VoroNet: A scalable object network based on Voronoi...
Transcript of VoroNet: A scalable object network based on Voronoi...
VoroNet: A scalable object networkbased on Voronoi tessellations
Olivier Beaumont, Anne-Marie Kermarrec,Loris Marchal and Etienne Riviere
SCALAPPLIX project, LaBRI (Bordeaux)
PARIS project, IRISA (Rennes)
GRAAL project, LIP, ENS Lyon
March 2006
Outline
1 Introduction
2 Description of VoroNet
3 Evaluation (simulation)
4 Perspectives, conclusion
Loris Marchal VoroNet 2/ 24
Introduction
Outline
1 Introduction
2 Description of VoroNet
3 Evaluation (simulation)
4 Perspectives, conclusion
Loris Marchal VoroNet 3/ 24
Introduction
Peer-to-peer overlay
What is peer-to-peer ?
paradigm to organize distributed ressources (peers)
overlay network: logical organization
core functionnality: search objects in the system
distributed hashtables (DHT) (Chord, Pastry,. . . )
hash function gives identifiers for peers and objects
choice of hash function to get uniform distributionmain goals:
scalability ,fault tolerance ,efficient search ,
but restricted to exact search /highly depends on the hash function /
Loris Marchal VoroNet 4/ 24
Introduction
Peer-to-peer overlay
What is peer-to-peer ?
paradigm to organize distributed ressources (peers)
overlay network: logical organization
core functionnality: search objects in the system
distributed hashtables (DHT) (Chord, Pastry,. . . )
hash function gives identifiers for peers and objects
choice of hash function to get uniform distributionmain goals:
scalability ,fault tolerance ,efficient search ,
but restricted to exact search /highly depends on the hash function /
Loris Marchal VoroNet 4/ 24
Introduction
Peer-to-peer overlay
What is peer-to-peer ?
paradigm to organize distributed ressources (peers)
overlay network: logical organization
core functionnality: search objects in the system
distributed hashtables (DHT) (Chord, Pastry,. . . )
hash function gives identifiers for peers and objects
choice of hash function to get uniform distributionmain goals:
scalability ,fault tolerance ,efficient search ,
but restricted to exact search /highly depends on the hash function /
Loris Marchal VoroNet 4/ 24
Introduction
Peer-to-peer overlay
Content-based topologies:
CAN (Content Adressable Network)I d-dimensional torus
I degree O(d)
I diameter O(N1/d)
I not really “content addressed”:location (of objects and peers) computedwith hash function (to ensurehomogeneous distribution)
?
Loris Marchal VoroNet 5/ 24
Introduction
Peer-to-peer overlay
Content-based topologies:
CAN (Content Adressable Network)I d-dimensional torus
I degree O(d)
I diameter O(N1/d)
I not really “content addressed”:location (of objects and peers) computedwith hash function (to ensurehomogeneous distribution)
?
Loris Marchal VoroNet 5/ 24
Introduction
Peer-to-peer overlay
Content-based topologies:
CAN (Content Adressable Network)I d-dimensional torus
I degree O(d)
I diameter O(N1/d)
I not really “content addressed”:location (of objects and peers) computedwith hash function (to ensurehomogeneous distribution)
?
