Supervised Peer-to-Peer Systems
Kishore Kothapalli and Christian Scheideler∗
Department of Computer Science
Johns Hopkins University
3400 N. Charles Street
Baltimore, MD 21218, USA
Email:{kishore,scheideler}@cs.jhu.edu
Abstract
In this paper we present a general methodology for designing
supervised peer-to-peer systems. A supervised peerto-
peer system is a system in which the overlay network
is formed by a supervisor but in which all other activities
can be performed on a peer-to-peer basis without involving
the supervisor. It can therefore be seen as being between
server-based systems and pure peer-to-peer systems.
The supervisor only has to store a constant amount of information
about the system at any time and only needs to
send a small constant number of messages to integrate or remove
a peer in a constant amount of time. Thus, with a minimum
amount of involvement from the supervisor, peer-topeer
systems can be maintained, for example, that can handle
large distributed computing tasks as well as tasks such
as file sharing and web crawling. Furthermore, our concept
extends easily to multiple supervisors so that peers can
join and leave the network massively in parallel. We also
show how to extend the basic system to provide robustness
guarantees under the presence of random faults and also
adaptive adversarial join/leave attacks. Hence, with our approach,
supervised peer-to-peer systems can share the benefits
of server-based and pure peer-to-peer systems without
inheriting their disadvantages.
1. Introduction
Peer-to-peer systems have recently attracted a signifi-
cant amount of attention inside and outside of the research
community. The advantage of peer-to-peer systems is that
they can scale to millions of sites with low-cost hardware
whereas the classical approach of using server-based systems
does not scale well, unless powerful servers are provided.
On the other hand, server-based systems can provide
∗ Supported by NSF grants CCR-0311121 and CCR-0311795.
guarantees and are therefore preferable for critical applications
that need a high level of reliability. The question is
whether it is possible to marry the two approaches in order
to share their benefits without sharing their disadvantages.
We propose supervised peer-to-peer systems as a possible
solution to this.
A supervised peer-to-peer system is a system in which
the overlay network is formed by a supervisor but in which
all other activities can be performed on a peer-to-peer basis
without involving the supervisor. That is, all peers that want
to join (or leave) the network have to contact the supervisor,
and the supervisor will then initiate their integration into (or
removal from) the network. All other operations, however,
may be executed without involving the supervisor. In order
for a supervised network to be highly scalable, two central
requirements have to be fulfilled:
1. The supervisor needs to store at most a polylogarithmic
amount of information about the system at any
time (i.e. if there are n peers in the system, storing contact
information about O(log2 n) of these peers would
be fine, for example), and
2. the supervisor needs at most a constant number ofmessages
to include a new peer into or exclude an old peer
from the network.
The second condition makes sure that the work of the supervisor
to include or exclude peers from the system is kept
at a minimum. Still, one may certainly wonder whether supervised
peer-to-peer systems are really as scalable as pure
peer-to-peer systems on the one hand and as reliable as
server-based systems on the other hand.
1.1. Motivation
First of all, remember that even pure peer-to-peer systems
need some kind of a “rendezvous point”, such as a
well-known host server [13] or well-known web address
(like gnutellahosts.com), which allows new peers to join the
system. The rendezvous point typically does not play any
role in the overall topology of the network but just acts as
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a bridge between new nodes and the existing network. This
means that nodes have to self-organize to form an overlay
network with good topological properties such as diameter,
degree and expansion.
We show that allowing the supervisor to oversee the
topology of the overlay network, apart from working as
the rendezvous point, tremendously simplifies the problem
of maintaining the above mentioned topological properties
of the overlay network. Hence, as long as the communication
effort of a supervisor for including or excluding a peer
is only a low constant, supervised designs should compete
well with pure peer-to-peer systems.
Our approach has many interesting applications in the
area of grid computing [6, 16, 19], WebTV, and massive
multi-player online gaming [9]. A supervisor may also
serve, for example, as a reliable anchor for code execution
rollback, which is important for failure recovery mechanisms
such as those used in the Time Warp system [7].
