HAL Id: hal-01865469
https://hal.archives-ouvertes.fr/hal-01865469
Preprint submitted on 31 Aug 2018
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entic research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diusion de documents
scientiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
Parallel Bayesian Search with no Coordination
Pierre Fraigniaud, Amos Korman, Yoav Rodeh
To cite this version:
Pierre Fraigniaud, Amos Korman, Yoav Rodeh. Parallel Bayesian Search with no Coordination . 2018.
�hal-01865469�
Parallel Bayesian Search with no Coordination
∗
Pierre Fraigniaud
†
Amos Korman
‡
Yoav Rodeh
§
Abstract
Coordinating the actions of agents (e.g., volunteers analyzing radio signals in SETI@home) yields
efficient search algorithms. However, such an efficiency is often at the cost of implementing complex
coordination mechanisms which may be expensive in term of communication and/or computation
overheads. Instead, non-coordinating algorithms, in which each agent operates independently from
the others, are typically very simple, and easy to implement. They are also inherently robust to slight
misbehaviors, or even crashes of agents. In this paper, we investigate the “price of non-coordinating”,
in term of search performance, and we show that this price is actually quite small. Specifically, we
consider a parallel version of a classical Bayesian search problem, where set of
k ≥
1 searchers are
looking for a treasure placed in one of the boxes indexed by positive integers, according to some
distribution
p
. Each searcher can open a random box at each step, and the objective is to find
the treasure in a minimum number of steps. We show that there is a very simple non-coordinating
algorithm which has expected running time at most 4(1
−
1
k+1
)
2
OPT
+ 10, where
OPT
is the expected
running time of the best fully coordinated algorithm. Our algorithm does not even use the precise
description of the distribution
p
, but only the relative likelihood of the boxes. We prove that, under
this restriction, our algorithm has the best possible competitive ratio with respect to
OPT
. For the
case where a complete description of the distribution
p
is given to the search algorithm, we describe
an optimal non-coordinating algorithm for Bayesian search. This latter algorithm can be twice as
fast as our former algorithm in practical scenarios such as uniform distributions. All these results
provide a complete characterization of non-coordinating Bayesian search. The take-away message
is that, for their simplicity and robustness, non-coordinating algorithms are viable alternatives to
complex coordinating mechanisms subject to significant overheads. Most of these results apply as
well to linear search, in which the indices of the boxes reflect their relative importance, and where
important boxes must be visited first.
∗
This work has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme (grant agreement No 648032).
†
IRIF, CNRS and University Paris Diderot, Paris, France. E-mail: Pierre.Fraigniaud@irif.fr.
‡
IRIF, CNRS and University Paris Diderot, Paris, France. E-mail: Amos.Korman@irif.fr.
§
Weizmann Institute of Science. E-mail: yoav.rodeh@gmail.com.
1 Introduction
BOINC [
22
] (Berkeley Open Infrastructure for Network Computing) is a platform for volunteer computing
supporting dozens of projects including the famous SETI@home analyzing radio signals for identifying
signs of extra terrestrial intelligence. Most projects maintained at BOINC use parallel search mechanisms
where a central server controls, and distributes the work to volunteers. The framework in this paper is an
abstraction for projects operated at platforms similar to BOINC with hundreds of thousands distributed
searchers. We address the following question: how to distribute the work among the searchers with
respect to the amount of coordination between them provided by the central server? This paper drives to
the conclusion that no coordination might actually be a quite viable strategy, both efficient and robust.
Specifically, we consider a parallel variant of the classical Bayesian search problem, typically attributed
to Blackwell [
6
]. A treasure is placed according to some distribution
p
in one of infinitely many boxes,
indexed by the positive integers. The search for the treasure is performed in parallel by
k ≥
1 agents,
also called searchers, which means that at each time step each searcher can “peek” into a box to check
whether the treasure is present there. The goal is to minimize the expected time until the first searcher
finds the treasure. We assume that the number
k
of searchers is known to the algorithm. We will
consider two cases, whether
p
is given to the algorithm, or not. However, in the latter case, we assume
that the algorithm is aware of the relative likelihood of the boxes. In both cases, we can assume, w.l.o.g.,
that the boxes B
x
, x ≥ 1, are ordered so that p is non-increasing, i.e., p(x + 1) ≤ p(x) for every x ≥ 1.
