An Ordering ofConvex Topological Relations
Matthew P. Dube Max J. Egenhofer
School of Computing and Information Science
[email protected]@umit.maine.edu
A World of Languages
with a truckload of expressions that
don’t exist in all languages... BogusławskiWierzbickaGoddard
чужбина чужбина buitenland ???
Lost in Translation...
Translation vs. Circumscription
•Translation: one word for another word
•Circumscription: one word to many potential words
.
.n..
Outline• Topological Spatial Relations
• Properties of Conceptual Neighborhood Graphs
• Connectivity
• Distance Hereditary
• Convexity
• Visual Example of Algorithm
• Mapping Language back to Theory
• Translation/Circumscription Example
• A Look to the Future
Topological Spatial Relations in
S2
Egenhofer
embraceentwined
overlap
attach
meetdisjoint equal covers containscoveredBy inside
A-Neighborhood
EgenhoferFreksa
embrace
entwined
overlap
attach
meet
disjoint
equal
covers
contains
coveredBy
inside
Al-Taha
Disjunctions of Relations
embrace
entwined
overlap
attach
meet
disjoint
equal
covers
contains
coveredBy
inside
BennettCohn
Disjunctions of Relations
embrace
entwined
overlap
attach
meet
disjoint
equal
covers
contains
coveredBy
inside
Disjunctions of Relations
embrace
entwined
overlap
attach
meet
disjoint
equal
covers
contains
coveredBy
inside
Disjunctions of Relations
embrace
entwined
overlap
attach
meet
disjoint
equal
covers
contains
coveredBy
inside
Howorka
Disjunctions of Relations
embrace
entwined
overlap
attach
meet
disjoint
equal
covers
contains
coveredBy
inside
Ligozat
ConvexityConvex objects maintain all of their shortest paths
Concave objects lack at least one shortest pathArtigas SzwarcfiterDourado
Convex SubgraphsConnected
•Can all nodes be reached from all other nodes?
Distance Hereditary•Can all nodes be reached in regularly optimal
cost?
Multi-Path Preserving•Can all nodes be reached in as many regularly
optimal ways?
Adjacency Matrix
embrace
entwined
overlap
attach
meet
disjoint
equal
covers
contains
coveredBy
inside
Adjacency Matrix
d(A,B) + d(B,C) ≥ d(A,C)
Distance Matrix
Distance Matrix
33
d(A,B) + d(B,C) ≥ d(A,C)
Distance Matrix
d(A,B) + d(B,C) ≥ d(A,C)EgenhoferAl-Taha
embrace
entwined
overlap
attach
meet
disjoint
equal
covers
contains
coveredBy
inside
Connected Subgraphs
embrace
entwined
overlap
attach
meet
disjoint
equal
covers
contains
coveredBy
inside
embrace
entwined
overlap
attach
meet
disjoint
equal
covers
contains
coveredBy
inside
Connected Subgraphs
0 1 0
1 0 1
0 1 0
embrace
entwined
overlap
attach
meet
disjoint
equal
covers
contains
coveredBy
inside
embrace
entwined
overlap
attach
meet
disjoint
equal
covers
contains
coveredBy
inside
embrace
entwined
overlap
attach
meet
disjoint
equal
covers
contains
coveredBy
inside
Distance Hereditary Subgraphs
embrace
entwined
overlap
attach
meet
disjoint
equal
covers
contains
coveredBy
inside
Howorka
embrace
entwined
overlap
attach
meet
disjoint
equal
covers
contains
coveredBy
inside
Distance Hereditary Subgraphs
embrace
entwined
overlap
attach
meet
disjoint
equal
covers
contains
coveredBy
inside
Multi-Path Preserving Subgraphs
embrace
entwined
overlap
attach
meet
disjoint
equal
covers
contains
coveredBy
inside
If a node v is on a shortest path
between nodes u and w, it must
satisfy:
d(u,v) + d(v,w) = d(u,w)
Multi-Path Preserving Subgraphs
If a node v is on a shortest path
between nodes u and w, it must
satisfy:
d(u,v) + d(v,w) = d(u,w)
Multi-Path Preserving Subgraphs
If a node v is on a shortest path
between nodes u and w, it must
satisfy:
d(u,v) + d(v,w) = d(u,w)
Multi-Path Preserving Subgraphs
If a node v is on a shortest path