Loris Marchal VoroNet 5/ 24
Introduction
VoroNet
Object-based peer-to-peer overlayI objects are linked rather than peersI an object is held by the node which published it
Content-based topology:I not based on a DHTI objects with “close” attributes will be neighbors
d-dimensional attribute space
VoroNet topology is inspired from:I Voronoi diagram in the attribute spaceI Kleinberg’s small world routing algorithm designed for grids
Loris Marchal VoroNet 6/ 24
Introduction
VoroNet
Object-based peer-to-peer overlayI objects are linked rather than peersI an object is held by the node which published it
Content-based topology:I not based on a DHTI objects with “close” attributes will be neighbors
d-dimensional attribute space
VoroNet topology is inspired from:I Voronoi diagram in the attribute spaceI Kleinberg’s small world routing algorithm designed for grids
Loris Marchal VoroNet 6/ 24
Introduction
VoroNet
Object-based peer-to-peer overlayI objects are linked rather than peersI an object is held by the node which published it
Content-based topology:I not based on a DHTI objects with “close” attributes will be neighbors
d-dimensional attribute space
VoroNet topology is inspired from:I Voronoi diagram in the attribute spaceI Kleinberg’s small world routing algorithm designed for grids
Loris Marchal VoroNet 6/ 24
Introduction
VoroNet
Object-based peer-to-peer overlayI objects are linked rather than peersI an object is held by the node which published it
Content-based topology:I not based on a DHTI objects with “close” attributes will be neighbors
d-dimensional attribute space we consider for now: d = 2VoroNet topology is inspired from:
I Voronoi diagram in the attribute spaceI Kleinberg’s small world routing algorithm designed for grids
Loris Marchal VoroNet 6/ 24
Introduction
VoroNet
Object-based peer-to-peer overlayI objects are linked rather than peersI an object is held by the node which published it
Content-based topology:I not based on a DHTI objects with “close” attributes will be neighbors
d-dimensional attribute space we consider for now: d = 2VoroNet topology is inspired from:
I Voronoi diagram in the attribute spaceI Kleinberg’s small world routing algorithm designed for grids
Loris Marchal VoroNet 6/ 24
Introduction
Voronoi tessellations
set of points in R2
consider object at point M
region of points closer from Mthan
do the same for all objects
Voronoi neighbors: when cellsshare a border
graph of Voronoi neighbors:Delaunay triangularization
Loris Marchal VoroNet 7/ 24
Introduction
Voronoi tessellations
M
set of points in R2
consider object at point M
region of points closer from Mthan
do the same for all objects
Voronoi neighbors: when cellsshare a border
graph of Voronoi neighbors:Delaunay triangularization
Loris Marchal VoroNet 7/ 24
Introduction
Voronoi tessellations
A
M
set of points in R2
consider object at point M
region of points closer from Mthan
do the same for all objects
Voronoi neighbors: when cellsshare a border
graph of Voronoi neighbors:Delaunay triangularization
Loris Marchal VoroNet 7/ 24
Introduction
Voronoi tessellations
A
M
set of points in R2
consider object at point M
region of points closer from Mthan from A
do the same for all objects
Voronoi neighbors: when cellsshare a border
graph of Voronoi neighbors:Delaunay triangularization
Loris Marchal VoroNet 7/ 24
Introduction
Voronoi tessellations
BA
M
set of points in R2
consider object at point M
region of points closer from Mthan from A
do the same for all objects
Voronoi neighbors: when cellsshare a border
graph of Voronoi neighbors:Delaunay triangularization
Loris Marchal VoroNet 7/ 24
Introduction
Voronoi tessellations
BA
M
set of points in R2
consider object at point M
region of points closer from Mthan from A and B
do the same for all objects
Voronoi neighbors: when cellsshare a border
graph of Voronoi neighbors:Delaunay triangularization
Loris Marchal VoroNet 7/ 24
Introduction
Voronoi tessellations
M
set of points in R2
consider object at point M
region of points closer from Mthan from any other object:Voronoi cell of M (or region)
do the same for all objects
Voronoi neighbors: when cellsshare a border
graph of Voronoi neighbors:Delaunay triangularization
Loris Marchal VoroNet 7/ 24
Introduction
Voronoi tessellations
M
set of points in R2
consider object at point M
region of points closer from Mthan from any other object:Voronoi cell of M (or region)
do the same for all objects
Voronoi neighbors: when cellsshare a border
graph of Voronoi neighbors:Delaunay triangularization
Loris Marchal VoroNet 7/ 24
Introduction
Voronoi tessellations
M
set of points in R2
