Travel Time Estimation of a Path using Sparse Trajectories
Yilun Wang
1,2,*
, Yu Zheng
1,+
, Yexiang Xue
1,3,*
1
Microsoft Research, No.5 Danling Street, Haidian District, Beijing 100080, China
2
College of Computer Science, Zhejiang Univeristy
3
Department of Computer Science, Cornell University
{v-yilwan, yuzheng}@microsoft.com, yexiang@cs.cornell.edu
ABSTRACT
In this paper, we propose a citywide and real-time model for
estimating the travel time of any path (represented as a sequence of
connected road segments) in real time in a city, based on the GPS
trajectories of vehicles received in current time slots and over a period
of history as well as map data sources. Though this is a strategically
important task in many traffic monitoring and routing systems, the
problem has not been well solved yet given the following three
challenges. The first is the data sparsity problem, i.e., many road
segments may not be traveled by any GPS-equipped vehicles in
present time slot. In most cases, we cannot find a trajectory exactly
traversing a query path either. Second, for the fragment of a path with
trajectories, they are multiple ways of using (or combining) the
trajectories to estimate the corresponding travel time. Finding an
optimal combination is a challenging problem, subject to a tradeoff
between the length of a path and the number of trajectories traversing
the path (i.e., support). Third, we need to instantly answer users’
queries which may occur in any part of a given city. This calls for an
efficient, scalable and effective solution that can enable a citywide and
real-time travel time estimation. To address these challenges, we
model different drivers’ travel times on different road segments in
different time slots with a three dimension tensor. Combined with
geospatial, temporal and historical contexts learned from trajectories
and map data, we fill in the tensor’s missing values through a context-
aware tensor decomposition approach. We then devise and prove an
object function to model the aforementioned tradeoff, with which we
find the most optimal concatenation of trajectories for an estimate
through a dynamic programming solution. In addition, we propose
using frequent trajectory patterns (mined from historical trajectories)
to scale down the candidates of concatenation and a suffix-tree-based
index to manage the trajectories received in the present time slot. We
evaluate our method based on extensive experiments, using GPS
trajectories generated by more than 32,000 taxis over a period of two
months. The results demonstrate the effectiveness, efficiency and
scalability of our method beyond baseline approaches.
Categories and Subject Descriptors
H.2.8 [Database Management]: Database Applications - data
mining, Spatial databases and GIS;
Keywords
Travel time estimation; tensor; trajectories; urban computing;
1. INTRODUCTION
Real-time estimation of the travel time of a path, which is represented
by a sequence of connected road segments, is of great importance for
traffic monitoring [1], finding driving directions [20], ridesharing [13]
and taxi dispatching [22]. Existing solutions, e.g., using loop sensors,
usually tell people the travel speed of an individual road segment
rather than the travel time of an entire path. The latter’s value is not a
simple summation of the travel time of each individual road segment,
as a path also contains road intersections (sometimes with traffic
lights) where a driver needs to slow down or wait for a while.
Explicitly modeling the time delay at an intersection is not easy [8]. In
addition, these methods have limited coverage, as many streets do not
have a loop sensor embedded.
An alternative method is to use floating car data (e.g., GPS trajectories
of vehicles) to estimate the travel time of a path. For example, as
shown in Figure 1, we estimate the travel time of path
, using four trajectories
,
,
, and
. Unfortunately,
there are three major issues remaining unsolved in existing methods.
They are as follows:
Figure 1. Problem demonstration
1) Data sparsity: For example,
is not traversed by any trajectory in
the previous 30 minutes. Using an average of
’s historical travel
times is not accurate enough (since its traffic conditions change over
time of day and day of the week). Sometimes, the road may never be
traversed by any trajectories (even in history) in our dataset, as in
practice we only have the data of a sample of vehicles.
2) Trajectory concatenation: For the sub-path (e.g.,
)
with trajectories, how to combine these trajectories effectively to
achieve an accurate estimate is still a challenging problem. Clearly,
there are multiple ways of using the four trajectories shown in Figure
1. For instance, we can calculate the travel time of
solely based on
. Or, we can compute the travel time for
(based
on
and
),
(based on
,
and
), and
(using
,
and
), separately. Later, the travel time of
can
be obtained by summing the travel times of each road segment. We
can also use
and
to estimate the travel time of
, then
concatenating it with that of
; or, do
first based on
and
, then concatenating it with
.
