MULTIDIMENSIONAL VECTOR ROUTING IN A P2P NETWORK
Laurent Yeh, Georges Gardarin and Florin Dragan
PRiSM Laboratory, Versailles University, 45 avenue des Etats Unis, Versailles, France
Keywords:
DHT, P2P, Routing, Multidimensional indexing.
Abstract:
P2P systems tend to be largely accepted as a common support for deploying massively distributed data man-
agement applications. Many multidimensional data P2P indexing techniques suffer from severe limitations
regarding the number of data dimensions to be indexed. In this paper, we propose a new approach for indexing
multidimensional data in a P2P architecture. It is based on an efficient query overlay network (the so-called
routing layer) built from a new data structure named skip-zone. We index data pieces of large dimensions as
vectors and we adopt the polyhedral algebra for splitting the domain of possible values into sub-domains. To
manage the overlay network meta-data required for tracking the network evolution or for routing data at each
peer, we propose an efficient distributed meta-data layer that works in cooperation with the routing layer. An
evaluation outlines the main properties of our architecture versus those of similar systems. The insensibility
of the vector dimension in our data model is a key advantage of our approach.
1 INTRODUCTION
While first used for simple queries, such as search-
ing for filenames (or document names), Peer-to-Peer
(P2P) networks represent a possible basis for imple-
menting large scale database systems. Among the
main qualities that distinguish P2P networks, we re-
member: robustness, scalability, reliability, no central
administration, no control over data placement. How-
ever, one of the problems of P2P data sharing sys-
tems is the weak data storage and retrieval capabilities
when faced with anything more complex than simple
values.
A recent problem that attracts the attention of the
database community is how P2P networks can be used
for efficient management of multidimensionnal data
(e.g., vectors). Many vector-based applications re-
quire the indexing of long vectors. Long vectors can
be used for representing image descriptors (for ex-
ample, color descriptors for images are usually com-
posed of at least 64 values). Yet another example
is indexing XML documents with long paths. Each
element name can be mapped to a value, the se-
quence of steps constituting a vector. The existing
solutions (S. Abiteboul, 2005) (Cai et al., 2003) (Li
et al., 2003)(Shu et al., 2005) have several limitations
among we remember: due to the hierarchical orga-
nization of the zone encoding, wild-card queries re-
quire less or more flooding requests for each bit with-
out determined values (0 or 1). The second problem
is related to the key length of the DHT. For instance,
Chord uses 160 bits; this is not enough for supporting
long vectors. For a vector of dimension N and for a
zone enclosing the point at level L, N ∗ L bits are re-
quired. Thus, the DHT entails a limit for the vector
length.
The main contribution of this paper is a new P2P
routing method suitable for vector data. The method
is not sensitive to the dimension of vectors to be in-
dexed (as shown in the section on experimental eval-
uation). The method is based on two basic ideas: the
first is the use of polyhedral geometry (Laidlaw et al.,
1986) that allows us to route independently of the
considered dimension; the second is an extension of
the Skip-List (Pugh, 1989)(Aspnes and Shah, 2003)
data structure for fast routing. We name this exten-
sion Skip-Zone.
Our solution is based on a P2P architecture di-
486
Yeh L., Gardarin G. and Dragan F. (2007).
MULTIDIMENSIONAL VECTOR ROUTING IN A P2P NETWORK.
In Proceedings of the Ninth International Conference on Enterprise Information Systems - DISI, pages 486-489
DOI: 10.5220/0002382104860489
Copyright
c
 SciTePress

vided in two layers: i) The first layer (named meta-
data layer) manages the metadata for the whole net-
work. The main purpose of this layer is to compute
the specific zone (a domain of values) controlled by
each peer. ii) The second layer is a routing layer
organized according to the new notion of skip-zone.
These two layers cooperate for the network evolution:
the first layer provides metadata information for every
peer that connects and disconnects into the network,
or splits its content; the second layer efficiently routes
the requests to the relevant vectors. The main advan-
tage of our solution is that peer evolution (splitting
and merging contents) involves only local metadata.
In contrast, other algoithms publish the splitting meta-
data to every peer in the overlay network for each peer
splitting event. This appears as a bottleneck for the
network scalability.
2 NETWORK STRUCTURE
We propose a network overlay divided in two layers.
The first (metadata layer) deals with metadata; it in-
sures the splitting of the value domain in a sequence
of zones covering a continuous range of values. The
second layer deals with query routing.
2.1 Network Architecture
routing layer
H5
H6
Meta-data layer
= hub
= peer
Routing to
index peer
Fetching
data
Sending
data
routing
peer R
index
peer I
client
peer C
data source
peer S
H3
P32
H0
H2H1
Hub storage peer
H for hub H
2
Figure 1: A Network Instance.
