the time dependencies between con-
secutive locations. [8] shows how the space of possible worlds
(i.e., trajectories between consecutive observations) can be effi-
ciently analyzed by multiplying Markov-Chain transition matrices
and that probabilistic query evaluation can be facilitated by inte-
grating pruning mechanisms into the Markov Chain matrices. All
these are sufficient for the case where there are few queried objects,
following similar movements; however, if there is a large number
of objects in the database, with different movements, evaluating a
probabilistic spatio-temporal query directly against each object in-
dividually (i.e., a scan-based approach) would be very expensive.
In this paper, we propose an indexing framework to efficiently
cope with large spatio-temporal data bases. Our work also assumes
that the movement of each object follows a Markov-Chain model
(described in Section 2). The objective of our index (described in
Section 4) is to minimize the number of objects for which exact
probabilistic evaluation has to be performed. To achieve this goal,
in Section 3, we propose a number of uncertain spatio-temporal
(UST) object approximations, which are stored in the index and a
set of corresponding pruning methods, which use the approxima-
tions to efficiently eliminate objects that may not qualify a given
probabilistic query. Section 5 presents an extensive experimental
evaluation, which demonstrates the effectiveness of our indexing
approach. Related work is discussed in Section 6. Finally, Section
7 concludes the paper.
2. PRELIMINARIES
This section formally defines the type of spatio-temporal data
that we index, the stochastic model that we use for uncertain trajec-
tories derived from the data, and the query types that we handle.
Data: We consider a discrete time and space domain, i.e., the com-
mon assumption of many existing works (e.g. [18, 1, 10, 8]), where
S = {s
1
, ...s
|S|
} ⊆ R
D
is a finite set of possible locations, which
we call states, in a D-dimensional space and T = N
+
0
is the time
domain. Given this spatio-temporal domain, the (certain) move-
ment of an object o corresponds to a trajectory represented as func-
tion o : T → S of time defining the location o(t) ∈ S of o at a
certain point of time t ∈ T . We consider incomplete (and/or im-
precise) spatio-temporal data, where the motion of an object is not
recorded by a crisp trajectory. Instead, we are only given a set
o.T
obs
of (location, time) observations for each object o. At any
time t /∈ o.T
obs
, the position of o is uncertain, i.e., a random vari-
able. In many applications, a stochastic model can be built and used
to infer knowledge about this uncertainty.
Uncertain Data Model: We refer to the spatio-temporal approxi-
mation of a trajectory of an object o in a time interval spanned by
two consecutive observations of o at t
i
and t
j
as a bead or dia-
mond (o, t
i
, t
j
). The diamond can be computed by considering
the maximum and minimum (singed) velocities of the object in
each dimension [25]. The whole approximation of the trajectory
based on a set T
obs
of observations (i.e., a chain of beads) is re-
ferred to as a necklace. For example, the movement of the object in
Figure 1 is bounded by a chain of diamonds.
Existing studies on modeling uncertain trajectories ([23, 25, 24,
26, 15]) naively consider all possible trajectories bounded by a
necklace equi-probable. However, given two consecutive obser-
vations o(t
i
) and o(t
j
) of object o, there are time dependencies
between consecutive locations between o(t
i
) and o(t
j
), which ren-
der some locations in the corresponding diamond (e.g., those near
the line segment that connects o(t
i
) and o(t
j
)) more probable to be
visited by o than others (e.g., those near the boundary of the dia-
mond). Therefore, these models possibly yield incorrect inferences
resulting in incorrect answers to probabilistic queries. To over-
come this problem, in our proposal each uncertain spatio-temporal
(UST) object o ∈ D is associated with an uncertain object trajec-
tory, which is represented by a stochastic process. The stochastic
process assigns to each t ∈ T exactly one random variable; random
variables at consecutive time moments can be mutually dependent.
This dependency is vital in most applications, where the locations
of an object at two close points of time are highly correlated. Thus,
the uncertain trajectory o(t) of object o comprises a set of (crisp)
trajectories, each assigned with a probability indicating its likeli-
hood to be the true trajectory of o. Thereby, each trajectory with
a non-zero probability is called a possible world of o. Assuming
that objects are mutually independent, our semantics comply with
the classic possible worlds model [6]. If t ∈ o.T
obs
(i.e., the exact
location of o at time t has been observed), then o(t) corresponds to
a (trivial) random variable having one possible location (i.e., state)
with probability 1.