Different concatenations have their own advantages and
disadvantages, subject to a trade-off between their support and
length. The ideal situation is to estimate the travel time of
using many trajectories like
covering the entire
path. Such trajectories reflect the traffic conditions of an entire
path, including intersections, traffic lights and direction turns,
hence, no need to model these complex factors separately and
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+ Yu Zheng is the correspondence author of this paper.
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KDD’14, August 24–27, 2014, New York, New York, USA.
Copyright © 2014 ACM 978-1-4503-2956-9/14/08…$15.00
http://dx.doi.org/10.1145/2623330.2623656
*The paper was done when the first and third authors were interns in
Microsoft Research under the supervision of the second author who
contributed the main idea and algorithms of this paper.
explicitly. However, as the length of a path increases, the number
of trajectories (i.e., the support) traveling on the path decreases
(refer to Figure 10 A) for details). Consequently, the confidence
of the travel time (derived from few drivers) decreases. For
example, what if
is generated by an uncommon driver or in an
unusual situation like pedestrians crossing a street? Furthermore,
in many cases, we cannot even find a trajectory passing an entire
path. On the other hand, using the concatenation of shorter sub-
paths can have more occurrences of trajectories on each sub-path
(i.e., having a high confidence in the derived travel time for each
sub-path). But this results in more fragments, across which the
aforementioned complex factors are difficult to model. The more
fragments a concatenation contains, the more inaccuracy a path’s
travel time could involve.
3) Tradeoff among Scalability, effectiveness and efficiency: As
users can query any path in a city, we need to model the traffic
conditions with a city scale, which usually contains tens of
thousands of road segments. In the meantime, we have to answer
users’ query instantly. So, a good solution should be scalable,
effective and efficient, all simultaneously. This requirement fails
some complex models that work well on a particular road.
In this paper, we propose a model for instant Path Travel Time
Estimation (PTTE), based on sparse trajectories generated by a
sample of vehicles (e.g., some GPS equipped taxicabs) in the
recent time slots as well as in history. Our model is comprised of
two major components. One is to estimate the travel time for road
segments without being traversed by trajectories through a
context-aware tensor decomposition (CATD) approach. The
second is to find the most optimal concatenation (OC) of
trajectories to estimate a path’s travel time using a dynamic
programing solution. Our work has three primary contributions:
Dealing with the missing values: We model different drivers’
travel times on different road segments in different time slots
with a three dimensional tensor. Combined with geospatial,
temporal and historical contexts learned from other data
sources, we fill in the tensor’s missing values through a
context-aware tensor decomposition approach. To expedite
the inference, we partition a city into disjoint geo-regions and
carry out the decomposition for each region in parallel.
Optimal concatenation: We devise and prove an object
function that can model the tradeoff between the support and
length of a concatenation. Using a dynamic programming
solution, we find the most optimal concatenation of
trajectories for estimating a path’s travel time. In addition,
we use frequent trajectory patterns mined in advance to scale
down the candidates of concatenation and propose a suffix-
tree-based index to manage the recently received trajectories,
improving the efficiency of our model.
Evaluation: We evaluate our model with the real trajectories
generated by over 32,000 taxis over a period of 2 month on
Beijing’s road network. The results of extensive experiments
demonstrate the advantages of our model. A sample of the
data has been released at [25].
The rest of the paper is organized as follows: Section 2 overviews
our model. Section 3 elaborates on the method for inferring the
travel time of road segments without trajectories. Section 3
introduces the method that searches for the most optimal
concatenation. Section 4 presents the experiments and Section 5
summarizes related work. We conclude the paper in Section 6.
2. OVERVIEW
Definition 1: Road Network. A road network is comprised of a
set of road segments connected among each other in a graph
format. Each road segment is a directed edge with two terminal
points, a list of intermediate points describing the segment, a
length ., a level . (e.g. a highway or a street), a direction
. (e.g. one-way or bi-directional) and the number of lanes ..
Definition 2: Trajectory. A spatial trajectory is a sequence of
time-ordered points, :
, where each point
has a geospatial coordinate set and a timestamp, ,,.
Definition 3: Path. A path is represented by a sequence of
connected road segments, e.g., :
, in an .
Definition 4: Trajectory pattern. A trajectory pattern is a
sequential pattern of road segments with a support over a
threshold, calculated by the number of trajectories traversing these
road segments. If we set support as 2,
and
in
Figure 1 are trajectory patterns, while
is not eligible.