Figure 1 presents an instance of the P2P network
where the metadata layer is portrayed in the upper part
and the routing layer in the lower one. The metadata
layer is arranged in a tree-like topology and manages
how data is distributed to peers. The routing layer
manages the routing of data requests to peers with rel-
evant data. It integrates the data source peers as tree
leaves. Peers are here data storage elements. In this
organization, each peer is placed according to its in-
dex role (e.g., peer I). A peer is located in the network
under the index node determined by the domain of
values it manages. The localization of a peer index is
done by the metadata layer as explained in the sequel.
2.2 Metadata Layer
The Metadata layer deals with the problem of splitting
the domain of possible values into sub-zones. Hubs
are organized in a hierarchy and each leave zone is
allocated to an index peer.
Thus, zones are divided along the descending
paths in the hub hierarchy; the zones from upper lev-
els cover zones from lower levels. For example ac-
cording to Figure 1 the zone indexed at the hub H
0
describes all the domain of values. The union of
zones described by the hubs H
1
, H
2
, H
3
represents the
zone described by the root hub H
0
. An other example
is Zone(H
3
) = (Zone(H
5
) ∪ Zone(P
32
) ∪ Zone(H
6
)).
Recursively, each hub describes a sub-zone of its par-
ent one. By using a tree topology, zones are split con-
tinuously. Each index peer can store any value de-
scribed by its immediate parent hub.
Structure of a hub: A hub is a metadata-layer com-
ponent that helps to define the domains of values for
every peers. Each hub has N children (that are other
hubs or peers) and one parent hub, except the root
hub. A hub of size N has a single input and N-outputs
and is characterized by a set of zones {Z
1
, Z
2
, .., Z
n
}.
Each zone is defined by a polyhedron (Laidlaw
et al., 1986); a polyhedron is able to describe a ”vol-
ume” in any dimension; it provides the ∈ operator
for testing if a point is included in a volume. Mathe-
matically, a polyhedron is defined by a set of inequa-
tions. An example in dimension 4 is: x
1
≥ 10 ∧ x
1
≤
20∧ x
2
≥ 8∧ x
2
≤ 167∧ 2∗x
3
+ x
1
≥ 20∧ x
3
≤ 120∧
x
4
+ 5 ∗ x
2
≥ 10 ∧ x
4
≤ 20. By hub-dimension, we
refer to the maximum number of children that can be
connected to a hub. The hub-dimension is a parameter
that is considered to be fixed for all hubs in the meta-
data layer (a strategy with a variable hub-dimension
can also be considered).
Two important properties of a set of zones stored
in hubs are:
(i) Z
1
(HUB
k
) is
adjacent Z
2
(HUB
k
) is adjacent
...is
adjacent Z
n
(HUB
k
).
(ii) ∃ j such that (Z
1
(HUB
k
) ∪ Z
2
(HUB
k
). . . ∪
Z
n
(HUB
k
)) ⊂ Z
j
(HUB
f
) where Parent(HUB
k
) =
HUB
f
.
The first property (adjacency property) assures the
continuity of zones, while the second states that each
set of zones described by a parent hub is included in
a zone described by a hub at a higher level.
Dynamicity of the Metadata Layer: The metadata
layer evolves in terms of hub splitting and grouping.
MULTIDIMENSIONAL VECTOR ROUTING IN A P2P NETWORK
487

A hub is split when its load is too high. Two twin
hubs are tentatively grouped when the load of one is
below a threshold. A peer can trigger either a split or
a group without centralized decision. Depending on
the data load of their peer children, hubs may split or
may fusion.
Peer index split: When a peer P is saturated, its load
must be distributed to other peers. For that P invites
an available peer from the network to share a part of
its load, then the peer P shares its load with the new
index peer. If the hub-dimension of the hub H
i
allows
to receive a new zone, this new peer is simply linked
to the old hub. Otherwise, a new hub is created and
the two previous peers are connected to it.
Computation of New Zones: Splitting a peer load
with load balancing control is an important problem.
The problem is to find a set of inequations that divides
the set of n multidimensional points indexed by a peer
in two sets of approximately equal size. Splitting in
two even subsets is not an easy task. One simple so-
lution is to iterate through the set of vectors for each
dimension until a set of good splitting equations is
found. Two polyhedra containing approximately the
same number of points must be created from the orig-
inal one. As an example, consider a network index-
ing vectors of 3 dimensions < x1, x2, x3 > and a peer
Pi indexing the set of 4 points: {v
1
, v
2
, v
3
, v
4
}={<
0, 0, 0 >, < 0, 0, 2 >, < 0,2,0 >, < 2,0,0 >}. Con-
sidering the values of the first dimension, a possible
split is defined by x
1
= 2. In this case, the decom-
position generates a set of 3 values and a set of one
value. As it can be observed the values are not evenly
distributed and another decomposition must be found.
Similarly, we can find decompositions on the second
and on the third dimensions with the same (uneven)
distribution of values. To find a good solution, at least
two dimensions must be considered.