The main challenge in answering a probabilistic spatio-temporal
query is to correctly consider the possible worlds semantics in the
model. In other words, the query results should comply with the
possible worlds model. Naively, this can be done by evaluating the
query predicate on each possible world, and summing up the prob-
abilities of possible worlds satisfying the query predicate. In gen-
eral, the number of possible worlds to be considered is O(|S|
T
);
exhaustively examining all of them requires exponential time, even
for finite time and space domains. Clearly, any naive approach that
enumerates all possible worlds, is not feasible.
In this work, we model the uncertain movement of an object
within a diamond, using a first-order Markov-chain model. This
approach models the movement between successive points in time,
based on background knowledge (e.g., physical laws) and has proven
capable of effectively capturing the behavior of real objects in prac-
tice. For instance, [1] and [10] show how Markov-models can ef-
fectively capture the movement of vehicles on road networks for
prediction purpose. In [18], it is shown that Markov-models can
also be used to model the indoor movement of people, as tracked
in RFID applications.
DEFINITION 1. A stochastic process o(t), t ∈ T , is called a
Markov-Chain if and only if
∀t ∈ N
0
∀s
j
, s
i
, s
t−1
, ...s
0
∈ S :
P (o(t + 1) = s
j
|o(t) = s
i
, o(t − 1) = s
t−1
, ..., o(0) = s
0
) =
P (o(t + 1) = s
j
|o(t) = s
i
),
where the conditional probability
o.P
i,j
(t) := P (o(t + 1) = s
j
|o(t) = s
i
)
is the (single-step) transition probability of object o from state s
i
to state s
j
at time t. The matrix o.M(t) := (o.P
i,j
)
i,j
is called
transition matrix. Let o.P (t) = (p
1
, . . . , p
|S
) be the distribution
vector of an object o at time t, where p
i
corresponds to the proba-
bility that o is located at state s
i
at time t. The distribution vector
o.P (t + 1) can be inferred from o.P (t) as follows:
o.P (t + 1) = o.P (t) · o.M(t)
Queries: Within the scope of this paper, we focus on selection
queries specified by the following parameters: (i) a spatial window
S
2
⊆ S, (ii) a contiguous time window T
2
⊆ T , and (iii) a prob-
ability threshold τ . In the remainder, we use Q
2
= S
2
× T
2
to
denote the search space of a query. The most intuitive definition of
a probabilistic spatio-temporal query is given below:
DEFINITION 2. [Probabilistic Spatio-Temporal τ Exists Query]
Given a query window S
2
in space and a query window T
2
in
time, a probabilistic τ spatio-temporal exists query (PSTτ ∃Q), re-
trieves all objects o ∈ D such that P (∃t ∈ T
2
: o(t) ∈ S
2
) ≥ τ;
i.e., the trajectory of o intersects the query window Q
2
with prob-
ability at least τ.
For example, consider the trajectory of Figure 1 and assume that
we only know for certain its observed locations at {t
s
, t
i
, t
j
, t
e
};
a PSTτ∃Q query defined by rectangle Q
2
would return the de-
picted trajectory, only if the probability that the trajectory inter-
sects S
2
at any time within T
2
exceeds τ. Another query type is
the Probabilistic Spatio-Temporal τ ForAll Query ([8]), denoted as
(PSTτ∀Q), which requires an object to remain in the spatial win-
dow S
2
for the whole time window T
2
. Due to space constraints,
we will not discuss this query in this work, but note that our tech-
niques proposed for PSTτ∃Q queries can easily be adapted.
By modeling the movement within a diamond using a Markov-
chain model, the true probability that an object o satisfies a PSTτ∃
query, can be computed in PTIME [8], exploiting that the matrix
M is generally sparse (only a few states are directly connected to
a single state). Still, query evaluation remains too expensive over
a large spatio-temporal database, where we have to compute the
qualification probabilities of all objects. In view of this, we define
a set of approximations of uncertain object trajectories enabling
spatio-temporal and probabilistic filtering in Section 3. We then
show in Secion 4 how we can organize these approximations in an
index in order to perform efficient query evaluation.