This would make supervised peer-to-peer systems particularly
interesting for grid computing. Though supervised
peer-to-peer systems are not as stable as server-based systems
with powerful servers, their advantage is that because
the supervisor only takes care of the topology but may not
be involved at all in peer-to-peer activities, it is from a legal
point of view a much safer design than the server-based
design.
1.2. Our contribution
In Section 2, we show howto combine known techniques
in the pure peer-to-peer world such as the hierarchical decomposition
approach of CAN [14] and the continuousdiscrete
approach [12] in a novel way to obtain a general
framework for the design of supervised peer-to-peer systems
that only requires the supervisor to store a constant
amount of information about the system at any time and to
only send and receive a low constant number of messages
in order to integrate or remove a peer from the system. We
demonstrate our approach by showing howto maintain a supervised
hypercube network and a supervised de Bruijn network
with it. Our scheme can also be extended to allow concurrent
join/leave operations or allow multiple supervisors
as outlined in Section 3. In Section 4 we look at robustness
issues and discuss how our supervised design can be
extended to handle random or adversarial faults.
1.3. Related work
Special cases of supervised peer-to-peer systems have already
been formally investigated in [13, 16, 17], but to the
best of our knowledge a general framework for supervised
peer-to-peer systems has not been presented yet. In [13],
the authors consider a special node called the host server
that is contacted by all new peers that join the system. The
overlay network maintained by the host server is close to
a random-looking graph. As shown by the authors, under a
stochastic model of join/leave requests the overlay network
can, with high probability, guarantee connectivity, low diameter,
and low degree. Alternative designs were later proposed
in [16, 17]. [17] shows how to maintain supervised
trees for guaranteed broadcasting and [16] shows how to
maintain a supervised de Bruijn graph for grid computing.
In this work, we propose a unified model that enables one
to create a large class of supervised overlay networks.
Most of the distributed systems are either server-based
or peer-to-peer. For example, Napster is rather server-based
because all peer requests are handled at a single location.
Also systems like SETI@home [19], Folding@home [8],
and distributed.net [6] are heavily server-oriented because
they do not allow peer-to-peer interactions. Other systems
such as the IBM OptimalGrid allow communication between
peers but it still uses a star topology and therefore
is still closer to being server-based than supervised.
The line of research that is probably closest to our approach
is the work on overlay networks in the area of
application-layer multicasting. Among them are SpreadIt
[5], NICE [1], Overcast [10], and PRM [2], to name a few.
However, these systems only focus on specific topologies
such as trees, and they do not seem to be generalizable to
a universal approach for supervised systems. Other protocols
for application-layer multicasting such as Scribe [4],
Bayeux [23], I3 [20], Borg [22], SplitStream [3], and CANMulticast
[15] are rather extensions of a pure peer-to-peer
system.
2. A general framework for supervised peerto-
peer systems
Our general framework for supervised peer-to-peer systems
needs several ingredients, including a hierarchical
decomposition technique [14], a continuous-discrete technique
[12], and a recursive labeling technique. After presenting
these techniques we show how to glue them together
in an appropriate way so that we obtain a universal
approach for supervised peer-to-peer systems. Afterwards,
we give some examples that demonstrate how to apply this
approach.
2.1. The hierarchical decomposition technique
Consider any space U = [0, 1)d for some d ≥ 1. The
decomposition tree T (U) of U is an infinite binary tree in
which the root represents U and for every node v representing
the subcubeU_ in U, the children of v represent two subcubes
U__ and U___, where U__ and U___ are the result of cutting
U_ in the middle at the smallest dimension in which U_
has a maximum side length. Let every edge to a left child
in T (U) be labeled with 0 and every edge to a right child
in T (U) be labeled with 1. Then the label of a node v, _v,
is the sequence of all edge labels encountered when moving
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along the unique path from the root of T (U) downwards to
v.
Our goal for the supervised peer-to-peer system will be
to map the peers to nodes of T (U) so that
• the subcubes of the (nodes assigned to the) peers are
disjoint,
• the union of the subcubes of the peers gives the entire
set U, and
• the peers are only distributed among nodes of two consecutive
levels in T (U).