consider object at point M
region of points closer from Mthan from any other object:Voronoi cell of M (or region)
do the same for all objects
Voronoi neighbors: when cellsshare a border
graph of Voronoi neighbors:Delaunay triangularization
Loris Marchal VoroNet 7/ 24
Introduction
Voronoi tessellations
M
set of points in R2
consider object at point M
region of points closer from Mthan from any other object:Voronoi cell of M (or region)
do the same for all objects
Voronoi neighbors: when cellsshare a border
graph of Voronoi neighbors:Delaunay triangularization
Loris Marchal VoroNet 7/ 24
Introduction
Kleinberg’s small-world
N nodes in a 2D grid(√
N ×√
N)I routing in O(
√N)
add random long range links:I probability for a long link to be
at distance l: ∝ 1l2
I use greedy routing algorithmI then routing in O(ln2 N)
can be extended to anydimension d, with proba ∝ 1
ld
Loris Marchal VoroNet 8/ 24
Introduction
Kleinberg’s small-world
N nodes in a 2D grid(√
N ×√
N)I routing in O(
√N)
add random long range links:I probability for a long link to be
at distance l: ∝ 1l2
I use greedy routing algorithmI then routing in O(ln2 N)
can be extended to anydimension d, with proba ∝ 1
ld
Loris Marchal VoroNet 8/ 24
Introduction
Kleinberg’s small-world
N nodes in a 2D grid(√
N ×√
N)I routing in O(
√N)
add random long range links:I probability for a long link to be
at distance l: ∝ 1l2
I use greedy routing algorithmI then routing in O(ln2 N)
can be extended to anydimension d, with proba ∝ 1
ld
Loris Marchal VoroNet 8/ 24
Description of VoroNet
Outline
1 Introduction
2 Description of VoroNet
3 Evaluation (simulation)
4 Perspectives, conclusion
Loris Marchal VoroNet 9/ 24
Description of VoroNet
VoroNet neighborhood
Voronoi neighborslong range neighbors:
I randomly chose a target point tI the long range neighbors is the object “responsible” for point tI keep a back pointer for overlay maintenance
close neighbors (within distance dmin) for convergence
Loris Marchal VoroNet 10/ 24
Description of VoroNet
VoroNet neighborhood
Voronoi neighborslong range neighbors:
I randomly chose a target point tI the long range neighbors is the object “responsible” for point tI keep a back pointer for overlay maintenance
close neighbors (within distance dmin) for convergence
Loris Marchal VoroNet 10/ 24
Description of VoroNet
VoroNet neighborhood
t
Voronoi neighborslong range neighbors:
I randomly chose a target point tI the long range neighbors is the object “responsible” for point tI keep a back pointer for overlay maintenance
close neighbors (within distance dmin) for convergence
Loris Marchal VoroNet 10/ 24
Description of VoroNet
VoroNet neighborhood
t
Voronoi neighborslong range neighbors:
I randomly chose a target point tI the long range neighbors is the object “responsible” for point tI keep a back pointer for overlay maintenance
close neighbors (within distance dmin) for convergence
Loris Marchal VoroNet 10/ 24
Description of VoroNet
VoroNet neighborhood
dmin
Voronoi neighborslong range neighbors:
I randomly chose a target point tI the long range neighbors is the object “responsible” for point tI keep a back pointer for overlay maintenance
close neighbors (within distance dmin) for convergence
Loris Marchal VoroNet 10/ 24
Description of VoroNet
VoroNet neighborhood – details
Space: 2-dimensional torus: [0, 1]× [0, 1]Long link target of object x: distribution in 1/d2
Pr[target(x) ∈ B(y, dr)
]= α
πr2
d(x, y)2
Link always points on the closest object from target.
Close neighbors: within dmin = 1πNmax
Nmax = maximal number of nodes for which we have an efficientrouting
Loris Marchal VoroNet 11/ 24
Description of VoroNet
VoroNet neighborhood – details
Space: 2-dimensional torus: [0, 1]× [0, 1]Long link target of object x: distribution in 1/d2
Pr[target(x) ∈ B(y, dr)
]= α
πr2
d(x, y)2
Link always points on the closest object from target.
Close neighbors: within dmin = 1πNmax
Nmax = maximal number of nodes for which we have an efficientrouting
Loris Marchal VoroNet 11/ 24
Description of VoroNet
VoroNet neighborhood – details
Space: 2-dimensional torus: [0, 1]× [0, 1]Long link target of object x: distribution in 1/d2
Pr[target(x) ∈ B(y, dr)
]= α
πr2
d(x, y)2
Link always points on the closest object from target.
Close neighbors: within dmin = 1πNmax
Nmax = maximal number of nodes for which we have an efficientrouting
Loris Marchal VoroNet 11/ 24
Description of VoroNet
VoroNet neighborhood – details
Space: 2-dimensional torus: [0, 1]× [0, 1]Long link target of object x: distribution in 1/d2
Pr[target(x) ∈ B(y, dr)
]= α
πr2
d(x, y)2
Link always points on the closest object from target.