3. APPROXIMATING UST-OBJECTS
In this section, we introduce (conservative) spatio-temporal as
well as probabilistic (conservative) spatio-temporal (UST-) object
approximations, which serve as building blocks for our proposed
index, to be presented in Section 4.
3.1 Spatio-Temporal Approximation
To bound the possible locations of an object o between two sub-
sequent observations (o(t
i
), o(t
j
)), we need to determine all state-
time pairs (s, t) ∈ S × T, t
i
≤ t ≤ t
j
such that o has a non-zero
probability of being at state s at time t. This is done by consider-
ing all possible paths between state o(t
i
) at time t
i
and state o(t
j
)
at time t
j
. An example of a small set of such paths is depicted
in Figure 2(a). Here, we can see a set of five possible trajecto-
ries of an object o, i.e., all possible (state, time) pairs of o in
the time interval [t
i
, t
j
]. In practice, the number of possible paths
becomes very large. Nonetheless, we can efficiently compute the
set of possible (state, time) pairs using the Markov-chain model:
The set of state-time pairs S
i
reachable from o(t
i
) can be com-
puted by performing t
j
− t
i
transitions using the Markov chain
o.M(t) of o, starting from state o(t
i
) and memorizing all reachable
(state, time) pairs. Similarly, we can compute S
j
as all state-time
pairs (s, t) ∈ S × T , t
i
≤ t ≤ t
j
such that o can reach state o(t
j
)
at time t
j
by starting from state s at time t. S
j
can be computed in
a similar fashion, starting from state o(t
j
) and using the transposed
Markov chain o.M(t)
T
. The intersection S
i,j
= S
i
∩ S
j
yields all
state-time pairs which are consistent with both observations. Let us
note that in practice, it is more efficient to compute S
i
and S
j
in a
parallel fashion, to reduce the explored space. When the computa-
tion of S
i
and S
j
meet at some time t
i
≤ t ≤ t
j
, we can prune any
states which are not reachable by both s(t
i
) at time t
i
and s(t
j
) at
time t
j
. However, the number |S
i,j
| of possible state-time pairs in
S
i,j
can grow very large, so it is impractical for our index structure
(proposed in Section 4) to store all S
i,j
for each o ∈ DB in our
index structure. Thus, we propose to conservatively approximate
S
i,j
. The issue is to determine an appropriate approximation of
S
i,j
which tightly covers S
i,j
, while keeping the representation as
simple as possible. The basic idea is to build the approximation by
means of both object observations o(t
i
) and o(t
j
) with the corre-
sponding velocity of propagation in each dimension. To do so, we
first compute for the set of state-time pairs S
i
to derive the maxi-
mum and minimum possible velocity in the time interval [t
i
, t
j
]:
v
0
d
:= max
(s,t)∈S
i
(
s[d] − o(t
i
)[d])
t − t
i
)
v
6
d
:= min
(s,t)∈S
i
(
s[d] − o(t
i
)[d])
t − t
i
)
x
d
d
v
d
v
d
<
>
o(t
i
)
o(
t
)
d
o(
t
j
)
v
d
v
d
<
>
t
t
i
t
j
(a) Approximating trajectories
x
i
S
ij
(
)
S
i
,
j
(
o,t
i
,t
j
)
(
ot
i
t
j
)
t
i
t
j
(
o
,
t
i
,
t
j
)
t
(b) Diamond vs MBR
Figure 2: Spatio-Temporal Approximation.