Whereas CAN-based peer-to-peer systems usually satisfy
the first two properties, they have problems with the third
property. But as we will see, it will be easy for our supervised
peer-to-peer approach to also maintain the third property.
2.2. The continuous-discrete technique
The basic idea underlying the continuous-discrete approach
[12] is to define a continuous model of graphs and
to apply this continuous model to the discrete setting of a fi-
nite set of peers.
Consider any d-dimensional space U = [0, 1)d, and suppose
that we have a set F of functions fi : U → U.
Then we define EF as the set of all pairs (x, y) ∈ U2
with y = fi(x) for some i. Given any subset S ⊆ U, let
Γ(S) = {y ∈ U \ S | ∃x ∈ S : (x, y) ∈ EF }. The expansion
α of F is defined as α = minS⊆U, |S|≤|U|/2
|Γ(S)|
|S|
where |S| denotes the volume of a set S. F is called mixing
ifα > 0 and rapidly mixing if α is a constant. If F does
not mix, then there are disconnected areas in U.
Consider now any set of peers V , and let R(v) be the
region in U that has been assigned to peer v. Let GF (V )
be the graph with node set V that contains an edge (v,w)
for every pair of nodes v and w for which there is an edge
(x, y) ∈ EF with x ∈ R(v) and y ∈ R(w). It can be
seen that if F is mixing and ∪vR(v) = U then GF (v) is
connected. Moreover, the following theorem holds implying
that the expansion in the continuous case can be preserved
in the discrete case.
Theorem 2.1 If F has an expansion of α and regions have
been assigned to the peers so that all three demands stated
in the hierarchical decomposition approach are satisfied,
then GF (V ) has an expansion of Ω(α).
2.3. The recursive labeling technique
In the recursive labeling approach, the supervisor assigns
a label to every peer that wants to join the system. The labels
are represented as binary strings and are generated in the
order: 0, 1, 01, 11, 001, 011, 101, 111, 0001, 0011, . . .. Formally,
consider the mapping _ : IN0 → {0, 1}∗ with
the property that for every x ∈ IN0 with binary representation
(xd . . . x0)2 (where d is minimum possible),
_(x) = (xd−1 . . . x0xd). Then _ generates the sequence
of labels displayed above. In the following, it will also
be helpful to view labels as real numbers in [0, 1). Let
the function r : {0, 1}∗ → [0, 1) be defined so that
for every label _ = (_1_2 . . . _d) ∈ {0, 1}∗, r(_) =
_d
i=1
_i
2i . Then the sequence of labels above translates into
0, 1/2, 1/4, 3/4, 1/8, 3/8, 5/8, 7/8, 1/16, 3/16, . . ..
When using the recursive approach, the supervisor aims to
maintain the following invariant at any time:
Invariant 2.2 The set of labels used by the peers is
{_(0), _(1), . . . , _(n − 1)}, where n is the current number
of peers in the system.
This above invariant can be preserved when using the
simple strategy of assigning the label _(n) to a new peer
that wants to join the system and increasing n by 1. Similarly,
when a peer w with label _ is leaving the system, the
supervisor asks node with label _(n − 1) to take over the
role and label of w and decrements n by 1.
2.4. Putting all pieces together
Now we are ready to put the pieces together.We assume
that we have a single supervisor for maintaining the overlay
network. In the following, the label assigned to some peer
v will be denoted as _v. Given n peers with unique labels,
we define the predecessor pred(v) of peer v as the peer w
for which r(_w) is closest from below to r(_v), and we de-
fine the successor succ(v) of peer v as the peer w for which
r(_w) is closest fromabove to r(_v) (viewing [0, 1) as a ring
in both cases). Given two peers v and w, we define their distance
as δ(v,w) = min{(1 + r(_v) − r(_w)) mod 1, (1 +
r(_w)−r(_v)) mod 1}. In order to maintain a doubly linked
cycle among the peers, we simply have to maintain the following
invariant:
Invariant 2.3 Every peer v in the system is connected to
pred(v) and succ(v).