Close neighbors: within dmin = 1πNmax
Nmax = maximal number of nodes for which we have an efficientrouting
Loris Marchal VoroNet 11/ 24
Description of VoroNet
Size of the neighborhood
Voronoi neighbors:I graph of the Voronoi neighbors is planar ⇒ average degree 6 6I O(1) sizeI mean value 6 6 (max = N)
Close neighbors: number of objects in B(o, dmin)I O(1) size for a “reasonable” distributionI mean value = 1 (max = N)
Long range neighbors:I one long range neighbor per objectI O(1) backward links for a “reasonable” distributionI mean value = 1 (max = N)
size of data stored at each node: O(1) mean value 6 9(we will check this property in the experiments)
Loris Marchal VoroNet 12/ 24
Description of VoroNet
Size of the neighborhood
Voronoi neighbors:I graph of the Voronoi neighbors is planar ⇒ average degree 6 6I O(1) sizeI mean value 6 6 (max = N)
Close neighbors: number of objects in B(o, dmin)I O(1) size for a “reasonable” distributionI mean value = 1 (max = N)
Long range neighbors:I one long range neighbor per objectI O(1) backward links for a “reasonable” distributionI mean value = 1 (max = N)
size of data stored at each node: O(1) mean value 6 9(we will check this property in the experiments)
Loris Marchal VoroNet 12/ 24
Description of VoroNet
Size of the neighborhood
Voronoi neighbors:I graph of the Voronoi neighbors is planar ⇒ average degree 6 6I O(1) sizeI mean value 6 6 (max = N)
Close neighbors: number of objects in B(o, dmin)I O(1) size for a “reasonable” distributionI mean value = 1 (max = N)
Long range neighbors:I one long range neighbor per objectI O(1) backward links for a “reasonable” distributionI mean value = 1 (max = N)
size of data stored at each node: O(1) mean value 6 9(we will check this property in the experiments)
Loris Marchal VoroNet 12/ 24
Description of VoroNet
Size of the neighborhood
Voronoi neighbors:I graph of the Voronoi neighbors is planar ⇒ average degree 6 6I O(1) sizeI mean value 6 6 (max = N)
Close neighbors: number of objects in B(o, dmin)I O(1) size for a “reasonable” distributionI mean value = 1 (max = N)
Long range neighbors:I one long range neighbor per objectI O(1) backward links for a “reasonable” distributionI mean value = 1 (max = N)
size of data stored at each node: O(1) mean value 6 9(we will check this property in the experiments)
Loris Marchal VoroNet 12/ 24
Description of VoroNet
Overlay maintenance
How to insert an object x ?
Update the Voronoi diagram:I Find the closest existing object (route to x)I Add a new Voronoi cellI Find and update the Voronoi neighbors
Find and the close neighborssufficient to consider close neighbors of Voronoi neighbors
create a long range target point t, find the corresponding object:=⇒ route a JOIN message to t
Loris Marchal VoroNet 13/ 24
Description of VoroNet
Overlay maintenance
How to insert an object x ?
Update the Voronoi diagram:I Find the closest existing object (route to x)I Add a new Voronoi cellI Find and update the Voronoi neighbors
Find and the close neighborssufficient to consider close neighbors of Voronoi neighbors
6 dmin
M
z
x y
create a long range target point t, find the corresponding object:=⇒ route a JOIN message to t
Loris Marchal VoroNet 13/ 24
Description of VoroNet
Overlay maintenance
How to insert an object x ?
Update the Voronoi diagram:I Find the closest existing object (route to x)I Add a new Voronoi cellI Find and update the Voronoi neighbors
Find and the close neighborssufficient to consider close neighbors of Voronoi neighbors
6 dmin
M
z
x y
create a long range target point t, find the corresponding object:=⇒ route a JOIN message to t
Loris Marchal VoroNet 13/ 24
Description of VoroNet
General routing scheme
Route (Target):find the object responsible for the Voronoi cell where Target is.
Route(Target)z = DistanceToRegion(Target)if d(z,Target) > 1
3d(Target ,CurrentObject) andd(Target ,CurrentObject) > dmin then
Spawn(Route,Target ,GreedyNeighbor(Target))else
AddVoronoiRegion(z)AddVoronoiRegion(Target)Perform some local computations depending on the operation at zRemoveVoronoiRegion(z)(depending on the operation,RemoveVoronoiRegion(Target))
return
Loris Marchal VoroNet 14/ 24
Description of VoroNet
Polylogarithmic routing - sketch of proof - 1/2
Lemma 1
The probability for the long link of x to be chosen in a disk of center
y and radius fr, where r = d(x, y) is lower bounded by πf2
K(1+f)2.