Let
s
1
, . . . , s
k
be the
k
searchers at hand. If coordination is allowed, let
A
coord
be the algorithm that
lets searcher
s
i
peek into box
B
(t−1)k+i
at time
t
. A simple application of the rearrangement inequality
shows that
A
coord
is an optimal algorithm, that is, it minimizes the expected time until one searcher
finds the treasure. This time is
P
x≥1
p
(
x
)
dx/ke
since the box
B
x
is opened at time
dx/ke
in
A
coord
, and
this box has probability
p
(
x
) to contain the treasure. In particular, the optimal expected time to find
the treasure with a single searcher is
P
x≥1
x p
(
x
). Therefore, if coordination is allowed,
k
searchers
essentially allow to find the treasure
k
times faster than one searcher alone, in expectation. (Specifically,
the speedup resulting from using
k
searchers approaches
k
when the expectation of the distribution
p
grows to infinity). However, as simple as this algorithm is,
A
coord
is very sensitive to faults of all sorts. For
example, if one searcher crashes at some point during the execution then the searchers may completely
miss the treasure, unless the protocol employs some mechanism for detecting such faults. Indeed, in
A
coord
, each box is eventually opened by just one searcher. Namely, box
B
(t−1)k+i
is opened only by
searcher s
i
, for every t ≥ 1 and 1 ≤ i ≤ k.
In this paper, we highlight the usefulness of a class of search algorithms, called non-coordinating,
which is inherently robust. In such algorithms, all searchers operate independently, executing the same
protocol, differing only in the outcome of the flips of their private random coins. A canonical example is
the case of multiple random walkers that search a graph [
1
]. Although many search problems cannot be
efficiently parallelized without coordination, when such parallelization can be achieved, the benefit can
potentially be high, not only in terms of saving in communication and overhead in computation, but also
in terms of robustness. To get some intuition, observe that when executing a non-coordinating algorithm,
the correct operation as well as the running time can only improve if more searchers than planned are
actually being used. Suppose for instance that an oblivious adversary is allowed to crash at most
k
0
out of the
k
searchers at arbitrary times during the execution. To overcome the presence of
k
0
faults,
one can simply run the non-coordinating algorithm that is designed for the case of
k − k
0
searchers. If
the running time of the non-coordinating algorithm for
x
searchers without crashes is
T
(
x
), then the
running time of the new robust (non-coordinating) algorithm would be at most
T
(
k − k
0
). Note that
even when coordination is allowed, one cannot expect to obtain robustness at a cost less than
b
T
(
k − k
0
)
in the worst case, where
b
T
(
x
) denotes the cost of an optimal coordinating algorithm for
x
searchers
1
without crashes, since the number of searchers that remain alive is in the worst case
k − k
0
. Hence, if
T (·) and
b
T (·) are close, we get robustness almost for free by using a non-coordinating algorithm.
In this paper, we are interested in computing how much we lose in term of performance when using
non-coordinating algorithms. Specifically, let
k ≥
1, and let us denote by
T
k
(
A, x
) the expected time for
an algorithm
A
to find the treasure with
k
searchers running in parallel if this treasure is placed at box
x
. Further, given a distribution
p
over the placement of the treasure in the boxes, let
T
p,k
(
A
) denote the
expected time for A to find the treasure when it is placed in one of the boxes according to p. We have
T
p,k
(A) =
X
x≥1
p(x)T
k
(A, x). (1)
With this notation, the expected running time of the optimal coordinating algorithm is
T
p,k
(
A
coord
) =
P
x≥1
p
(
x
)
dx/ke
. We are interested in comparing these two terms, i.e.,
T
p,k
(
A
) for a non-coordinating
algorithm
A
versus
T
p,k
(
A
coord
), the complexity of the best search algorithm with full coordination. For
this purpose, we use competitive analysis, and say that an algorithm
A
is
c
-competitive for
k
searchers
looking for a treasure placed according to p if there is a constant b such that
T
p,k
(A) ≤ c T
p,k
(A
coord
) + b.
We show that there is a non-coordinating algorithm with small competitive ratio, hence establishing
that indeed one does not lose much in using non-coordinating algorithms.
Before going into the details of our results, let us observe that although the random placement of the
treasure is the common setting in Bayesian search problems, yielding Eq.
(1)
for defining the complexity
of a search algorithm, there is another abstract search setting which deserves to be investigated, that we
call linear search. Indeed, searching for a proper divisor of a given number
n
, the typical approach to
solve the problem consists of enumerating the candidate divisors in increasing order, from 2 to
√
n
, and
checking them one by one. This is because the probability that a random number is divisible by a given
prime is inversely proportional to this prime. Similarly, in cryptography, an attack is better proceeded
by systematically checking smaller keys than longer ones, because the time to check a key is typically
exponential in its size. There are thus several contexts in which the search space can be ordered in a
way such that, given that the previous trials were not successful, the next candidate according to the
order is either the most preferable, or most likely to be valid, or the easiest to check. This led us to
consider another measure of complexity, comparing the search time of an algorithm
A
to the search time
of the algorithm with one searcher opening the boxes sequentially in order of their indices, namely
T
k
(A) = max
x≥1
T
k
(A, x)/x. (2)
Again, we say that a search algorithm
A
is
c
-competitive for
k
searchers looking for a treasure arbitrarily
placed in one box if there is a constant b such that
T
k
(A) ≤ c T
k
(A
coord
) + b.
