OPTIMAL
Christos
COTERIESt
H. Papadimitrioul
in
the protocol
connected subgraph
implied
by a coterie,
of the network
set of nodes contains
functions
vote assignments
a
if its
one of the sets in the family.
way to implement
coteries is by voting.
We study the following
problem:
Given a network
with weigh ts, find the coterie that maximizes the expected number of operating
nodes.
We show that
and
respect
to two
number
of node (edge)
partition.
Finally,
techniques are described for dynamically
votes upon node or edge failures.
Our measure
of the desirability
of a coterie
is the expected number of operating
nodes (possibly
weighted).
In the literature
more crude measures are
nodes and links
nodes must continue
In [GMB85]
posed as a protocol
is the minimum
with
node (edge) vulnera-
that result in no operating
manner.
may fail,
and the re-
functioning.
If however
the network has been partitioned
into isolated components, different partitions
may arrive at conflicting
decisions.
bility,
which
namely
In this paper we study the problem of designing
the optimal
coterie for a network.
We assume independent node failures, all with equal probability
P;
edge failures and independent
node failures with different probabilities
can be handled in a very similar
1. INTRODUCTION
maining
measures,
in [BGM89]
reassigning
this problem,
although hopelessly intractable
in general, can be solved when the network is a cactus (all
biconnected
components
are either edges or cycles).
In a network
are compared
performance
failures
One simple
Even the case of a cycle with equal probabilities
weights is quite nontrivial
and interesting.
Sideri2
H. Garcia–Molina
and D. Barbara have studied
In [GMB85]
it is pointed
coteries in some detail.
out (in two theorems attributed
to M. Yannakakis)
that coteries are much more general, than vote assignments, since there are at most 2n different vote
assignments,
and at least 22’” coteries. In [BGM86]
ABSTRACT:
In a network processors may faiL It
is desirable that the surviving
nodes continue to operate, but no two disjoint
partitions
operate independently.
One SOIUtion for this problem
is based
on the notion of coteries.
A coterie is a family of
sets of nodes such that any two sets in the family
intersect.
and Martha
the concept
for allowing
of a coterie is pro
studied, such as the probability
that some node is
number of nodes that hit
erating, or the minimum
coteries. To our knowledge the present measure
not been studied in the past. We feel that it is
opall
has
ob-
viously
the
expected
at most one partition
a more
accurate
throughput
each node receives
measure
as it captures
of the system
a stream
(assuming
of requests
that
of intensity
to operate.
In this protocol
we fix a set of subsets
of nodes (this set of subsets is called a coterie);
a
partition
continues to operate if and only if it contains one of these subsets. Obviously
any two such
subsets must intersect and any subset must induce
a connected subgraph of the graph representing
the
equal to its weight).
network.
An easy and practical
way to implement
coteries is by voting:
Each node gets a number of
votes, and a partition
can operate if it has a quo-
is easily shown to be intractable.)
Interestingly,
all
coteries shown optimal
in our work are vote assignments.
It is an open problem
whether
there is a
rum.
network
Besides, our results
can be eas-
ily extended to the cases of the more crude measures
mentioned
above. Furthermore,
we show that several interesting
cases of the problem are analytically
tractable,
in that the optimum
coterie can be calculated (although
the problem for general networks
in which
the optimum
coterie
is not a vote
assignment.
tion
tReseamh
supported
by the ESPRIT
Basic
No. 3075 ALCOM,
and an NSF Grant.
1 University
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0-89791-439-2/91/0007/0075
specific
ultimate
positive
result
is a polynomial-
time algorithm
for computing
the optimum
coterie
(a graph no two cycles of which share
of any cactus
Greece-
fee all or part of this material
the ACM
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at San Diego.
Institute,
that the copies are not made or distributed
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comof the
is by
To copy
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$1.50
75
coteries at most one such component
exists).
If no
such component
exists then w(F) = O. That is,
an edge), possibly weighted.