where s[d] (o(t
i
)[d]) denotes the projection of state s (o(t
i
)) to
the d-th dimension. By definition, we can guarantee that for any
t
i
≤ t ≤ t
j
it holds that
o(t)[d] ≤ o(t
i
)[d] + (t − t
i
) · v
0
d
and
o(t)[d] ≥ o(t
i
)[d] + (t − t
i
) · v
6
d
Furthermore, we bound the velocity of propagation at which o
can have reached state o(s
j
) at time t
j
from each location in the
state-space S
j
:
v
1
d
:= max
(s,t)∈S
j
(
o(t
j
)[d] − s[d]
t
j
− t
)
v
>
d
:= min
(s,t)∈S
j
(
o(t
j
)[d] − s[d]
t
j
− t
)
Again, we can bound the position of o in dimension 1 ≤ d ≤ D
at time t
i
≤ t ≤ t
j
as follows:
o(t)[d] ≤ o(t
j
)[d] − (t
j
− t) · v
1
d
, and
o(t)[d] ≥ o(t
j
)[d] − (t
j
− t) · v
>
d
In summary, using the positions o(t
i
) at time t
i
and o(t
j
) at time
t
j
, and using velocities v
0
d
, v
6
d
, v
>
d
, v
1
d
, we can bound the random
variable of the position o(t) of o at time t
i
≤ t ≤ t
j
by the interval
o(t)[d] ∈ I
d
(t) := [max(o(t
i
)[d]+(t−t
i
)·v
6
d
, o(t
j
)[d]−(t
j
−t)·v
>
d
)),
min(o(t
i
)[d] + (t − t
i
) · v
0
d
, o(t
j
)[d] − (t
j
− t) · v
1
d
)] (1)
Deriving these intervals for each dimension, yields an axis-parallel
rectangle, approximating all possible positions of o at time t. In the
following, we will call this time dependent spatial approximation of
o(t) in the time interval [t
i
, t
j
] between two observations o(t
i
) and
o(t
j
) a spatio-temporal diamond, denoted as (o, t
i
, t
j
). A nice ge-
ometric property of this approximation is that computing the inter-
section with the query window at each time t is very fast. Another
advantage is that existing spatial access methods (e.g., R-trees) can
be easily used to efficiently organize these approximations. To
store the approximation, we only need to store the 4 · D real val-
ues v
0
d
, v
6
d
, v
1
d
, v
>
d
, 1 ≤ d ≤ D. A diamond is reminiscent to a
time-parameterized rectangle, used to model the worst-case MBR
for a set of moving objects in [19]; however, the way of deriving
velocities is different in our case. As an example, Figure 2(a) shows
for one dimension d ∈ D, positions o(t
i
) at time t
i
and o(t
j
) at
time t
j
. The diamond formed by the velocity bounds v
0
d
, v
6
d
, v
>
d
and v
1
d
conservatively approximates the possible (location, time)
pairs. Note that it is possible to use a minimal bounding rect-
angle 2(o, t
i
, t
j
) instead of the diamond (o, t
i
, t
j
) to conserva-
tively approximate the (location, time) space S
i,j
. In cases, how-
ever, where the movement of an object in one dimension is biased
x
d
d
Q
v
d
v
<
>
v
d
I
cov,d
I
v
d
v
d
<
>
t
I
int,d
(a) query intersection
2
d
3
I
I
3
1
2
I
1
I
2
t
(b) dimension-wise intersections
Figure 3: Intersection between query and diamond
in one direction, a rectangle may yield a very bad approximation
(see Figure 2(b) for an example). Our index employs both ap-
proximations 2(o, t
i
, t
j
) and (o, t
i
, t
j
) for spatio-temporal prun-
ing; 2(o, t
i
, t
j
) is used for high-level indexing and filtering, while
(o, t
i
, t
j
) is used as a second-level filter.
3.2 Spatio-Temporal Filter
Based on the spatio-temporal approximation of an uncertain ob-
ject as described in the previous section, it is possible to perform
filtering during query processing.
If none of the diamonds assigned to an object o ∈ D intersects
the query window, then o is safely pruned. In turn, if a diamond
of o is inside the query window S
2
in space, i.e. fully covered by
S
2
, at any point of time t ∈ T
2
, then o is a true hit and, thus, o
can be immediately reported as result of the query. In order to em-
ploy the above spatio-temporal pruning conditions, for a diamond
(o, t
i
, t
j
) of an object o we need to determine the points of time
when it intersect the query window S
2
in space, as well as the
points of time when (o, t
i
, t
j
) is fully covered by S
2
. For this
purpose, it is helpful to focus on the spatial domain S and interpret
a diamond as well as the query as a time-parameterized (moving)
rectangle. By doing so, we can adapt the techniques proposed in
[19]: In general, a rectangle R
1
intersects (covers) another rectan-
gle R
2
, if and only if R
1
intersects (covers) R
2
in each dimension.
Thus, for each spatial dimension d (d ∈ {1, . . . D}), we compute
the points of time when the extents of the rectangles intersect in that
dimension and the points of time when the extents of the diamond
rectangle are fully covered by the query rectangle S
2
.
For a single dimension d, with Equation 1 the query window
given by Q
2
d
intersects the diamond given by o(t
i
)[d], o(t
j
)[d],
v
0
d
, v
6
d
, v
1
d
and v
>
d
within the points of time
I
int,d
:= {t ∈ (T
2
∩ [t
i
, t
j
]|I
d
(t) ∩ S
2
d
}.
Similarly, Q
2
d
fully covers the diamond within the points of time
I
cov,d
:= {t ∈ T
2
∩ [t
i
, t
j
]|I
d
(t) ⊆ S
2
d
}.
An example is illustrated in Figure 3(a). To compute both sets
I
int,d
and I
cov,d
, we intersect the margins of the diamond with
the query window resulting in a set of time intervals, which subse-
quently have to be intersected accordingly in order to derive I
int,d
and I
cov,d
. Now, let us consider the overall intersection time in-
terval I
int
=
T
D
d=1
I
int,d
(e.g., see Figure 3(b)) and the overall
points of covering time I
cov
=
T
D
d=1
I
cov,d
.
If, for an object o ∈ D, there is no diamond yielding a non-
empty set I
int
, o can be safely pruned. If any diamond of o yields
a non-empty set I
cov
, o can be reported as result.
In summary, the spatio-temporal filter can be used to identify
uncertain object trajectories having a probability of 100% or 0%
intersecting (remaining in) the query region Q
2
. Still, the proba-
bility threshold τ of the query is not considered by this filter. In
addition, the object approximation may cover a lot of dead space
if there exist outlier state-time pairs which determine one or more
of the velocities, despite having a very low probability. In the fol-
lowing, we show how to exclude such unlikely outliers in order to
shrink the approximation, while maintaining probabilistic guaran-
tees that employ the probability threshold τ.
3.3 Probabilistic UST-Object Approximation
We now propose a tighter approximation, based on the intuition
that the set of possible paths within a diamond is generally not uni-
formly distributed: paths that are close to the direct connection be-
tween the observed locations often are more likely than extreme
paths along the edges of the diamond. Therefore, given a query
with threshold τ, we can take advantage of a tighter approximation,
which bounds all paths with cumulative probabilities τ to perform
more effective pruning.
Based on this idea, we exploit the Markov-chain model in order
to compute new diamonds, which are spatio-temporal subregions,
called subdiamonds, of the (full) diamond (o, t
i
, t
j
), as depicted
in Figure 4(a). For each such subdiamond, we will then show how
to compute the cumulative probability of all possible trajectories of
o passing only through this subdiamond. Let us focus on restricting
the diamond at one direction of one dimension; we choose one di-
mension d ∈ D, and one direction dir ∈ {∧, ∨}. Direction ∧ (∨)
corresponds to the two diamond sides v
0
d
and v
1
d
(v
6
d
and v
>
d
). To
obtain the subdiamond, we scale the corresponding sides by a factor
λ ∈ [0, 1] relative to the average velocity v
avg
d
=
o(t
j
)[d]−o(t
i
)[d]
t
j
−t
i
.
We obtain the adjusted velocity values for direction ∧ as follows:
v
0
λ
d
= ((v
0
d
− v
avg
d
) · λ) + v
avg
d
= v
0
d
· λ + v
avg
d
· (1 − λ)
and
v
1
λ
d
= ((v
1
d
− v
avg
d
) · λ) + v
avg
d
= v
1
d
· λ + v
avg
d
· (1 − λ)
The adjusted velocity values for direction ∨ can be computed anal-
ogously. Thus, for a given diamond (o, t
i
, t
j
), dimension d ∈ D,
direction dir ∈ {∧, ∨} and scalar λ ∈ [0, 1], we obtain a smaller
diamond (o, t
i
, t
j
, d, dir, λ), derived from (o, t
i
, t
j
) by scaling
direction dir in dimension d by a factor of λ. Figure 4(b) illus-
trates some subdiamonds for one dimension, the ∧ direction and
for various values of λ.
To use such subdiamonds for probabilistic pruning, we first need
to compute the probability P (inside(o, (o, t
i
, t
j
, d, dir, λ))) that
object o will remain within (o, t
i
, t
j
, d, dir, λ) for the whole time
interval [t
i
, t
j
], in a correct and efficient way. The main challenge
for correctness, is to cope with temporal dependencies, i.e. the fact
that the random variables o(t
i
) and o(t
i
+ δt) are highly corre-
lated. Thus, we cannot simply treat all random variables o(t) as
mutually independent and aggregate their individual distributions.
To illustrate this issue, consider Figure 4(a), where one subdia-
mond is depicted. Assume that each of the five possible trajec-
tories has a probability of 0.2. We can see that three trajectories are
completely contained in the subdiamond, so that the probability
P (inside(o, (o, t
i
, t
j
, d, dir, λ))) that o fully remains in the sub-
diamond (o, t
i
, t
j
, d, dir, λ) is 60%. However, multiplying for all
time instants t ∈ [t
i
, t
j
] the individual probabilities that o is lo-
cated in (o, t
i
, t
j
, d, dir, λ) at time t produces an arbitrarily small
and incorrect result, as time dependencies are ignored. Further-
x
d
d
t
(a) A sub-diamond
x
d
d
λ
1
0
t
(b) Possible sub-diamonds
p
1
λ
0
0
1
(c) linear optimization
x
d
Q
1
Q
d
Q
1
Q
2
c
c
opt
t
(d) Computing λ
opt
p
λ
1
0
0
1
λ
opt
f(λ
opt
)
(e) conservative approx.
Figure 4: Probabilistic Diamonds
more, due to the generally exponential number of possible trajecto-
ries, P (inside(o, (o, t
i
, t
j
, d, dir, λ))) is too expensive to com-
pute by iterating over all possible trajectories. Instead, we compute
this probability efficiently and correctly, as follows.
To compute the probability of possible trajectories between o(t
i
)
at t
i
and o(t
j
) at t
j
that are completely contained in (o, t
i
, t
j
, d, dir, λ),
an intuitive approach is to start at o(t
i
) at time t
i
and perform t
j
−t
i
transitions using the Markov-chain o.M (t). After each transition,
we identify states that are outside (o, t
i
, t
j
, d, dir, λ). Any possi-
ble trajectory which reaches such a state is flagged. Upon reaching
t
j
, we only need to consider possible trajectories in state o(t
j
),
since all other worlds have become impossible due to the observa-
tion of o at t
j
. The fraction of un-flagged worlds at state o(t
j
) at
time t
j
yields the probability that o does not completely remain in
(o, t
i
, t
j
, d, dir, λ).
To formalize the above approach, we first rewrite the probability
P (inside(o, )|o(t
i
), o(t
j
))) that the trajectory of o remains in the
subdiamond ,
1
given the observations o(t
i
), o(t
j
) at times t
i
, t
j
∈
o.T
obs
, using the definition of conditional probability:
P (inside(o, )|o(t
i
), o(t
j
)) =
P (o(t
j
)|inside(o, ), o(t
i
))
P (o
t
j
|o
t
i
)
,
where P (o(t
j
)|inside(o, ), o(t
i
)) denotes the probability that o
reaches the state o(t
j
) observed by observation o(t
j
), given that o,
starting at o(t
i
) at time t
i
remains inside . P (o(t
j
)|o(t
i
)) denotes
the probability that state o(t
j
) at time t
j
is reached, given that o
starts at o(t
i
) at time t
i
, regardless whether o remains in .
3.4 Finding the optimal Probabilistic Diamond
In the previous section, we described, how to compute the prob-
ability of a probabilistic diamond (o, t
i
, t
j
, d, dir, λ) from a dia-
mond (o, t
i
, t
j
), dimension d, direction dir, and scaling factor λ.
In this section we will show how to find, for a given query window
Q
2
and a given query predicate the subdiamond with the highest
pruning power. Let us focus on PSTτ∃ queries first. That is, our
aim is to find a value for d, dir and λ, such that the resulting subdi-
amond (o, t
i
, t
j
, d, dir, λ) does not intersect Q
2
, and at the same
time it has a high probability P (inside(o, )). This probability
can be used to prune o as we will show later. Formally, we want
to efficiently determine
argmax
d∈D,dir∈{∨,∧},λ∈[0,1]
[P (inside(o, (o, t
i
, t
j
, d, dir, λ))]
constrained to Q
2
∩ (o, t
i
, t
j
, d, dir, λ) = ∅.
For a single dimension d, and the north direction, a possible situ-
ation is depicted in Figure 4(d). Here, the projection
d
(o, t
i
, t
j
) of
the full diamond (o, t
i
, t
j
) to the d-th dimension and the projec-
tions Q
2
1
[d] and Q
2
2
[d] of two query windows Q
2
1
and Q
2
2
are de-
1
Since the context is clear, we simply use to denote
(o, t
i
, t
j
, d, dir, λ).
picted. The aim is to find the largest values λ
opt
of λ, such that the
corresponding probabilistic diamond (o, t
i
, t
j
, d, ∧, λ
∃
opt
) which
we call optimal subdiamond, does not intersect Q
2
1
(Q
2
2
). To solve
this problem, we distinguish between the following cases.
Case 1: the direct line between observations (o(t
i
), t
i
) and (o(t
j
), t
j
)
in dimension d intersects Q
2
[d]. In this case, there cannot exist any
λ ∈ [0, 1] such that Q
2
∩ (o, t
i
, t
j
, d, dir, λ) = ∅. Therefore, our
problem has no solution in dimension d, and d is ignored.
Case 2: the direct line between (o(t
i
), t
i
) and (o(t
j
), t
j
) does
not intersect Q
2
[d], and we assume without loss of generality that
Q
2
[d] is located above this line.
2
In addition, in this case, the
time value of the north corner c of
d
(o, t
i
, t
j
) is located in the
interval T
2
(e.g., see Q
2
2
in Figure 4(d)).
3
In this case, the edge
v
0
opt
of the optimal subdiamond (o, t
i
, t
j
, d, dim, λ
∃
opt
) is given
by (o(t
i
), t
i
) and (s, t) where s corresponds to the lower bound of
S
2
[d] and t equals to the time component of c.
Case 3: Q
2
is above the direct line between (o(t
i
), t
i
) and (o(t
j
), t
j
)
(as in Case 2), but the time value of the north corner c of
d
(o, t
i
, t
j
)
is not located in the time interval T
2
(e.g. Q
2
1
of Figure 4(d)). In
this case, the optimal subdiamond must touch a corner of Q
2
[d]
due to convexity of both Q
2
[d] and any diamond. If Q
2
[d] is lo-
cated to the left of c (the right direction is handled symmetrically),
then the edge v
0
opt
of the optimal subdiamond is given by the line
between (o(t
i
), t
i
) and the lower right corner of Q
2
[d] (e.g., see
Figure 4(d)).
The optimal value λ
∃
opt
for cases 2 and 3 equals the quotient
v
0
opt
−v
avg
v
0
−v
avg
, i.e., the fraction of the maximum velocity of the opti-
mal subdiamond and the maximum velocity of the full diamond,
both normalized by the average velocity v
avg
=
s(t
j
)[d]−s(t
i
)[d]
t
j
−t
i
.
After identifying the value for λ
∃
opt
, for a dimension d and a di-
rection dir, we can compute the probability of the corresponding
subdiamond (o, t
i
, t
j
, d, dir, λ
∃
opt
). Since we can guarantee, that
any path in this subdiamond does not intersect the query window,
we can obtain a lower bound
P
LB
(never(o, t
i
, t
j
, Q
2
))=P (inside(o, (o, t
i
, t
j
, d, dir, λ
∃
opt
)))
(2)
of the event that o never intersects the query window in the time
interval [t
i
, t
j
]. This directly yields an upper bound
P
UB
(sometimes(o, t
i
, t
j
, Q
2
)) =
1 − P (inside(o, (o, t
i
, t
j
, d, dir, λ
∃
opt
))) (3)
of the probability that the reverse event that o intersects the query
2
If Q
2
[d] is below the line, we consider direction dir = ∨ sym-
metrically.
3
Corner c is given by the intersection of lines (o(t
i
), t
i
) + v
0
and
(o(t
j
), t
j
) + v
1
.