An Efficient Approach for the Generation of Allen Relations

An Eﬃcient Approach for the Generation of Allen
Relations
Kleanthi Georgala and Mohamed Ahmed Sherif and Axel-Cyrille Ngonga
Ngomo
University of Leipzig
Institute for Applied Informatics
September 2nd, 2016
The Hague, Netherlands
Georgala Sherif Ngonga Ngomo (InfAI) AEGLE September 14, 2016 1 / 1

Overview

Why Link Discovery between events?

:E1 rdfs:label "Engine failure"@en
:E1 rdf:type :Error
:E1 :beginDate :"2015-04-22T11:39:35"
:E1 :endDate :"2015-04-22T11:39:37"

:E1 rdfs:label "Engine failure"@en
:E1 rdf:type :Error
:E1 :beginDate :"2015-04-22T11:39:35"
:E1 :endDate :"2015-04-22T11:39:37"
:E2 rdfs:label "Car accident"@en
:E2 rdf:type :Accident
:E2 :beginDate :"2015-06-28T11:45:22"
:E2 :endDate :"2015-06-28T11:45:24"

Link Discovery
Linked Data 4th principle: Include links to other URIs so that they can
discover more things.
Deﬁnition (Link Discovery)
Given sets S and T of resources and relation R
Find M = {(s, t) ∈ S × T : R(s, t)}
Example: R = :failureType

Do you have time to talk about.. time?
What if R = :startsBefore ?
No dedicated approaches for LD between event data
Silk scalability issues
Time complexity
Quadratic a-priori runtime
Completeness
Missing links
Scalability
Diverse KBs

Event Deﬁnition
Deﬁnition (Event)
Events can be modeled as time intervals: v = (b(v), e(v))
b(v) is the beginning time (:beginDate)
e(v) is the end time (:endDate)
b(v) < e(v)

Allen’s Interval Algebra
Relation Notation Inverse Illustration
X before Y bf (X, Y ) bfi(X, Y )
X
Y
X meets Y m(X, Y ) mi(X, Y )
X
Y
X finishes Y f (X, Y ) fi(X, Y )
X
Y
X starts Y st(X, Y ) sti(X, Y )
X
Y
X during Y d(X, Y ) di(X, Y )
X
Y
X equal Y eq(X, Y ) eq(X, Y )
X
Y
X overlaps with Y ov(X, Y ) ovi(X, Y )
X
Y

Our Solution
Aegle: Allen’s intErval alGebra for Link
discovEry
Eﬃcient computation of temporal
relations between events
Allen’s Interval Algebra: distinct,
exhaustive, and qualitative relations
between time intervals

Our Contribution
Eﬃcient Link Discovery between Events by:
1 Expressing 13 Allen relations using 8 atomic relations
2 Time is ordered: Find matching entities using two sorted lists

Express st(s, t) using atomic relations
s
t
b(s) = b(t)

s
t
b(s) = b(t)
s
t
b(s) < e(t)

s
t
b(s) = b(t)
s
t
b(s) < e(t)
s
t
e(s) > b(t)

s
t
b(s) = b(t)
s
t
b(s) < e(t)
s
t
e(s) > b(t)
s
t
e(s) < e(t)

AEGLE
Compute 8 atomic Boolean relations between begin and end points
BeginBegin (BB) for b(s), b(t):
BB1(s, t) ⇔ (b(s) < b(t))
BB0(s, t) ⇔ (b(s) = b(t))
BB−1(s, t) ⇔ (b(s) > b(t)) ⇔ ¬(BB1(s, t) ∨ BB0(s, t))
BeginEnd(BE) for b(s), e(t)
EndBegin(EB) for e(s), b(t)
EndEnd(EE) for e(s), e(t)

Combine the atomic relations
s
t
st(s, t) ⇔ BB0
(s, t) ∧ BE1
(s, t) ∧ EB−1
(s, t) ∧ EE1
(s, t) ⇔ {BB0
(s, t) ∧ EE1
(s, t) }

Combine the atomic relations
s
t
st(s, t) ⇔ BB0
(s, t) ∧ BE1
(s, t) ∧ EB−1
(s, t) ∧ EE1
(s, t) ⇔ {BB0
(s, t) ∧ EE1
(s, t) }
t
s
sti(s, t) ⇔ BB0
(s, t) ∧ BE1
(s, t) ∧ EB−1
(s, t) ∧ EE−1
(s, t) ⇔
{BB0
(s, t) ∧ EE−1
(s, t)} =
{ BB0
(s, t) ∧¬(EE0
(s, t)∨ EE1
(s, t))}

Algorithm for st, sti
Source
s1
s2
Target
t1
t2

Source
s1
s2
Target
t1
t2
For st:

Source
s1
s2
Target
t1
t2
For st:
Compute BB0
:

Source
s1
s2
Target
t1
t2
For st:
Compute BB0
:
s1 s2 t1 t2
{(s1, t1), (s2, t1)}

Source
s1
s2
Target
t1
t2
For st:
Compute BB0
:
s1 s2 t1 t2
{(s1, t1), (s2, t1)}
Compute EE1
:
s1 s2 t1 t2
{(s1, t1), (s2, t1), (s1, t2)}

Source
s1
s2
Target
t1
t2
For st:
Compute BB0
:
s1 s2 t1 t2
{(s1, t1), (s2, t1)}
Compute EE1
:
s1 s2 t1 t2
{(s1, t1), (s2, t1), (s1, t2)}
Intersection between BB0
and EE1
:
{(s1, t1), (s2, t1)}

Source
s1
s2
Target
t1
t2
For sti:

Source
s1
s2
Target
t1
t2
For sti:
Retrieve BB0
and EE1
:

Source
s1
s2
Target
t1
t2
For sti:
Retrieve BB0
and EE1
:
Compute EE0
:
s1 s2 t1 t2
{(s2, t2)}

Source
s1
s2
Target
t1
t2
For sti:
Retrieve BB0
and EE1
:
Compute EE0
:
s1 s2 t1 t2
{(s2, t2)}
Union between EE0
and EE1
:
{(s1, t1), (s2, t1), (s2, t1), (s2, t2)}

Source
s1
s2
Target
t1
t2
For sti:
Retrieve BB0
and EE1
:
Compute EE0
:
s1 s2 t1 t2
{(s2, t2)}
Union between EE0
and EE1
:
{(s1, t1), (s2, t1), (s2, t1), (s2, t2)}
Diﬀerence between BB0
and EE0
, EE1
:
{(s1, t2), (s2, t2)}

Experimental Set-Up
Datasets: S = T
Log Type Dataset name Size Unique b(s) Unique e(s)
Machinery
3KMachines 3,154 960 960
30KMachines 28,869 960 960
300KMachines 288,690 960 960
Query
3KQueries 3,888 3,636 3,638
30KQueries 30,635 3,070 3,070
300KQueries 303,991 184 184

Experimental Set-Up
Datasets: S = T
Machinery
3KMachines 3,154 960 960
30KMachines 28,869 960 960
300KMachines 288,690 960 960
Query
3KQueries 3,888 3,636 3,638
30KQueries 30,635 3,070 3,070
300KQueries 303,991 184 184
State-of-the-art:
Silk extended to deal with spatio-temporal data
Baseline for eq using brute-force

Experimental Set-Up
Datasets: S = T
Machinery
3KMachines 3,154 960 960
30KMachines 28,869 960 960
300KMachines 288,690 960 960
Query
3KQueries 3,888 3,636 3,638
30KQueries 30,635 3,070 3,070
300KQueries 303,991 184 184
State-of-the-art:
Silk extended to deal with spatio-temporal data
Baseline for eq using brute-force
Evaluation measures:
atomic runtime of each of the atomic relations
relation runtime required to compute each Allen’s relation
total runtime required to compute all 13 relations

Results
Q1: Does the reduction of Allen relations to 8 atomic relations inﬂuence the
overall runtime of the approach?

Results
Q2: How does Aegle perform when compared with the state of the art in
terms of time eﬃciency?
Log Type Dataset Name
Total Runtime
Aegle Aegle * Silk
Machine
3KMachines 11.26 5.51 294.00
30KMachines 1,016.21 437.79 29,846.00
300KMachines 189,442.16 78,416.61 NA
Query
3KQueries 26.94 17.91 541.00
30KQueries 988.78 463.27 33,502.00
300KQueries 211,996.88 86,884.98 NA

Results
Machine Query
Relation Approach 3KMachines 30KMachines 300KMachines 3KQueries 30KQueries 300KQueries
m
Aegle 0.02 0.19 3.42 0.02 0.21 3.89
Silk 23.00 2,219.00 NA 41.00 2,466.00 NA
eq
Aegle 0.05 0.79 49.84 0.05 0.45 348.51
Silk 23.00 2,250.00 NA 41.00 2,473.00 NA
baseline 2.05 171.10 23,436.30 3.15 196.09 31,452.54
ovi
Aegle 3.16 222.27 38,226.32 11.97 257.59 42,121.68
Silk 22.00 2,189.00 NA 42.00 2,503.00 NA

Conclusions
Aegle: reduction of 13 Allen Interval relations to 8 atomic relations
eﬃciency: simple sorting with complexity O(n log n)
scalable LD
outperforms the state-of-the-art
Future Work:
Implement Aegle in parallel
Incremental computation of temporal links on streams of data

Thank you!
Visit http://aksw.org/Projects/LIMES.html
Questions?
Kleanthi Georgala
AKSW Research Group
Augustusplatz 10, Room P905
04109 Leipzig, Germany
georgala@informatik.uni-leipzig.de
http://aksw.org/KleanthiGeorgala.html

An Efficient Approach for the Generation of Allen Relations

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