Now, suppose that the labels of the peers are generated
via the recursive strategy above. Then we have the following
properties:
Lemma 2.4 Let n be the current number of peers in the
system, and let ˉn = 2_log n_. Then for every peer v ∈ V ,
|_v| ≤ _log n and δ(v, pred(v)) ∈ {1/(2ˉn), 1/ˉn}.
So the peers are approximately evenly distributed in
[0, 1) and the number of bits for storing a label is as low
as it can be without violating the uniqueness requirement.
Now, recall the hierarchical decomposition approach.
The supervisor will assign every peer p to the unique node
v in T (U) at level log(1/δ(p, pred(p))) with _v being equal
to _p (padded with 0’s to the right so that |_v| = |_p|).
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As an example, if we have 4 peers currently in the system,
then the mapping of peer labels to node labels is
0 → 00, 1 → 10, 01 → 01, 11 → 11. With this strategy,
it follows from Lemma 2.4 that all three demands formulated
in the hierarchical decomposition approach are satisfied.
Consider now any family F of functions acting on some
space U = [0, 1)d and let C(p) be the subcube of the node
in T (U) that p has been assigned to. Then the goal of the supervisor
is to maintain the following invariant at any time.
Invariant 2.5 For the current set V of peers in the system
it holds that
1. the set of labels used by the peers is
{_(0), _(1), . . . , _(n − 1)}, where n = |V |,
2. every peer v in the system is connected to pred(v) and
succ(v), and
3. there are bidirectional connections {v,w} for every
pair of peers v and w for which there is an edge
(x, y) ∈ EF with x ∈ C(v) and y ∈ C(w).
2.5. Maintaining Invariant 2.5
Next we describe the actions that the supervisor has to
perform in order to maintain Invariant 2.5 during an isolated
join or leave operation. For simplicity, we assume that
all nodes are reliable and trustworthy and also that peers depart
gracefully. We also assume that each message sent by
the supervisor can contain up to a constant number of node
addresses and labels. We start with the following important
fact which can be easily shown.
Fact 2.6 Whenever a new peer v enters the system, then
pred(v) has all the connectivity information v needs to satisfy
Invariant 2.5(3), and whenever an old peer w leaves
the system, then it suffices that it transfers all of its connectivity
information to pred(w) in order to maintain Invariant
2.5(3).
Thus, if the peers take care of the connections in Invariant
2.5(3), the only part that the supervisor has to take care
of is maintaining the cycle. For this we require the following
invariant.
Invariant 2.7 At any time, the supervisor stores the contact
information of pred(v), v, succ(v), and succ(succ(v))
where v is the peer with label _(n − 1).
We now describe how to maintain the invariant during
any join or leave operation. In the following, S denotes the
supervisor.
Join: If a new peer w joins, in order to satisfy Invariant 2.7,
the following actions are performed.
• S informs w that _(n) is its label, succ(v) is its predecessor,
and succ(succ(v)) is its successor.
• S informs succ(v) that w is its new successor.
• S informs succ(succ(v)) that w is its new predecessor.
• S asks succ(succ(v)) to send its successor information
to the supervisor, and
• S asks v which is now pred(w) to send the connectivity
information according to F to node w.
• S sets n = n + 1.
Leave: If an old node w reports _w, pred(w) and succ(w)
before it leaves, then the following actions can be performed
in order to maintain Invariant 2.7.
• S informs v (the node with label _(n − 1)) that _w
is its new label, pred(w) is its new predecessor, and
succ(w) is its new successor.
• S informs pred(w) that its new successor is v and
succ(w) that its new predecessor is v.
• S informs pred(v) that succ(v) is its new successor
and succ(v) that pred(v) is its new predecessor.
• S asks pred(v) to send its predecessor information to
the supervisor and to ask pred(pred(v)) to send its
predecessor information to the supervisor, and
• S sets n = n − 1.
Thus, the supervisor only needs to handle a constant number
of messages, at most 8, for each join/leave of a peer.
2.6. Examples
For a supervised hypercubic network, simply select F
as the family of functions on [0, 1) with fi(x) = x +
1/2i (mod 1) for every i ≥ 1. Using our framework, this
gives an overlay network with degree O(log n), diameter
O(log n), and expansionO(1/√log n), which is better than
what pure hypercubic peer-to-peer systems like Chord [21]
can achieve.