2.3 Routing Layer
This section describes the routing layer which is run
for routing a data request through the network. It does
not require the metadata layer. Every peer encloses
a data structure named skip-zone that can work au-
tonomously.
2.3.1 Skip-Zone Structure for Routing
Figure 2 illustrates the architecture of the vector rout-
ing layer. It is inspired from traditional skip-list. The
routing layer is built over all peers in the lower part of
the network of hubs as illustrated in figure 1.
Every peer has an autonomous Skip-Zone. A
Skip-Zone uses the notion of zone again as illustrated
in Figure 2. Every peer has 2 ∗ L zones (L zones for
P5 P7P6
Zl1
Zl2
Zr1
Zr2
Zr3
P8 P9 P10P4P2 P3P1
V1 V2
Polyhedron
V  ?
…
(1)
Figure 2: The Routing Layer.
the left part and L zone for the right part). L is the
number of levels in a skip-zone structure, which is a
fixed parameter for the entire network. According to
the Figure 2, the skip-zone defines three levels. On
each side, a zone covers a set of peers. Notice that
every peer is localized in the network according to
the position given by the metadata layer. For a peer,
a zone defines a domain (space) of multidimensional
values indexed by all peers from its position to a peer
at a distance D
k
, where k ∈ 1..L.
Thus, the first level covers all peers at a distance
D
1
from the original peer (e.g. Zl
1
, respectively Zr
1
).
The second level covers a distance of D
2
, the third
level for a distance D
3
, and so on. The choice of D
is arbitrary, but it is fixed for the entire network. Ac-
cording to Figure 2, and based on the ordering rela-
tion, the Zr
1
, (respectively Zl
1
) zone is included in
the zone represented by Zr
2
(respectively Zl
2
). We
have the following relationship: Zl
i
⊆ Zl
j
, Zr
i
⊆ Zr
j
for i < j.
Each zone is composed of a polyhedron and a
pointer to the peer at the extremity of the zone. For
examples, the zone Zr
2
has a pointer to the peer P6,
and the zone Zr
3
has a pointer to the peer P10.
The number of levels for a skip-zone could be
high. Indeed, the storage cost for each polyhedron
is very low (2 inequations for each dimension). The
computation of each zone is inexpensive as described
in the above section. Thus, for a network, it makes
sense that a Skip-Zone can cover the entire network
or a large part of it.
3 MULTIDIMENSIONAL POINT
ROUTING
Before presenting the routing principles, notice that
according to our previous description of the metadata
layer, if a point exist, it must be in one and only one
index peer (or in other words, in a single zone).
Considering the example in Figure 2, for a point V
to be routed to the relevant peer index, the algorithm
ICEIS 2007 - International Conference on Enterprise Information Systems
488

must firstly checks whether it is in the zone Zr
3
. If
it is true, the immediate superior level (Zr
2
) is used
to determine if the point V is at the left or right of P6.
For example, in the case that the point is at the right of
P6, the request is propagated to P6 which recursively
processes the algorithm. Notice that in a given skip-
zone, the search cost is logarithmic.
On the contrary, if the answer of the previous
check in Zr
3
is false, the point is tested again with
the left highest level Zl
3
with the process described
in the previous paragraph. In the case that the check
is again false, the request is forwarded to the extrem-
ity of Zl
3
and Zr
3
(peer P
8
). The network clones the
original request in two requests but with a notable dif-
ference: Each request has a direction (go left or right).
In our example, only P8 will receive the propagation
of the request. At P
8
, due to the direction, the check
is only done on the right part of the skip-zone owned
by P8. If the test is again false, the request is once
more propagated to the skip-zone at the highest right
extremity.
4 DIMENSION INSENSIBILITY
In contrast to other works, in which the DHT rout-
ing key size represents a bottleneck, our routing ap-
proach is not very sensible to the dimension of the
searched vector. Indeed, our routing principle is based
on the (∈) operation that checks if a point is in a zone
(volume in a N-dimension space) or if two volumes
intersect. This check is processed according to the
polyhedral principle that requires 2*D tests. Each
check is the test of a set of inequations of the type
a
d
x
d
+ a
d−1
x
d−1
... < 0 which is solved with a very
low CPU cost compared to messages transmission de-
lay or disk access. Thus, each new dimension requires
to test at least two additional inequations. Solving for
data of 1 dimension or N dimensions differs only by
the CPU time required to check each inequation. That
is why our approach is almost not sensible to the di-
mension of vectors.
5 CONCLUSION
In this paper, we introduced a P2P system suitable
for indexing and routing vector data. Compared to
other works, our originality is twofold. First we use
polyedra for describing data zone managed by each
peer. This is done in a first overlay network named
metadata layer. Secondly, we extend the principle of
skip-list for working with data of any dimension. We
named this new structure ”skip-zone”.
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