We start by examining the more pure cases of a tree and a cycle. In
the case of a tree we show that the optimum
coterie
is a monarchy
(a coterie that
which contains a single node,
optimum
center can be easily
Interestingly,
the location
of
the probability
of failure
contains only one set,
the “center”)
and the
computed analytically.
the center depends on
w(F)
=
Finally
pected
w(v)
{&C
the performance
weighted number
in the the case of a ring with no weights (and
probabilities
of failure equal to p), the situation
complex.
we characterize
exactly
By a sequence
the
P(S)
all
is
imately
is the odd oligarchy
equally
M 6 S;
of coterie S is P(S)
of working nodes.
the ex-
Our goal is given G, p, and w to design a coterie S
is as large as possible. It is obvious
that we only need to consider coteries which have the
following
properties:
of lemmata
optimum
coterie
(2k + 1 nodes,
spaced along
the ring,
on a
(2) Connectivity,
that
is for any M c S, Gl~
is con-
nected.
(3) Nonredundancy,
that
is Ml,
M2 c S ~
Ml
~
S – {M}
U
M2 .
approx-
(4) Minimality,
are assigned
one vote, and all other nodes O votes).
mum k decreases with p and eventually
= &FW(F)
such that P(S)
ring with n nodes. If the expected number of failed
nodes, np, is less than 2, then the optimum
coterie
is the democracy
(each node has a vote and, if the
number of nodes is even, one node is assigned a tiebreaking second vote). As pn increases, the optimum
coterie
C~
p.
When the network
has cycles, computing
the
best coterie is quite a bit more complicated.
Even
unexpectedly
if 3C ~ C(F),
otherwise
{M’}
The optik becomes
that
violates
(5) Maximality,
is M’
C M E S ~
(l)–(3).
that
is VM
@ S, S
U {Al}
violates
(l)-(4).
O, that is the optimum
coterie is again the monarchy. For practicaJ purposes, our calculations
show
that the optimum
coterie of a ring is approximated
within
l% by choosing the best among the democracy, the monarchy and the troika (k = 1).
Coteries that do not have properties
(2)-(5)
can be
replaced by coteries that have better performance
as follows:
If (2) is not satisfied, then add to S all
trees that span M on G, and delete M. If (3) is not
When the ring is weighted, the optimum
coterie
is still an odd oligarchy, but the voting nodes are not
satisfied, omit M2. If (4) is not satisfied,
and add M’. If (5) is not satisfied, add M.
necessarily
trivial
equally
0(n4)
spaced.
algorithm
We present
for computing
a highly
non-
our results
For example,
on trees in the process).
(1) VM1, M,e
S, M1n
condition
holds:
be
A vote
function
node v
to node
ciate a
assignment
in a network
G = (V, E) is a
v assigning a nonnegative
integer to each
of V. v(v) is the number of votes aesigned
v. With each vote assignment we can assocoterie S which contains those sets M such
I+z.cv
J4v)
that ~VGM v(v) >
That is, S contains all majorities
of the v~te assignment.
S satisfies
Let
F ~ V be a random variable denoting
the set
of failed nodes.
The probability
that F occurs is
pl~l(l–p)lv-~l.
Let C(F) = {Cl, Ch,...,
C’~} be the
connected components
the sum of the weights
but exhaust
failures).
Our results in Sections 3, 4, and 5 can
easily be extended to the more general definition.
M,#O
Following
the terminology
in [GMB85],
we shall
referring to sets of nodes in a coterie as groups.
are disjoint
of any two groups intersect all cuts of the graph. We
preferred
to adopt the simpler,
standard
definition
above (which cannot be relaxed in the case of edge
Let G = (V, E) be a network.
For each node v E V
we have a weight w(v).
A set of sets S G 2V is a
on G if the following
if two groups
the nodes of the graph, then obviously no node failure
can make them independent
partitions
of the graph.
More generally, we could just require that the union
2. PRELIMINARIES
coterie
M
Remark:
Strictly
speaking, in the context of node
failures coteries can be a little more general than (l).
the optimum
oligarchy in the case of a weighted ring. Finally, we
show how to reduce the case of a cactus to that of a
ring (using
omit
of GI v-F.
Let w(F) denote
of the nodes of the connected
component
C~ that contains some group in S, if such
a component
exists. (Obviously
by the definition
of
the basic condition
(1) of coteries, and can be made
to satisfy conditions
(2)–(5).
It was pointed out in
76
[BGM85]
that
(at most 2n’)
there
than
are far fewer vote assignments
there
are coteries
the optimal position of the center, can be easily found
if we calculate (using equation (1)) the performance
of all nodes of the tree. Using dynamic programming,
we can show:
(at least 220”).