For f = 1/6: probability lower bounded by:1
98 ln(√
2πNmax)X: number of hops necessary to reach the disk of center Targetand radius d
6 .
E(X) =+∞∑i=1
Pr[X > i] 6+∞∑i=1
(1− 1
98 ln(√
2πNmax)
)i−1
E(X) 6 98 ln(√
2πNmax).
but link target 6= link destination
This accounts for a super-step.Loris Marchal VoroNet 15/ 24
Description of VoroNet
Polylogarithmic routing - sketch of proof - 1/2
Lemma 1
The probability for the long link of x to be chosen in a disk of center
y and radius fr, where r = d(x, y) is lower bounded by πf2
K(1+f)2.
For f = 1/6: probability lower bounded by:1
98 ln(√
2πNmax)X: number of hops necessary to reach the disk of center Targetand radius d
6 .
E(X) =+∞∑i=1
Pr[X > i] 6+∞∑i=1
(1− 1
98 ln(√
2πNmax)
)i−1
E(X) 6 98 ln(√
2πNmax).
but link target 6= link destination
This accounts for a super-step.Loris Marchal VoroNet 15/ 24
Description of VoroNet
Polylogarithmic routing - sketch of proof - 1/2
Lemma 1
The probability for the long link of x to be chosen in a disk of center
y and radius fr, where r = d(x, y) is lower bounded by πf2
K(1+f)2.
For f = 1/6: probability lower bounded by:1
98 ln(√
2πNmax)X: number of hops necessary to reach the disk of center Targetand radius d
6 .
E(X) =+∞∑i=1
Pr[X > i] 6+∞∑i=1
(1− 1
98 ln(√
2πNmax)
)i−1
E(X) 6 98 ln(√
2πNmax).
but link target 6= link destination
This accounts for a super-step.Loris Marchal VoroNet 15/ 24
Description of VoroNet
Polylogarithmic routing - sketch of proof - 1/2
Lemma 1
The probability for the long link of x to be chosen in a disk of center
y and radius fr, where r = d(x, y) is lower bounded by πf2
K(1+f)2.
For f = 1/6: probability lower bounded by:1
98 ln(√
2πNmax)X: number of hops necessary to reach the disk of center Targetand radius d
6 .
E(X) =+∞∑i=1
Pr[X > i] 6+∞∑i=1
(1− 1
98 ln(√
2πNmax)
)i−1
E(X) 6 98 ln(√
2πNmax).
but link target 6= link destination
This accounts for a super-step.Loris Marchal VoroNet 15/ 24
Description of VoroNet
Polylogarithmic routing - sketch of proof - 1/2
Lemma 1
The probability for the long link of x to be chosen in a disk of center
y and radius fr, where r = d(x, y) is lower bounded by πf2
K(1+f)2.
For f = 1/6: probability lower bounded by:1
98 ln(√
2πNmax)X: number of hops necessary to reach the disk of center Targetand radius d
6 .
E(X) =+∞∑i=1
Pr[X > i] 6+∞∑i=1
(1− 1
98 ln(√
2πNmax)
)i−1
E(X) 6 98 ln(√
2πNmax).
but link target 6= link destination
This accounts for a super-step.Loris Marchal VoroNet 15/ 24
Description of VoroNet
Polylogarithmic routing - sketch of proof - 1/2
Lemma 1
The probability for the long link target of x to be chosen in a disk of
center y and radius fr, where r = d(x, y) is lower bounded by πf2
K(1+f)2.
For f = 1/6: probability lower bounded by:1
98 ln(√
2πNmax)X: number of hops necessary to reach an object whose long linktarget belongs to the disk of center Target and radius d
6 .
E(X) =+∞∑i=1
Pr[X > i] 6+∞∑i=1
(1− 1
98 ln(√
2πNmax)
)i−1
E(X) 6 98 ln(√
2πNmax).
but link target 6= link destination
This accounts for a super-step.Loris Marchal VoroNet 15/ 24
Description of VoroNet
Polylogarithmic routing - sketch of proof - 1/2
Lemma 1
The probability for the long link target of x to be chosen in a disk of
center y and radius fr, where r = d(x, y) is lower bounded by πf2
K(1+f)2.