Definition 5: Concatenation. A path can be decomposed into
different concatenations ( || ) of its sub-paths,
|| ...||
…||
…||
, 1,, ,
.
For instance,
can be formed by
||
, or
||
, or
||
. Thus, the
travel time of can be obtained via the summation of different
concatenations, e.g.,
+
, or
,
or
.
Definition 6: Travel Time. A driver ’s travel time on a road
segment in time slot is defined as
,,
. Likewise,
,,
denotes ’s travel time on path in time slot .
Figure 2. Framework of our model
Figure 2 presents the framework of our model which is comprised of
two major parts. In the above part, we project each trajectory received
in a current time slot onto a road network, using a map-matching
algorithm [21]. The trajectories (combined with road network data)
are then used to construct a 3D tensor
where the three dimensions
stand for road segments, time slots and drivers, respectively. Each
entry is the travel time of a particular driver on a particular road
segment in a specific time slot. We partition a day into several time
slots based on a certain time interval (e.g., we divide a day into 48
time slots with 30 minutes each in the experiments). Clearly, the
tensor is very sparse (i.e., having many entries without values), as a
driver can only travel a few road segments in a time slot. To deal with
the data sparsity problem, we extract three categories of features,
consisting of geospatial, temporal, and historical contexts, from the
road network data and trajectories. The first two feature sets are stored
in two matrices, respectively, and the historical context is represented
by another tensor
. The two matrices and
are then factorized
with
collaboratively, helping fill
’s missing entries in a current
time slot (i.e., inferring the travel time of road segments without being
traveled by trajectories in the current time slot). The general idea is
that road segments with similar contexts could have a similar travel
time. The context matrices and tensor reveal the similarity and with a
more proportion of non-zero entries than
, thereby reducing the
factorization error and improving the inference accuracy. After filling
Map-
Matching
Tensor
Construction
Tensor
Decomposition
Road
Networks
Trajectory
Database
Frequent Trajectory
Pattern Mining
Optim al
Concatenation
Features
Path
Context Feature
Extract ion
Trajectories
Patterns
cost
A
rec
A
r
the missing entries in
, we obtain the travel time of any driver on
any road segment in current time slot (stored in
).
In the bottom part, given a query path , we estimate its travel time in
the current time slot, based on
, the trajectories received in the
time slot and trajectory patterns. Specifically, we devise and prove an
objective function that can represent the tradeoff between the length
and support of a trajectory pattern. Based on the objective function,
we find the most optimal concatenation of trajectories for a path,
using a dynamic programing approach. In practice, it is not necessary
to try every possible concatenation of a path, as some sub-paths have
never been traversed by any trajectory. So, we mine frequent
trajectory patterns from historical trajectories in advance and study the
concatenation of these existing patterns to estimate the travel time of a
path. This reduces the online computational loads significantly, while
guaranteeing accuracy in travel time estimation. Note that we are not
using the historical travel time of a trajectory pattern. The patterns just
provide us with candidate schemes of subpaths for finding an optimal
concatenation of a path. Each trajectory pattern’s travel time in current
time slot is mainly calculated based on the trajectories received in the
time slot. If a pattern contains road segments without being traversed
by trajectories in the current time slot, we retrieve the inferred time
from
, according to the driver, road segment and time slot. For
instance, two drivers (
,
) travelled
, but nobody traveled
in a pattern
, in current time slot . That is,
,
,
, and
,
,
can be calculated from the present
trajectory data, while
,
,
and
,
,
are unknown. In this case,
we retrieve the latter two from
, calculating
,
,
,
,
,
,
, and
,
,
,
,
,
,
.
With
, we can estimate a driver’s travel time on a trajectory
pattern even if the recently received data is incomplete. The
dimension of drivers in
enables us to calculate the variance
among different drivers’ travel times on a road segment or a sub-
path. Intrinsically, different drivers travel the same road segment
with different times, majorly depending on the different traffic
conditions they experience. Thus, the variance implies the
complexity of traffic conditions on a road segment or a sub-path,
helping estimate a more accurate travel time of a path (elaborated
in Section 4.1). Finally, the travel time of a path is calculated as:
∑
∑
,,
||
; (1)
Where Ψ is the concatenation of path , represented by a set of
trajectory pattern s; is a collection of drivers traversing (or
partially traversing) a ; is the current time slot.