For a supervised de Bruijn network, simply select F as
the family of functions on [0, 1) with f0(x) = x/2 and
f1(x) = (1 +x)/2. Using our framework, this gives an
overlay network with degree O(1), diameter O(log n), and
expansion O(1/ log n), which is also better than the previous
pure de Bruijn peer-to-peer systems [11, 12].
3. Concurrency
In this section we extend our approach to concurrent join
and leave operations and also provide a way to allow multiple
supervisors.
3.1. Concurrent Join/Leave Operations
In order to be able to handle d join or leave requests in
parallel, we extend Invariant 2.5 with the following rule:
4. Every peer v in the system is connected to d predecessors
and predi(v) for i = 1, 2, • • • , d and d successors
succi(v) for i = 1, 2, • • • , d, where predi(v)
(resp. succi(v)) is the ith predecessor (resp. successor)
of v on the cycle.
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In addition to this, given that v is the node with label _(n − 1), Invariant 2.7 needs to be extended to:
Invariant 3.1 At any time, the supervisor stores the contact
information of v, the 2d successors of v, and the 3d predecessors
of v.
These invariants can be preserved during join/leave operations
and the following claim holds:
Claim 3.2 The supervisor needs at most O(d) work and
O(1) time (given that the work can be done in parallel) to
process d join or leave operations.
3.2. Multiple Supervisors
We now show how multiple supervisors can work together
in maintaining a single supervised peer-to-peer system.
In a network with k supervisors S0, S1, • • • Sk−1, the
[0, 1)-ring is split into the k regions Ri = [(i − 1)/k, i/k),
1 ≤ i ≤ k, and supervisor Si is responsible for region
Ri. Every supervisor manages its region as described for
a single supervisor above, and the borders are maintained
by communicating with the neighboring supervisors on the
ring. Each time a new node v wants to join the system via
some supervisor Si, Si forwards it to a random supervisor to
integrate v into the system. Each time a node v under some
supervisor Si wants to leave the system, Si contacts a random
supervisor (which may also be itself) to provide a replacement
node. Using Chernoff bounds we get:
Claim 3.3 Let n be the total number of nodes in the system.
Then it holds for every i ∈ {1, . . . , k} that the number
nodes currently placed in Ri is in the range n/k ± O(_(n/k) log k + logk), with high probability.
4. Robustness
In this section we show how to extend the basic scheme
to provide robustness guarantees against random node failures
and also adaptive adversarial join/leave attacks [18].
4.1. Robustness against random faults
We call a supervised network robust if as long as at most
a constant fraction of the nodes fail, the supervisor can still
recover the rest of the network. In order to be robust against
random faults, the supervisor uses the scheme of Section 3.1
with d = c log n for some constant c > 1. Thus, each node
is connected to c log n predecessors and successors. The supervisor
stores the contact information along the lines of Invariant
3.1. The following theorem can be shown.
Theorem 4.1 Even if every node in the network fails independently
with a constant probability, the scheme above in
which the supervisor only maintains a logarithmic amount
of information at any time suffices to fully restore the network
and each leave operation executed during the recovery
needs O(log n) work, with high probability.
4.2. Robustness against adversarial attacks
When considering adaptive adversarial attacks (e.g.,
[18]) it does not suffice that the supervisor maintain information
as in the previous subsection as the adversary
can place nodes at critical positions to effectively disconnect
the supervisor from the network or disrupt routing.
Formally, we allow the adversary to own up to _n of the
n nodes in the system for some sufficiently small constant
_ > 0. These nodes are also called adversarial nodes and
the rest are called honest nodes. The supervisor and the honest
nodes are oblivious to adversarial nodes, i.e., there is no
mechanism to distinguish at any time whether a particular
node is honest or not. To achieve robustness in the presence
of an adaptive adversary, we use the following scheme.