For f = 1/6: probability lower bounded by:1
98 ln(√
2πNmax)X: number of hops necessary to reach an object whose long linktarget belongs to the disk of center Target and radius d
6 .
E(X) =+∞∑i=1
Pr[X > i] 6+∞∑i=1
(1− 1
98 ln(√
2πNmax)
)i−1
E(X) 6 98 ln(√
2πNmax).
but link target 6= link destination
This accounts for a super-step.Loris Marchal VoroNet 15/ 24
Description of VoroNet
Polylogarithmic routing - sketch of proof - 1/2
During a super-step, the distance to the target is divided by 5/6Number of super-steps bounded by
ln(√
2dmin
)
ln(65)
=ln(√
2πNmax)ln(6
5)
Expectation of number of steps:
E(N) 6 α ln2(Nmax)
Loris Marchal VoroNet 16/ 24
Evaluation (simulation)
Outline
1 Introduction
2 Description of VoroNet
3 Evaluation (simulation)
4 Perspectives, conclusion
Loris Marchal VoroNet 17/ 24
Evaluation (simulation)
Experimental framework
SimulationsI 300.000 objectsI objects are not leaving the overlay (for now)
Distribution of objectI uniformI skewed (powerlaw with parameter α = 1, 2, 5)
We observe:I number of neighborsI polylogarithmic routingI what happens if we add several long range links
Loris Marchal VoroNet 18/ 24
Evaluation (simulation)
Outgoing degree
Uniform distribution
0
10000
20000
30000
40000
50000
60000
70000
80000
90000
0 2 4 6 8 10 12
Obj
ects
Out−degree
Skewed distribution (α = 5)
0
10000
20000
30000
40000
50000
60000
70000
80000
90000
0 2 4 6 8 10 12O
bjec
ts
Out−degree
Loris Marchal VoroNet 19/ 24
Evaluation (simulation)
Polylogarithmic routing
10
20
30
40
50
60
70
80
30025020015010050
Hop
s
Objects in the overlay (*1000)
UniformSparse (alpha = 1)Sparse (alpha = 2)Sparse (alpha = 5)
h =(lnn
)γ ⇔ ln(h) = ln((
lnn)γ
)= γ ln(lnn)
Loris Marchal VoroNet 20/ 24
Evaluation (simulation)
Polylogarithmic routing
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4
4.1
4.2
4.3
2.2 2.25 2.3 2.35 2.4 2.45 2.5 2.55
log(
Hop
s)
log(log(Objects in the overlay))
UniformSparse (alpha = 1)Sparse (alpha = 2)Sparse (alpha = 5)
h =(lnn
)γ ⇔ ln(h) = ln((
lnn)γ
)= γ ln(lnn)
Loris Marchal VoroNet 20/ 24
Evaluation (simulation)
Using several long links to improve routing
10
20
30
40
50
60
70
80
30025020015010050
Hop
s
Objects in the overlay (*1000)
123456789
10
Loris Marchal VoroNet 21/ 24
Perspectives, conclusion
Outline
1 Introduction
2 Description of VoroNet
3 Evaluation (simulation)
4 Perspectives, conclusion
Loris Marchal VoroNet 22/ 24
Perspectives, conclusion
Extension to higher dimensional cases
No bound on the number ofneighbors, O(1) data size ?
Extend small world propertyseems possible
Compute Voronoi diagram:geometric algorithms costlyand sensitive to computationerrors
No need to have completedescription of Voronoi cells,only compute neighborhood
Use other techniques:lifting in dimension d + 1 and LP
Loris Marchal VoroNet 23/ 24
Perspectives, conclusion
Extension to higher dimensional cases
No bound on the number ofneighbors, O(1) data size ?
Extend small world propertyseems possible
Compute Voronoi diagram:geometric algorithms costlyand sensitive to computationerrors
No need to have completedescription of Voronoi cells,only compute neighborhood
Use other techniques:lifting in dimension d + 1 and LP
Loris Marchal VoroNet 23/ 24
Perspectives, conclusion
Extension to higher dimensional cases
No bound on the number ofneighbors, O(1) data size ?