between nodes u and w, it must
satisfy:
d(u,v) + d(v,w) = d(u,w)
Multi-Path Preserving Subgraphs
If a node v is on a shortest path
between nodes u and w, it must
satisfy:
d(u,v) + d(v,w) = d(u,w)
Multi-Path Preserving Subgraphs
If a node v is on a shortest path
between nodes u and w, it must
satisfy:
d(u,v) + d(v,w) = d(u,w)
Multi-Path Preserving Subgraphs
If a node v is on a shortest path
between nodes u and w, it must
satisfy:
d(u,v) + d(v,w) = d(u,w)
Multi-Path Preserving Subgraphs
If a node v is on a shortest path
between nodes u and w, it must
satisfy:
d(u,v) + d(v,w) = d(u,w)
Multi-Path Preserving Subgraphs
If a node v is on a shortest path
between nodes u and w, it must
satisfy:
d(u,v) + d(v,w) = d(u,w)
Multi-Path Preserving Subgraphs
If a node v is on a shortest path
between nodes u and w, it must
satisfy:
d(u,v) + d(v,w) = d(u,w)
Multi-Path Preserving Subgraphs
If a node v is on a shortest path
between nodes u and w, it must
satisfy:
d(u,v) + d(v,w) = d(u,w)
Multi-Path Preserving Subgraphs
If a node v is on a shortest path
between nodes u and w, it must
satisfy:
d(u,v) + d(v,w) = d(u,w)
ifthen
Multi-Path Preserving Subgraphs
2 1 2
1 2 1
2 1 2
embrace
entwined
overlap
attach
meet
disjoint
equal
covers
contains
coveredBy
inside
Multi-Path Preserving Subgraphs
Convex Subgraphs of
the A-Neighborhood
embrace
entwined
overlap
attach
meet
disjoint
equal
covers
contains
coveredBy
inside
Connected
• All nodes can be reached from all other nodes
Distance Hereditary
• All nodes can be reached in regularly optimal cost
Multi-Path Preserving
• All nodes can be reached in as many regularly optimal ways
Convex Subgraphs of
the A-Neighborhood
Connected
• All nodes can be reached from all other nodes
Distance Hereditary
• All nodes can be reached in regularly optimal cost
Multi-Path Preserving
• All nodes can be reached in as many regularly optimal ways
Subset/Superset Ordering
Schilder
ContributionsAlgorithm for identifying subgraphs of arbitrary graphs that satisfy particular properties
•connected
•distance hereditary
•multi-path preserving
•convexity
Contributions
•104 subgraphs of the A-neighborhood identified as convex
•Partial ordering based on subset/superset relationship
Natural Spatial Language
Communication is often spoken or written
Spatial information is important
Often comes through the realm of prepositional phrases
“Fairly complete” list of 90 prepositions
LandauJackendoff
aboutaboveacrossafter
against
alongalongsideamidstamongst
around
atatop
behindbelowbeneathbesidebetweenbetwixtbeyondby
down
from
ininsideinto
nearnearby
offononto
opposite
outoutside
overpast
throughthroughoutto
towardunderunderneath
upupon
via
withwithinwithout
far from
in back ofin betweenin front ofin line with
on top of
to the left ofto the right ofto the side of
afterward
apart
awaybackbackward
downstairsdownward
east
forward
here
inward
left
homeward
north
outward
rightsideways
souththere
together
upstairsupward
westago
as
because of
before
despite
during
for
like
of
since
until
English Spatial Prepositions
aboutacrossagainstalongalongsideamidstamongst
aroundat
besideby
ininsidenearoutoutsidethroughthroughout
withwithin
far from
together
in back ofin betweenin front of
on top of
to the left ofto the right of
afterward
apart
away
backbackward
downstairsdownwardeastforward
here
inward
left
homeward
north
outward
rightsidewayssouththere
to the side of
upstairsupward
west
Topologically Driven Other Dimensionsaboveafter
atop
behindbelowbeneathbetweenbetwixtbeyonddown
from
into
nearby
offononto
opposite
over
past
totoward
underunderneathupupon
via