Still, vote assignments seem to be able to generate all
coteries of interest (for example, all these shown op-
timum in this paper), It is an open problem whether
there are situations in which the optimum coterie is
not obtainable by a vote assignment (this is of interest since vote assignments are so easy to implement,
compared to general coteries).
A particular simple kind of coterie is monarchy
Sj = {{c}};
c 6 V is called the center. Obviously,
monarchy corresponds to a vote assignment in which
a vote is assigned to the center, and the rest of the
nodes have zero votes.
The computational
per is the following:
Corolary.
The optimal coterie on a tree can be computed in time O(n). ❑
Interestingly, the optimal position
pends heavily on the probability of
ample, in the tree shown in Figure
values of p node 1 would be chosen
as center because
this node minimizes the average distance to all other
nodes (and this is an estimate of the performance
in
this case). For very large values of p, node 2 would
be chosen because it has the maximum
degree (and
for large values of p, obviously this is what matters).
problem studied in this P*
Finally,
OPTIMUM
COTEItIE:
Given G, p, w and b >0
there a coterie IS, such that P(S) ~ b?
of the center defailure p. For ex1, for very small
is
for intermediate
values of p node 3 might be
a reasonable degree and
chosen because it has both
is quite
centrally
located.
Regarding the complexity of this problem, we can
show the following by a reduction from the reliability
problem [Va181]:
Theorem
1. OPTIMUM
COTERIE
is #P-hard.
K
❑
In terms of upper bounds, it is obvious that OPexponenTIMUM COTERIE is in nondeterministic
tial time; if we restrict our attention to vote assignments (which, for all we know, may be an equivalent
problem), the complexity is “only” PSPACE...
3, THE
CASE
OF
TREES
Barbara and Garcia-Molina
[BGM86] have observed
that a coterie may contain only nodes of a single biconnected component of the graph representing the
network. In the case of trees we can prove the following slightly stronger fact:
Theorem
2. The optimal
Figure
4. OPTIMAL
coterie in a tree is always
Consider
and let 1<
that a node of
G be a tree, p the probability
the tree fails, and suppose that all nodes have equal
weight. In this case the performance equals is the
expected number of working nodes. A node v can
operate iff it is connected to the center, therefore,
the performance of the coterie & can be calculated
as follows:
Let
– py(”,’)
choice of the center.
All the above also hold when each node v of the
tree has weight
a monarchy.u
~(o = Z(1
1: Optimal
oligarchy
w(v).
COTERIES
a ring
with
2k + 1<
is a coterie
n nodes
ON
with
A RING
equal
n be an odd integer.
defined
as follows:
weights,
A 2k + l–
we choose
2k + 1< n nodes on the ring UI,..., u2~+I in cyclic
order. There are 2k + 1 arcs (i.e., paths) of the ring
that contain exactly k+ 1 of these nodes and have two
of these as endpoints.
The coterie defined by these
2k + 1 groups of nodes is called a 2k + l–oligarchy.
It is easy to check that it is indeed a coterie.
In
fact, all such oligarchies
can be implemented
by a
vote assignment,
where all distinguished
nodes get
(1)
VEV
one vote, and all other nodes zero votes.
A 2k + l-oligarchy
is canonical
if the lengths
of all 2k + 1 arcs are approximately
equal, that is
where d(v, c) is the distance of node v from the center
c. The coterie having the best performance,
that is
77
they differ by at most one. An easy way for constructing
canonical odd oligarchies is by placing the
by omiting
one of the intersection
arcs. The resulting set is still a coterie because the new group still
2k + 1 nodes approximately
equally spaced on the
ring so that all spaces are approximately
equal, and
to intercect
a group
this group
the original
groups,
impossible
intersects
all other
groups
of the coterie,
furthermore
the small spaces are as evenly spread as
possible (Figure 2(a)). This however is not the only
way (Figure 2(b)).
If n = 2k, then the canonical
2k + l-oligarchy
is defined to be all arcs of length
k that contain a particular
node v, and all arcs of
length k + 1 that do not. If 2k > n, the canonical
(property
3)).
