Hashing Methods for Temporal Data - Computer Science - Boston ...

10 IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING, VOL. XXX, NO. XX, XXXXXXX 2001
MAXTIME is set to 50,000 for all workloads. The value
used in the R-trees for now was 100,000.
Each workload is described by the distribution used to
generate the object lifespans,the number of different oids,
the total number of changes in the evolution n object
additions and deletions),the total number of object
additions NB,and the total number of queries.
4.3 Experiments
First,the behavior of all implementations was tested using a
basic Uniform workload. The number of lifespans per object
follows a uniform distribution between 20 and 40. The total
number of distinct oids was jUj ˆ8; 000,the number of real
changes n ˆ 466; 854,and NB ˆ 237; 606 object additions.
Hence,the average number of lifespans per oid was
NB 30 we refer to this workload as Uniform-30). The
number of queries was 115,878.
Fig. 3a presents the average number of pages accessed
per query by all methods. The PPLH methods have the best
performance,about two pages per query. The ALH approach
uses more query I/O about 1.5 times in this example)
because of the larger buckets it creates. The MVBT also uses
more I/O about 1.75 times) than the PPLH approaches since
a tree path is traversed per query. The R i uses more I/Os
per query than the MVBT about 11.5 I/Os),mainly due to
tree node overlapping and larger tree height its height
relates to the total number of oid lifespans while MVBT's
height corresponds to the alive oids at the time specified by
the query). The R p tree has the worse query performance an
average of 28.3 I/Os per query). The performance of the R-
tree methods has been truncated in Fig. 3a to fit the graph.
While using a separate dimension for the two endpoints of a
lifespan interval allows for better clustering see also the
space usage in Fig. 3c),it makes it more difficult to check
whether an interval contains a query time instant.
Fig. 3b shows the average number of I/Os per update.
The best update performance was given by the PPLH-s
method. In PPLH-l,the NT array implementation inside
each page limits the actual page area assigned for storing
oids and,thus,increases the number of pages used per
bucket. The MVBT update is longer than PPLH-s since the
MVBT traverses a tree for each update instead of quickly
finding the location of the updated element through
hashing). The update of R i follows; it is larger than the
MVBT since the size of the tree traversed is related to all oid
lifespans while the size of the MVBT structure traversed is
related to the number of alive oids at the time of the update).
The R p tree uses larger update processing than the R i
because of the overhead to store an interval as two points.
The ALH had the worse update processing since all
lifespans with the same oid are thrown on the same bucket,
creating large buckets that must be searched serially.
The space consumed by each method appears in Fig. 3c.
The ALH approach uses the smallest space since it stores a
single record per oid lifespan and uses ªcontrolledº splits
with high utilization. The PPLH-s method has also very
good space utilization,very close to ALH. The R-tree
methods follow; R p uses slightly less space than the R i
because paginating intervals putting them into bounding
rectangles) is more demanding than with points. Note that
similarly to ALH,both R* methods use a single record per
oid lifespan; the additional space is mainly because the
average R-tree page utilization is about 65 percent. PPLH-l
uses more space than PPLH-s because the NT array
implementation reduces page utilization. The MVBT has
the largest space requirements,about twice more space than
the ALH and PPLH-s methods.
To consider the effect of lifespan distribution,all
approaches were compared using five additional workloads
called the exponential,step,normal,Poisson,and uniformconsecutive).
These workloads had the same number of
distinct oids …jUj ˆ8; 000†,number of queries 115,878),and
similar n … 0:5M† and NB ˆ…30† parameters. The
Exponential workload generated the n k lifespans per oid
using an exponential distribution with probability density
function f…x† ˆexp… x† and mean 1= ˆ 30. The total
number of changes was n ˆ 487; 774,the total number of
object additions was NB ˆ 245; 562,and NB ˆ 30:7. In the
Step workload,the number of lifespans per oid follows a
step function. The first 500 oids have four lifespans,the next
500 have eight lifespans,and so on,i.e.,for every 500 oids,
the number of lifespans advances by four. In this workload,
we had n ˆ 540; 425, NB ˆ 272; 064,and NB ˆ 34. The
Normal workload used a normal distribution with ˆ 30
and 2 ˆ 25. Here,the parameters were n ˆ 470; 485,
NB ˆ 237; 043,and NB ˆ 29:6.
For the Poisson workload the first lifespan for every oid
was generated randomly between time instants 1 and 500.
The length of a lifespan was generated using a Poisson
distribution with mean 1,100. Each next start time for a
given oid was also generated by a Poisson distribution with
mean value 500. For this workload,we had n ˆ 498; 914,
NB ˆ 251; 404,and NB ˆ 31. The main characteristic of the
Poisson workload is that the number of alive oids over time
can vary from a very small number to a large proportion of
jUj,i.e.,there are time instants where the number of alive
oids is some hundreds and other time instants where almost
all distinct oids are alive.
The special characteristic of the Uniform-consecutive
workload is that it contains objects with multiple but
consecutive lifespans. This scenario occurs when objects are
updated frequently during their lifetime. Each update is
seen as the deletion of the object followed by the insertion of
the updated object at the same time. Since the object retains
its oid through updates,this process creates consecutive
lifespans for the same object the end of one lifespan is the
start of the next lifespan). This workload was based on the
Uniform-30 workload and had n ˆ 468; 715, NB ˆ 236; 155,
and NB ˆ 30. An object has a single lifetime which is cut
into consecutive lifespans. The start_times of an object's
lifespans are chosen uniformly.
Fig. 4 presents the query,update and space performance
under the new workloads. The results resemble the
Uniform-30 workload. For brevity,we have excluded the
R-tree-based methods from the remaining discussion as
they consistently had much worse query performance; the
interested reader can find the detailed performance in
[17]). As before,the PPLH-s approach has the best overall
performance using slightly more space than the ªminimalº
space of ALH. PPLH-l has the same query performance
with PPLH-s,but uses more updating and space. Note that