In the linear search setting, the aforementioned algorithm
A
coord
is also optimal. We show that, as for
Bayesian search, one does not lose much in using non-coordinating algorithms in linear search.
1.1 Our Results
First, we design and analyze an optimal non-coordinating algorithm for Bayesian search, where
p
is given.
Our algorithm, called
A
?
, has optimal expected running time among all non-coordinating algorithms.
Specifically, for every distribution p, every k ≥ 1, and every non-coordinating algorithm A,
T
p,k
(A
?
) ≤ T
p,k
(A).
2
A remarkable property satisfied by our non-coordinating algorithm
A
?
is that, at any time
t >
1 during
its execution, all boxes that received a positive probability to be checked at some time
t
0
< t
, are now
going to be checked at time
t
with equal probability. The design of
A
?
is complex. However, when
p
is the uniform distribution over a finite domain,
A
?
becomes simple to describe: at each step, each
searcher running
A
?
chooses a box uniformly among those it did not check at previous step. This natural
algorithm for the uniform setting is optimal among all non-coordinating algorithms, and is shown to be
at most 2 times slower than A
coord
.
Next, we focus on the notion of order-invariant algorithms, that is, algorithms assuming only the
knowledge of the relative likelihood of the boxes (and not knowing the exact probability of finding the
treasure in each box). Such algorithms are appealing because they are “universal”, in the sense that,
once the boxes have been reordered such that
B
x
is not less likely to contain the treasure than
B
x+1
, any
order-invariant algorithm acts the same for all distributions. We present a very simple yet highly efficient
non-coordinating order-invariant algorithm, called
A
order
. In this algorithm, at step
t
, each searcher
checks a box uniformly chosen among those it did not check yet in
{
1
, . . . , d
t
2
e
(
k
+ 1)
}
. The performance
of
A
order
is essentially at most 4 times the expected running time of the best fully coordinated algorithm
A
coord
. Precisely, for every distribution p, and every k ≥ 1,
T
p,k
(A
order
) ≤ 4
1 −
1
k + 1
2
T
p,k
(A
coord
) + 10. (3)
Ignoring the constant additive term, the aforementioned upper bound implies that the cost paid for
not coordinating is just at most
16
/9
for two searchers,
9
/4
for three searchers, and approaches 4 as the
number of searchers goes to infinity. In fact we show that these costs are tight in a very strong sense, as,
for any given number of searchers, there is no non-coordinating order-invariant algorithm that achieves a
better competitive ratio. Specifically, for every distribution
p
, every
k ≥
1, and every order-invariant
non-coordinating algorithm A, if there exist b and c such that T
p,k
(A) ≤ c T
p,k
(A
coord
) + b, then
c ≥ 4
1 −
1
k + 1
2
. (4)
Algorithm
A
order
remembers all the boxes it checked, and so each searcher needs memory linear in the
running time of the algorithm. We also consider
A
obliv
which at step
t
chooses one box uniformly at
random in
{
1
, . . . , kd
t
2
e}
, hence potentially choosing many times the same box at different time steps.
This algorithm uses memory that is just logarithmic in its running time, but performs almost as well as
A
order
for large number of searchers. Precisely, for every distribution p, and every k ≥ 1,
T
p,k
(A
obliv
) ≤ 4 T
p,k
(A
coord
) + 2. (5)
All the aforementioned upper bound results on order-invariant algorithms are actually established by
considering the linear search setting, where the treasure is placed at an arbitrary box, and boxes are
ordered by importance, that is, when focussing on the complexity
T
k
(
A
) =
max
x≥1
T
k
(
A, x
)
/x
of any
algorithm
A
(cf. Eq.
(2)
). Indeed, it turns out that the two settings (Bayesian search and linear search)
are highly related, as far as order-invariant algorithms are concerned: an order-invariant algorithm that
works well against a treasure placed arbitrarily would also work well in any probabilistic setting (under
the assumption that the indices of the boxes are ordered according to their relative likelihood). In fact,
in the linear search setting,
A
order
and
A
obliv
have the same competitive ratio as those mentioned in
Eq.
(3)
and
(5)
, respectively. Moreover, the lower bound of Eq.
(4)
also holds in the linear search setting,
i.e., A
order
has also optimal competitive ratio in this latter setting.
3