Furthermore
we can prove
an endpoint
of group itf in the
another group iW such that ill
wise, we would replace A4 by M
2k + l–oligarchy
would
is called
democracy.
still
by canonicalization
the following:
If v is
coterie then there is
n M’ = {v}. Other– {v} and the result
be a coterie.
uz, ..., u~ be the endpoints
Letul,
(If it fails
is a subset of one of
of the groups
in S. We shall show that
m is an odd number
and each group
contains
k + 1 endpoints.
two groups
and M2 sharing
Ml
there can be no endpoint
2k + 1,
Consider
an endpoint.
out of their
First
union because a
group containing
this endpoint
would either contain
Ml or M2 or fail to intersect one of them, both impossible. If there is a third group using the remaining
endpoints,
then we prove our claim
Otherwise
all remaining
Ml and M2, which
number if endpoints
with
2k + 1 = 3.
groups must intersect both
implies that there is an equal
contained in Ml and M2.
To put the above argument
in a more intuitive
way, the only star shaped polygons
every side of
which intersects all others are the canonical odd star
polygons,
circle,
that
is, the ones with
and edges connecting
2k + 1 vertices
on a
every kth vertex.
So far, based only on the fact that S is a coterie
on the ring we have shown by a combinatorial
argument that it is an odd oligarchy.
To complete the
proof we shall show by analytical
coteries
Let
Figure
2: Constructi&
of canonical
Theorem
3. An opimum
odd oligarchies.
coterie on a ring, in which
weights,
is always
S be an optimum
coterie.
Consider
two
groups Ml and M2 of S of cardinality
11 and 12 respectively which intersect in a single node v. Let S
be the coterie that results if we replace Ml and M2
by two other groups that intersect in the node v’ to
the right of v (Figure 3). S“ is obtained
by moving
the intersection
point to the left of v. It easy to check
that both S’ and S’ are indeed coteries.
The significance of the canonical odd oligarchies
is established by the following
theorem.
all nodes have equal
means that optimal
must be canonical.
a canonical
Since S was assumed to be an optimum
we must have: P(S) > P(S’) or
odd oligarchy.
Proof:
Let S be the optimum
coterie of a ring. We
shall gradually
prove stronger and stronger properties of S eventually
proving that it is a canonical odd
oligarchy.
By connectivity
(property
2) all groups in
S are arcs of the ring. By minimality
(property
4)
any two groups intersect in only one arc. Because
suppose that two groups intersect in two arcs; then
we can replace one of these groups by a smaller one
~tu(qp(f’)
F
- ‘&(F’),(F)
coterie,
~ o
F
(in this csse w(F) is the number of working nodes
when the nodes in F have failed.) Most of the terms
in these two expressions cancel except for those when
F consists of one of v or v’ and nodes outside both
78
5’/+---? ./
probability
W
=
that
node j is operating,
2(2:;1,((1
k(l
- p)~’
is the following:
- (1 - p)’’’+’))+)+
– p)m~ – (k+
1)(1 - p)m(~+’)
(3)
To find the performance
P(S2h+l)
of the canonical
2k + l-oligarchy
we sum the probability
p(j) that a
node is operating over all nodes of the ring.
Formula (3) can be rewritten in terms of z =
pn, the number of expected failed nodes as follows:
Figure
groups.
3: Local
The inequality
Define II(2k + 1) = limn+m %
probability that a node is working).
optimality.
becomes:
Il(2k
+ 1) =
2(2k+l)
~
.*
(the average
_e-~)
(e
(4)
+ke-*
where f(k)
= ~~=Op2(l
– p)k+~(k
+ j)
1 (where
for S“ for P(S)
Differentiating
> P(S”)
~(k) twice,
~“(k)
>
tersect in a single node must be approximately
A simple
switching
argument
any two groups
value of z for which
(each node haa one vote, except for one node when n
is even). For z slightly above 2, smaller and smaller
values of 2k + 1 prevail (in fact, a differentiation
of
0, that is ~ is a convex function.
It follows that (1)
and (2) can hold only if 11 and 12 are approximately
equal. This establishes that any two groups that in-
prove that
Z2~+1 is the smallest
the 2k + l–oligarchy
is optimum).
It turns out that
the optimum
2k + 1 is a decreasing function
of z.
For z S 2 (that is when fewer than two nodes are
expected to fail) the optimum
2k + 1 is unbounded,
implying
that the optimum
coterie is the democracy
we obtain:
we can show that
l)e-~
We can now calculate, for any value of z, the optimum value of 2k + 1. The results are shown in Table
and m =
n+l–11–lz.
Similarly
– (k+
II(k) and a Taylor expansion suggests that 2k + 1
becomes optimum
near 2 + AZ).