3. DEALING WITH MISSING VALUES
3.1 Tensor Building and Feature Extraction
To model the traffic conditions of the current time slot, we
construct a tensor
, with the three dimensions
standing for road segments, drivers and time slots, respectively,
based on the GPS trajectories received in the most recent time
slots and the road network data. As shown in Figure 3, an entry
,,
denotes the th road segment is traveled by the th
driver with a time cost in time slot (e.g., 2-2:30pm). The last
time slot denotes the present time slot, combined with the -1
time slots right before it to formulate the tensor. Clearly, the
tensor is very sparse as a driver can only travel a few road
segments in a short time period. If we were able to fill in the
missing entries in terms of the values of non-zero entries, we can
know the travel time of any driver on any road segment in the
present time slot.
A common approach to this problem is to decompose a tensor into
the multiplication of a few (low-rank) matrices and a core tensor
(or just a few vectors), based on the tensor’s non-zero entries. For
example, we can decompose
into the multiplication of a core
tensor
and three matrices,
,
,
, if using a tucker decomposition model. An
objective function is defined as Equation 2 to control the errors.
,,,
,
(2)
where
·
denotes the
norm and
is a regularization of penalties to avoid over-fitting;
,
, and
are usually very small, denoting the number of latent factors.
is a parameter controlling the contributions of the regularization.
Afterwards, we can recover the missing values in
by
multiplying decomposed factors as
.
Figure 3. The model dealing with data sparsity
In our problem, however, the tensor is over sparse. For example, if
setting 30 minutes as a time slot, only 0.03% entries of
have
values. Decomposing
solely based on its own non-zero entries
is not accurate enough. To this end, we build another tensor
based on the historical trajectories over a long period of time (e.g.
one month). As shown in Figure 3,
has the same structure as
, while an entry
,,
denotes the th driver’s average
travel time on the th road segment in time slot in the history.
Intrinsically,
is much denser than
, denoting the historical
traffic patterns and drivers’ behavior on an entire road network. For
instance, using one-month trajectories and setting 30 minutes as a
time slot, the non-zero entries of
is about 0.4%. Decomposing
and
together reduces the error of supplementing
.
Besides
, we also construct another two matrices and to help
the decomposition of
. Specifically, as illustrated in Figure 4 A),
stores the geographical features
of each road segment, such as
., ., ., ., the number of neighbors (e.g.,
has 2
and 3 neighbors) at its terminals, and a tortuosity ratio (e.g.
.
.
⁄
), as well as the distribution of Point of Interests
(POIs)
around ’s terminals. While captures the similarity
between different road segments in geographic spaces, matrix
(consisting of
and
) represents the correlation between
different time slots in terms of the coarse-grained traffic conditions.
More specifically, we partition a city into disjoint and uniform grids
(e.g., 44 in Figure 4 B), each of which is comprised of many road
segments.
is built based on the recent trajectory data received
from
to
(e.g., 1pm-3pm), reflecting the present traffic
conditions on a road network. An entry of
denotes the number of
vehicles traversing a particular grid in a particular time slot. A row
of
represents coarse-grained traffic conditions in a city of a
particular time slot. Consequently, the similarity of two different
rows indicates the correlation of traffic flows between two time
slots. Additionally, in contrast to using the traffic flow on each
individual road segment in
,
can be filled densely, therefore
can help reduce the error of decomposing
.
has the same
structure as
, storing the historical average number of vehicles
traversing a grid from
to
. In other words,
and
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respectively correspond to the coarse-grained current and historical
traffic conditions in the same span of time of day. In the
implementation, we build
and
of an entire day in advance
and retrieve the entries according to current time (and the number of
time slots needed) when constructing and . For example, as
shown in Figure 4 C), the rows from
to
will be retrieved from
the prebuilt
to construct with
.
Figure 4. Constructing context matrices
3.2 Tensor Decomposition
To achieve a high accuracy of decomposition, we put together
and
(i.e.,
||
, as shown in Figure 3), decomposing
with context matrices and collaboratively. The objective
function is defined as Equation 3,
,,,,,
, (3)
where
and
, denotes the number of
grids;
, denotes the dimension of geographical
features;
,
,
and
are low rank latent factor matrices for time slots, grids, roads and
geographical features. Later, we can recover according to
.
,
, and
are parameters
controlling the contribution of different parts.