without
in line with
English Spatial Prepositions
aboutacrossagainstalongalongsideamidstamongst
aroundat
besideby
ininsidenearoutoutsidethroughthroughout
withwithin
far from
to the side ofapart
awaytogether
Topologically Driven
English Spatial Prepositions
embrace
entwined
overlap
attach
meet
disjoint
equal
covers
contains
coveredBy
inside
Along/Alongside
EgenhoferMarkShariff
Along/Alongside
Along/Alongside
embrace
entwined
overlap
attach
meet
disjoint
equal
covers
contains
coveredBy
inside
Along/Alongside
embrace
entwined
attach
equal
covers
contains
overlap
meet
disjoint
coveredBy
inside
Along/Alongside
C47
Along represents a convex subgraph
embrace
entwined
attach
equal
covers
contains
Topologically DrivenSpatial Prepositions
Prepositions Disjunctions of Topological Spatial Relations Convex Relation
about equal, coveredBy, inside C30
across overlap, coveredBy C19
against meet, attach C15
along/alongside disjoint, meet, overlap, coveredBy, inside C47
around disjoint, meet C13
at equal, coveredBy, inside C30
beside disjoint, meet, attach C29
by disjoint, meet, attach C29
far from/apart disjoint C2
in/inside/within coveredBy, inside C18
near disjoint, meet, attach C29
out/outside disjoint, meet, attach C29
through overlap C9
throughout equal, covers, contains C31
togetheroverlap, equal, coveredBy, inside, covers, contains, entwined,
embraceC91
to the side of disjoint, meet, attach C29
withoverlap, equal, coveredBy, inside, covers, contains, entwined,
embraceC91
English Spatial Prepositions
aboutacrossagainstalong
alongsideamidstamongst
aroundat
besideby
ininsidenear
outoutsidethroughthroughout
with
within
far from
to the side of
apart
away
together
aboutacrossagainstalong
alongsideamidstamongst
aroundat
besideby
ininsidenear
outoutsidethroughthroughout
with
within
far from
to the side of
apart
away
together
English Spatial Prepositions
English Spatial Preposition Ordering
aboutacross
against
alongalongside
amidstamongst
around
at
besideby
ininside
nearout
outside
through
throughout
with
within
far from
to the side of
apartaway
together
universal
nothing
How Can This Be Used?
Language Circumscription
The phrase “goes to” has been shown to be a problematic translation from English to Spanish and its related dialects
How can it be conveyed without drawing or showing it to the other person?
EgenhoferMark
чужбина
Bulgarian word
Foreigner’s land
Examples:
Norway (disjoint)
Greece (meet)
Not in Bulgaria (attach)
чужбина → “Outside”
aboutacross
against
alongalongside
amidstamongst
around
at
besideby
ininside
nearout
outside
through
throughout
with
within
far from
to the side of
apartaway
together
universal
nothing
Contributions
25 topologically driven English spatial prepositions
Partial ordering based on the ordering of convex relations
Mathematically-assisted translation and circumscription between languages
Conclusions•104 convex subgraphs identified
within the A-neighborhood of S2 region-region relations
•All topologically driven English terms from Landau and Jackendoff parsed to these subgraphs
•Translation and circumscription have the potential for mathematical analysis
A Look to the Future
•A mathematical strategy to circumscript between languages
•Convex subgraphs in other neighborhoods and for various sets of relations
•The impacts of convexity on relation composition = ?
Acknowledgments• Brian Lopez-Cornier
• University of Maine Upward Bound Math-Science Student
• DOE P047M080002
• Sensor Science, Engineering, and Informatics IGERT
• DGE 0504494 (PI: Kate Beard)
• NSF IIS 1016740 (PI: Max Egenhofer)
Lopez-Cornier
Beard
Questions?