In the following, a region is an interval of size 1/2i in
[0, 1) starting at an integer multiple of 1/2i for some i ≥ 0,
and a node v belongs to a region R if r(_v) ∈ R. Recall
that n = 2_log n_. The supervisor organizes the nodes into
regions so that each region contains between c log n and
2c log n nodes for some constant c > 1. Whenever these
bounds are violated in a region, the supervisor splits it or
merges it with a neighboring region. The n nodes are also
organized into 5 sets S1 to S5 and the following invariant is
maintained for these sets.
Invariant 4.2 At all times,
1. S1 has ˉn/8 nodes with labels _(0), • • • , _(n/8 − 1).
2. S2 has n/8 nodes with labels _(n/8), • • • , _(n/4 − 1).
3. S3 has n/4 nodes with labels _(n/4), • • • , _(n/2 − 1).
4. S4 has n/2 nodes with labels _(n/2), • • • , _(n − 1).
5. S5 has the remaining n − n nodes with labels
_(n), • • • , _(n − 1).
The following invariant describes the connections maintained
by the nodes in the various sets and the connections
maintained by the supervisor. To simplify notation, for a
real number x ∈ [0, 1), R(x) is the region that x belongs
to and Si(R) is the set of Si-nodes belonging to R. For every
region R, let SR = S1(R) ∪ S2(R) and ˉ SR = S3(R) ∪ S4(R) ∪ S5(R) if R precedes R(r(_(n))) and otherwise,
SR = S1(R) and ˉ SR = S2(R) ∪ S3(R) ∪ S4(R) ∪ S5(R).
Invariant 4.3 For all regions R, every SR-node is connected
to all nodes in SR ∪ ˉ SR. Every SR-node is also connected
to all nodes in the predecessor and successor regions
of R, denoted pred(R) and succ(R), and for every u ∈ SR
that has a connection to a node v ∈ SR_ according to Invariant
2.5(3), all SR-nodes are connected to all SR_-nodes.
The supervisor is connected to all the nodes in
SR in the regions R(r(_(n))), pred(R(r(_(n)))) and
succ(R(r(_(n)))).
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The set S1 is also referred to as the stable set. The goal of
the supervisor is to have the honest nodes in the majority in
every set S1(R), with high probability, since then quorum
strategies can be used to wash out adversarial behavior. The
set S2 is in a stage called the split-and-merge stage because
S2-nodes are merged into the stable set or removed from it
as nodes join or leave the system. The set S3 is in a stage
called mixing stage in which the supervisor performs random
transpositions to ensure that the nodes are well-mixed
before being integrated into the stable set. The set S4 is in
a reservoir stage. S4 is used to fill departed positions in the
sets S1 to S3 by selecting random nodes in S4 and filling
their positions with the last nodes in S5. Finally, the set S5
is in a filling stage where new nodes are added by assigning
them the label _(n).
The join and leave operations are extended as follows.
Join: The supervisor assigns to the new node the label
_(n) and integrates it so that the Invariants 4.2 and 4.3 are
satisfied. Each time a new node u is added where r(_(n))
is the successor of r(_v) for a node v at a position in S3
that has not performed a random transposition yet, a random
node w ∈ S3 is picked and the positions of v and w
are switched. (This is realized by the supervisor informing
all nodes in S1(R(_(n))) which positions are to be switched
so that this is reliably done without involving the supervisor.)
Each time a new node causes the supervisor to switch
from a region R to succ(R), the nodes in S2(R) are merged
into S1(R) as prescribed by Invariant 4.3.
Leave: If a node v leaves with v ∈ S4 ∪ S5, the supervisor
simply replaces it by the last node in S5. Otherwise,
the supervisor replaces v by a random node in S4 and
fills the position of that random node with the last node in
S5. (The supervisor initiates the leave operation for v only
if a majority of S1-nodes in v’s region notify it about that.
In this case, the supervisor has the necessary information
to correctly initiate the replacement.) Each time a departure
causes the supervisor to switch froma regionR to pred(R),
the nodes in S2(pred(R)) are split away from S1(R) as prescribed
by Invariant 4.3.
These operations yield the following result.
Theorem 4.4 For a sufficiently small constant _ > 0 it
holds that as long as the adversary owns at most _n nodes,
the above scheme guarantees that in every region R, the
honest nodes are in the majority in S1(R), with high probability.
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