In our model, and shares matrix , and and share matrix
. The dense representation of and helps generate a relatively
accurate and , which reduce the decomposition error of in
turn. Additionally, the combination of
and
reveals how
current coarse-grained traffic condition deviates from its historical
patterns. The information of the deviation is then propagated to ,
helping figure out the fine-grained deviation between current traffic
conditions and historical traffic patterns on each road segment. So,
our model considers both geospatial and temporal correlations. It
also incorporates the knowledge from present and historical traffic
data. As there is no closed-form solution for finding the most
optimal result of Equation 3, we use a numeric method, gradient
descent, to find a local optimization, as presented in Figure 5.
Algorithm 1: Tensor Decomposition
Input: tensor , matrix
, and matrix , an error threshold
Output: , , ,
1. Initialize
,
,
,
,
,
with small random values
2. Set as step size
3. While
4. Foreach
0
5.
;
6.
;
7.
;
8.
;
9.
;
10.
;
11.
;
12. Return , , ,
Figure 5. Algorithm for decomposing a tensor
The Symbol “” denotes the matrix multiplication;
stands for
the tensor-matrix multiplication, where the subscript stands for
the direction, e.g.,
is
∑
; is the
tensor outer product (also called Kronecker product);
the entries
of the th row of matrix are represented as
. More
specifically, we use an element-wise optimization algorithm
(instead of batch decomposition) [10], which updates the factors
independently (meaning they can be performed in parallel).
In reality, tensor is very large, given hundreds of thousands of
road segments and tens of thousands of drivers. Decomposing
such a big tensor is very time consuming, therefore reducing the
feasibility of our method in providing online services. To address
this issue, as illustrated in Figure 6, we partition a city into several
disjoint regions, building a tensor for each region based on the
data of the region. The matrices and are built in each smaller
region accordingly. By setting a proper splitting boundary, we try
to keep these small tensors a similar size. As a result, is
replaced by a few small tensors, which will be factorized in
parallel and more efficiently. We validate (in later experiments)
that the partition does not compromise the accuracy of the original
decomposition when choosing a proper number of partitions.
Figure 6. Spatial partition for expediting the tensor decomposition
4. OPTIMAL CONCATENATION (OC)
4.1 Objective Function
Given a path covered by trajectories, we need to find the best
concatenation that results in an accurate travel time estimation.
Intuitively, the best decomposition is the one that achieves the
lowest empirical risk between the estimate and true travel time
.
Suppose is decomposed as
||
||||
, where the estimated
travel time is
, the squared empirical risk is
then wrote as,
,
,
,,
, (4)
Hence, our problem is to search for the best concatenation which
yields the least empirical risk, formally defined as,
argmin
,
,,
,
,
,,
,
subject to
||
||||
. (5)
To come up with a computable form of
,
,
,,
, we relate
,
,
,,
with
, where
is the true travel
time of sub-path
. It is fair to assume if
||
||||
,
then
. Hence we have,
,
,
,,
∑
∑∑
∑
∑∑
If assuming
and
are independent, we have
=0, Therefore,
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B
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,
,
,,
∑
. (6)
Further,
∑
,
∑
,
∑
,
,
, (7)
where
is the number of drivers passing
, and
,
denotes
the th driver’s travel time on
;
,
is the variance of
these drivers’ travel times. Then, Equation 5 can be represented as:
argmin
,
,,
∑
,
subject to
||
||||
(8)
Equation 8 well reflects the aforementioned tradeoff between the
support and length of a concatenation. On one hand, it is easier to
find more drivers traveling a shorter sub-path. The more the
drivers pass a sub-path (i.e. support is higher,
is bigger), the
smaller the error of the inferred travel time is. On the other hand,
the shorter a sub-path is, the bigger the variance in travel time
would be. There are a lot of uncertainties of traveling a short path.
E.g., if only traveling one road segment, the travel time will be
significantly influenced by a traffic light. As a result, different
drivers’ travel times could be dramatically different.
4.2 Dynamic Programing Solution
To solve the optimization problem shown in Equation 8, we
propose a dynamic programing solution. Suppose a path :
,
, , denote
,
, then the optimization problem of can be
represented as Equation 9.
argmin
,
,,
∑
subject to
||
||||
. (9)
Let
be the minimal value of to the above problem, then the
minimal value of the squared empirical risk function of is
.