Extend small world propertyseems possible
Compute Voronoi diagram:geometric algorithms costlyand sensitive to computationerrors
No need to have completedescription of Voronoi cells,only compute neighborhood
Use other techniques:lifting in dimension d + 1 and LP
Loris Marchal VoroNet 23/ 24
Perspectives, conclusion
Extension to higher dimensional cases
No bound on the number ofneighbors, O(1) data size ?
Extend small world propertyseems possible
Compute Voronoi diagram:geometric algorithms costlyand sensitive to computationerrors
No need to have completedescription of Voronoi cells,only compute neighborhood
Use other techniques:lifting in dimension d + 1 and LP
Loris Marchal VoroNet 23/ 24
Perspectives, conclusion
Extension to higher dimensional cases
No bound on the number ofneighbors, O(1) data size ?
Extend small world propertyseems possible
Compute Voronoi diagram:geometric algorithms costlyand sensitive to computationerrors
No need to have completedescription of Voronoi cells,only compute neighborhood
Use other techniques:lifting in dimension d + 1 and LP
Loris Marchal VoroNet 23/ 24
Perspectives, conclusion
Extension to higher dimensional cases
No bound on the number ofneighbors, O(1) data size ?
Extend small world propertyseems possible
Compute Voronoi diagram:geometric algorithms costlyand sensitive to computationerrors
No need to have completedescription of Voronoi cells,only compute neighborhood
Use other techniques:lifting in dimension d + 1 and LP
Loris Marchal VoroNet 23/ 24
Perspectives, conclusion
Extension to higher dimensional cases
No bound on the number ofneighbors, O(1) data size ?
Extend small world propertyseems possible
Compute Voronoi diagram:geometric algorithms costlyand sensitive to computationerrors
No need to have completedescription of Voronoi cells,only compute neighborhood
Use other techniques:lifting in dimension d + 1 and LP
z = x2 + y2
Loris Marchal VoroNet 23/ 24
Perspectives, conclusion
Extension to higher dimensional cases
No bound on the number ofneighbors, O(1) data size ?
Extend small world propertyseems possible
Compute Voronoi diagram:geometric algorithms costlyand sensitive to computationerrors
No need to have completedescription of Voronoi cells,only compute neighborhood
Use other techniques:lifting in dimension d + 1 and LP
Loris Marchal VoroNet 23/ 24
Perspectives, conclusion
Extension to higher dimensional cases
No bound on the number ofneighbors, O(1) data size ?
Extend small world propertyseems possible
Compute Voronoi diagram:geometric algorithms costlyand sensitive to computationerrors
No need to have completedescription of Voronoi cells,only compute neighborhood
Use other techniques:lifting in dimension d + 1 and LP
Loris Marchal VoroNet 23/ 24
Perspectives, conclusion
Extension to higher dimensional cases
No bound on the number ofneighbors, O(1) data size ?
Extend small world propertyseems possible
Compute Voronoi diagram:geometric algorithms costlyand sensitive to computationerrors
No need to have completedescription of Voronoi cells,only compute neighborhood
Use other techniques:lifting in dimension d + 1 and LP
Loris Marchal VoroNet 23/ 24
Perspectives, conclusion
Extension to higher dimensional cases
No bound on the number ofneighbors, O(1) data size ?
Extend small world propertyseems possible
Compute Voronoi diagram:geometric algorithms costlyand sensitive to computationerrors
No need to have completedescription of Voronoi cells,only compute neighborhood
Use other techniques:lifting in dimension d + 1 and LP
Loris Marchal VoroNet 23/ 24
Perspectives, conclusion
Conclusion
Perspectives:
Range queries
Queries by proximity: all objects within d from a given object
Fault tolerance ?
VoroNet in a nutshell:
Object-to-object overlay
Efficient routing
Distributed construction and management
Reasonable size of neighborhood
Insensitive to object distribution
Base for complex requests
Loris Marchal VoroNet 24/ 24
Perspectives, conclusion
Conclusion
Perspectives:
Range queries
Queries by proximity: all objects within d from a given object
Fault tolerance ?
VoroNet in a nutshell:
Object-to-object overlay
Efficient routing
Distributed construction and management
Reasonable size of neighborhood
Insensitive to object distribution
Base for complex requests
Loris Marchal VoroNet 24/ 24