Finally, after
equal.
can now be used to
are approximately
X3 = 2.0312
equal.
monarchy
the troika
becomes
Z1 = 2.2881.
after
optimum,
The
and the
values of 13 sug-
A simple way for designing
canonical
2k + 1oligarchies is by placing 2k -I- 1 nodes on the ring at
approximately
equal distances from each other and
spreading
the larger distances as much as possible
gest that, for practical purposes, we can say that the
democracy
is optimum
for values of z up to 2, the
troika is optimum
for values of z up to 2.288 and
then the monarchy is optimum;
the resulting error is
(see Figure 2(a) for an example of a canonical
7oligarchy).
However, this is by no means the only
way (see Figure 2(b)).
less than
2k+l
All canonical 2k + l-oligarchies
achieve the same
value of P. The question is what is the optimum
value of 2k + 1. The answer depends on p. We
demonstrate
this interesting
dependence by studying
the csse where the number of nodes of the ring becomes very large. So, we assume that n = rn(2k + 1)
and calculate l’($z~+l),
where S2~+1 is the canonical 2k + l–oIigarchy.
be done by considering
In this case the calculation
the probability
that
lYo.
can
a single
node is connected to an intact group, and adding over
all nodes. The result of this calculation,
that is the
79
Xzk+l
1
2.288070
3
5
7
9
11
2.031120
2.030000
2.006599
2.004199
2.002850
21
2.000840
41
2.000230
0.683949
0.730063
0.730268
0.734546
3
0.734998
0.735235
0.735604
0.735716
5. WEIGHTED
Theorem
4.
ring with
The
RINGS
optimum
AND
coterie
n nodes can be computed
is either the monarchy (with any node as the center),
or a canonical 2k + l-oligarchy in one of the cycles of
the cactus, To deal with the first possibility is always
easy; and we can reduce the second to the case of a
weighted ring (solved aa in Theorem 4), only with
modified
weights that reflect the sub cacti hanging
CACTI
on a weighted
in time O(n4).
We can prove that the optimum coterie on a
weigted ring is a 2k + l-oligarchy by the same combinatorial arguments we used in the previous section.
However, since all nodes of the ring do not have equal
weights, we cannot use the local optimality argument
of equation (3) to prove that all groups are of equal
size. To prove Theorem 4, we must determine the
optimum endpoints of the groups. we start by determining the optimum common endpoint j of a pair
of groups (i, j) and (j, k) (say, clockwise), assuming i
and k fixed. Let opt(i, j, k) be a ternary relation giving for any pair (i, k) of group endpoints the optimal
common endpoint j.
Lemma
O(ns).
1.
opt(i,
j, k) can be computed
from
[GMB85]
(i, j) in the coterie.
to keep the algorithm
ted.
in time
is needed
are omit-
❑
Having solved OPTIMUM
COTERIE
for trees
and cycles, it is quite natural to attack cacti next. A
cactus is a set of edge-disjoint
cycles connected by
paths (possibly of length zero, see Figure 4). Cacti
are the most general family of networks on which we
know how to solve OPTIMUM
COTERIE
in polynomial time:
Figure
❑
and D, Barbara
“How
to As-
of
the ACM, Vol. 32, No. 4, 1985, pp. 841-860.
[BGM86]
D, Barbara and H. Garcia-Molina
“The Vulnerability of Vote Assignments,” ACM !fkansactions on Computer
Systems, Vol. 4, No. 3, 1986,
pp. 187-213.
[BGM89]
D. Barbara and H. Garcia-Molina
“Increasing
Availability Under Mutual Exclusion Constrains
with Dynamic Vote Reassignment ,“ ACM Tr. on
Computer Systems, Vol. 7, No. 4,1989, pp. 394426.
[Va181] L. G. Valiant “The Complexity of Reliability”
SICOMP 12, 1981.
the relation opt(i, j, k),
programming,
fixing a group
Care and cleverness
polynomial.
Details
H. Garcia-Molina
sign votes in a Distributed
Once we have computed
by dynamic
4).
REFERENCES
•1
we proceed
each node of the cyc~e (Figure
4: Cactus.
Theorem
5. The optimum
coterie of a weighted
cactus with n nodea can be computed in time 0(n4).
The basic observations
needed for the proof of
Theorem 5 are these two: First, the optimum
coterie
80
System,”
Journal