Additionally, we have a state transition function as Equation 10.
min
||
||
. (10)
Algorithm 2: Query path decomposition
Input: a query path
, a collection of trajectory
pattern s, a time slot , trajectories received in , and tensor
Output: Ψ
, the most optimal concatenation of path
1.
0, Ψ
;
2. For 1 to do
3.
∞; Ψ
;
4. For down to 1 do
5.
;
,,
0;
6. retrieve the drivers traversing (or partially traversing)
from the trajectory database
7. Foreach do
8.
,,
0;
9. Foreach
not traversed by ’s trajectory
10.
,,
,,
;
11.
,,
Calculate the time for the rest of based on ;
12.
,,
,,
;
13.
|
|
,,
;
14. If
15.
;
16.
||;
17. Return Ψ
;
Figure 7. Algorithm for finding the most optimal concatenation
Using Algorithm 2 shown in Figure 7, we solve this problem with a
complexity of
, where is the number of road segments
in and is the number of drivers passing a segment.
In practice, it is not necessary to check every concatenation of a
path, as many sub-paths may not be traversed by any trajectory in
the current time slot. To further improve the efficiency of our
solution, we mine frequent trajectory patterns from historical
trajectories in advance. Then, we just need to check the
concatenation of the trajectories patterns. Specifically, we can stop
the iteration at Line 4 in algorithm 2 if
is not a trajectory pattern.
We use a suffix-tree-based algorithm [18] to find the frequent
trajectory patterns. Specifically, after being map-matched, a
trajectory can be regarded as a string of road segment IDs. By
building a suffix tree, where a node denotes a road segment ID, a
trajectory is then represented as a path on the tree. For example,
the four trajectories shown in Figure 1 can be represented as the
tree depicted in Figure 8 A), where
is the most left
path of the tree.
and
are suffixes of
. The
number associated with each link stands for the number of the
trajectories passing the path (i.e., the support). If setting 2 as a
support, we find
,
, and
are patterns. In reality,
the suffix-tree is built based on historical trajectories over a long
period of time. As long trajectory patterns are very rare, we set the
maximum length of a pattern to 20 road segments.
Figure 8. Mining frequent trajectory patterns and used with tensor
4.3 Working with Tensor
Note that a query path may have some road segments that are not
traversed by any trajectory in the current time slot, though these
segments may belong to a trajectory pattern (in history). Following
the example shown in Figure 1, we demonstrate in Figure 8 B) how
is used with trajectory patterns to help the decomposition of a
query path. To estimate the travel time of a query path :
in time slot , we first search the suffix tree, which was
built based on the trajectory data over a long history (not the one
shown in the left part of Figure 8 A), for the trajectory patterns that
contains, e.g.,
and
. To calculate
defined in Equation 9, we need to know the travel time of each
driver passing
. However,
is not traversed by any
trajectory in time slot . That is
,,
is unknown for every
driver, though
,,
can be calculated based on the recently
received trajectories, i.e.
,
and
. To address this issue, we
retrieve
,
,
,
,
,
, and
,
,
from
and calculate
,,
for
,
,
, respectively, by Equation 11.
,,
,,
,,
, (11)
Having
,,
, we can calculate the most optimal concatenation
according to Equation 9, 10 and Algorithm 2. When the supplement
of an entry is negative, we resort to the historical average travel
time. The dimension of users in tensor
enables us to retrieve a
more accurate travel time for a particular driver, resulting in a better
estimate of the variance of travel times (as Equation 8). We validate
that this is more accurate than just using a historical average of
r
2
r
3
r
3
r
5
r
7
r
6
r
1
r
2
r
3
Root
r
2
r
6
r
7
2
1
1
2
1
1
1
1
1
3
3
1
r
3
r
6
1
2
r
2
→r
3,
r
3
→r
4
(u
2
, u
3
, u
4
)
P: r
1
→r
2
→r
3
→r
4
(1)
(3)
(3)
r
1
r
2
r
N
u
1
u
2
u
M
t
j
t
k
A
rec
(1)
,
,
Tr
2
,Tr
3
,Tr
4
t
r
4
,u
2
,k
t
r
4
,u
3
,k
t
r
4
,u
4
,k
t
r
3
→r
4
,u
2
,k
= t
r
3
,u
2
,k
+t
r
4
,u
2
,k
A) An example of suffix-tree B) Filling in the missing time for a pattern
Patterns:
