An Algorithmic Approach to Online Multi-Facility Location Problems
Christine Markarian
1
, Abdul-Nasser Kassar
2
and Manal Yunis
2
1
Department of Engineering and Information Technology, University of Dubai, U.A.E.
2
Department of Information Technology and Operations Management, Lebanese American University, Lebanon
Keywords:
Robustness, Multi-Facility Location, Online Algorithms, Competitive Analysis, Randomized Rounding.
Abstract:
Facility Location problems ask to optimally place facilities with respect to some objective so that all clients
requesting a facility service are served. These are one of the most well-studied optimization problems spanning
many research areas, such as operations research, computer science, and management science. Classical
algorithmic study of Facility Location problems is based on the assumption that clients need to be served
with one facility each. Nevertheless, in many real-world applications, facilities experience disruptions and
to overcome their failures, a robust service is desired. To obtain this, clients are served with more than one
facility, and this is commonly represented by an additional input parameter. The aim of the algorithm is then
to provide a robust service to all clients while minimizing costs. This is known as the Multi-Facility Location
problem (MFL), a well-known optimization problem, studied in the offline setting in which the entire input
sequence is known to the algorithm in advance. In this paper, we address MFL in the online setting, in which
client requests are not known in advance but revealed over time. We refer to it as the Online Multi-Facility
Location problem (OMFL) and study its metric and non-metric variants. We propose the first online algorithms
for these variants and measure their performance using the standard notion of competitive analysis. The latter
is a worst-case analysis that compares the cost of the online algorithm to that of the optimal offline algorithm
that is assumed to know all demands in advance.
1 INTRODUCTION
Facility Location (FL) is a classical NP-hard opti-
mization problem widely spread in the fields of com-
puter science and operations research (Drezner, 1995;
Manne, 1964). In its simplest form, we are given a set
of facilities and a set of clients. Each facility has an
opening cost and each client i has a connecting cost to
each facility j, which is the distance between i and j.
The goal is to open a subset of the facilities and con-
nect the clients to open facilities so as to minimize
the sum of the facility costs and the connecting costs.
FL is known as two versions, metric and non-metric.
In the metric version, the distances are assumed to be
symmetric and satisfy the triangle inequality.
In this paper, we focus on the algorithmic view of
FL problems and consider an online setting in which
clients are not known in advance but revealed to the
algorithm over time. As soon as one arrives, it needs
to be connected. Many real-world applications, that
contain FL as a sub-problem, have this online na-
ture, in which one is expected to react to present de-
mands whenever they arrive, without knowing about
future demands. Maintaining a given optimization
goal becomes more challenging in the face of this un-
certainty. This encourages the study of online algo-
rithms (Fiat and Woeginger, 1998) for FL problems.
The standard framework to measure online algorithms
is competitive analysis, in which demands and their
arrival order are selected by an oblivious adversary,
that is unaware of the choices of the algorithm. An
online algorithm is c-competitive or has competitive
ratio c if for all sequences of demands, the cost in-
curred by the algorithm is at most c times the cost in-
curred by an optimal offline algorithm, which knows
the entire sequence of demands in advance.
The study of Facility Location (FL) in the on-
line setting was initiated by Meyerson (Meyer-
son, 2001), who introduced the metric Online Fa-
cility Location problem (OFL), and proposed an
O(logn)-competitive randomized algorithm, where n
is the number of clients. Alon et al. (Alon et al.,
2006) studied the non-metric version and proposed
an O(log n log m)-competitive randomized algorithm,
where n is the number of clients and m is the num-
ber of facilities. Many other variations were known
for both metric and non-metric variants in the on-
line setting (Abshoff et al., 2016; Anagnostopoulos
et al., 2004; Div
´
eki and Imreh, 2011; Fotakis, 2007;
Fotakis, 2008; Markarian and Meyer auf der Heide,
2019).
Other variants include (Fotakis, 2011), in which
Markarian, C., Kassar, A. and Yunis, M.
An Algorithmic Approach to Online Multi-Facility Location Problems.
DOI: 10.5220/0010212200290035
In Proceedings of the 10th International Conference on Operations Research and Enterprise Systems (ICORES 2021), pages 29-35
ISBN: 978-989-758-485-5
Copyright
c
2021 by SCITEPRESS Science and Technology Publications, Lda. All rights reserved
29
the algorithm can merge existing facilities with each
other, and only the decision of assigning some de-
mands to the same facility is irrevocable. Another
variant by (Feldkord and Meyer auf der Heide, 2018)
allows the algorithm to adapt the position of the fa-
cilities for costs proportional to the distance by which
the position is changed.
All of these works assume that clients need to be
served with one facility each. In many real-world
applications, a robust service, in which a client is
served with more than one facility, is desirable (Sny-
der, 2006; Snyder and Daskin, 2006; Gerodimos,
1998). Facilities and/or connections to facilities may
be prone to failure and assigning clients to multi-
ple facilities would provide a fault-tolerant solution.
Such solutions are seeked in many applications, such
as providing replicated cash data in distributed net-
works (Byrka et al., 2010).
1.1 Our Contribution
In this paper, we explore a generalization of online
Facility Location (Alon et al., 2006; Fotakis, 2008), in
which we are additionally given a parameter k, which
is the number of facilities required to serve a client.
We refer to it as the Online Multi-Facility Location
problem, defined as follows.
Definition 1. (Online Multi-Facility Location) We are
given a collection of m facilities, n clients, and a
positive integer k. Each facility has an opening cost
and each client has a connecting cost to each facility.
Clients arrive over time. As soon as one arrives, it
needs to be connected to at least k open facilities. To
open a facility, we pay its opening cost. To connect a
client to a facility, we pay the corresponding connect-
ing cost. The goal is to minimize the total opening
and connecting costs.
We address the metric and non-metric Online
Multi-Facility Location problems. As far as we are
aware of, there are no online algorithms in the liter-
ature that solve these variants or can be trivially ex-
tended to solve them.
Lower Bounds. For the non-metric version, there
is a lower bound of (logmlogn), under the assump-
tion that NP 6⊆ BPP, where m is the number of facil-
ities and n is the number of clients, due to the lower
bound given for Online Set Cover (Korman, 2005).
As for the metric version, there is a lower bound
of (
logn
loglog n
), where n is the number of clients, due
to the lower bound given for metric OFL (Fotakis,
2008).
Our results can be summarized as follows.
1. We refer to the non-metric version as Online Non-
metric Multi-Facility Location (ONMFL). We
propose an online O(log(kn) log m)-competitive
randomized algorithm for ONMFL, where m is
the number of facilities; n is the number of clients;
and k is the number of required connections.
The latter uses a randomized rounding approach
that first constructs a fractional solution and then
rounds it into an integral one. Its competitive anal-
ysis is based on first comparing the fractional so-
lution constructed by the algorithm to the optimal
offline solution and then measuring the fractional
solution in terms of the integral one.
2. We refer to the metric version as Online Met-
ric Multi-Facility Location (OMMFL). We pro-
pose an online O(max{
f
max
f
min
,
c
max
c
min
} · k ·
logn
loglog n
)-
competitive deterministic algorithm for OMMFL,
where n is the number of clients; k is the num-
ber of required connections; c
max
and c
min
are
the maximum and minimum connecting costs, re-
spectively; f
max
and f
min
are the maximum and
minimum facility costs, respectively.
The idea of the algorithm is to ensure first that
each client is connected to one facility by running
an algorithm for metric Online Facility Location
(OFL). Then, the k 1 remaining connections are
made by choosing the cheapest possible facilities
so as not to worsen the competitive ratio by much.
Our approach can be seen as a general frame-
work that transforms any given online algorithm
for metric OFL into an algorithm for OMMFL, by
losing a bounded factor in the competitive ratio.
1.2 Outline
The remainder of this paper is structured as follows.
Section 2 gives an overview of literature related to
ONMFL and presents an algorithm for ONMFL along
with its competitive analyis. Section 3 gives an
overview of literature related to OMMFL and presents
an algorithm for OMMFL along with its competitive
analyis. Section 4 concludes the paper with some fu-
ture work.
2 ONLINE NON-METRIC
MULTI-FACILITY LOCATION
In this section, we start with an overview of works
related to ONMFL and some preliminaries. Then we
present an online randomized algorithm for ONMFL
and analyze its competitive ratio.
ICORES 2021 - 10th International Conference on Operations Research and Enterprise Systems
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2.1 Preliminaries & Related Work
ONMFL is a generalization of the Online Non-metric
Facility Location (ONFL) (Alon et al., 2006) with
k = 1. Alon et al. (Alon et al., 2006) gave an
O(lognlogm)-competitive randomized algorithm for
ONFL, where n is the number of clients and m is the
number of facilities.
A closely related problem is the Online Set k-
Multicover (OSMC) (Berman and DasGupta, 2008).
Given a universe U of n elements, a family S of m
subsets of V , each associated with a cost, and a pos-
itive integer k. A subset D U of elements arrives
over time. OSMC asks to select a collection C S of
subsets, of minimum cost, such that each arriving el-
ement belongs to at least k subsets of C . Berman and
DasGubta (Berman and DasGupta, 2008) proposed an
O(logmlogd)-competitive randomized algorithm for
OSMC, where m is the number of subsets and d is the
maximum set size.
Transformations between OSMC and ONMFL in-
stances can be made in both directions. An instance
of ONMFL can be transformed into an instance of
OSMC as follows. We associate each facility with
each of the 2
n
1 possible groups of clients, and let
each facility/group be a subset, with cost equal to the
sum of the cost of the facility and the connecting costs
of the clients in the group to the facility. We let each
client be an element. Following this transformation,
the algorithm of Berman and DasGubta would im-
ply a feasible algorithm for ONMFL, but with com-
petitive ratio O(log(m(2
n
1)) log n), where n is the
number of clients, and m is the number of facilities.
An instance of OSMC can be transformed into an in-
stance of ONMFL as follows. We represent each sub-
set by a facility and let the opening cost be the sub-
set cost. We represent each element by a client and
set the connecting cost to 0 if the client belongs to
the subset/facility and infinity otherwise.The offline
version of OSMC, in which all elements are given
in advance, has an O(log d)-approximation (Johnson,
1974; Virkumar, 2001), where d is the maximum set
size. Similar transformations can be made in the of-
fline setting. This implies an O(log n)-approximation
for the offline version of ONMFL, where n is the num-
ber of clients. These are the best achievable unless
NP DT IME(n
loglog n
) (Feige, 1998). The Online
Set Cover (OSC) (Alon et al., 2003) is a special case
of OSMC with k = 1. Korman’s lower bound on the
competitive ratio of OSC (Korman, 2005) implies an
(logmlogn) lower bound on the competitive ratio of
ONMFL, under the assumption that NP 6⊆ BPP, where
m is the number of facilities and n is the number of
clients.
For other online problems solved with similar
techniques as the ones used in this paper, see (Markar-
ian and Kassar, 2020; Hamann et al., 2018).
2.2 Online Algorithm
Our algorithm for ONMFL is based on constructing
a fractional solution first and then rounding it online
into an integral solution. Unlike (Alon et al., 2003)
and (Alon et al., 2006) for Online Set Cover (OSC)
and Online Non-metric Facility Location (ONFL), re-
spectively, in which a potential function is used in the
competitive analysis of the algorithms, our analysis is
based on simple arguments, similar to those we gave
in our previous work for a related problem, the Online
Node-weighted Steiner Forest (Markarian, 2018).
We start with giving the following graph formu-
lation for ONMFL. Given a root node r, m facility
nodes, and n client nodes. We add an edge from r to
each facility node j, with cost equal to the opening
cost of facility j. We add an edge from each facility
node j to each client node i, with cost equal to the con-
necting cost of client i to facility j. As soon as a client
i arrives, we need to purchase k disjoint paths between
r and i. The goal is to minimize the total costs of the
paths purchased, where the cost of a path is the cost
of its edges. To output a solution for ONMFL, each
facility whose corresponding edge is purchased will
be opened, and each client whose corresponding edge
to an open facility is purchased will be connected to
that facility.
The algorithm initially knows n, the number of
clients; k, the number of required connections; and
the opening costs of facilities. A subset D of n
0
n
clients arrives over time. As soon as a client i arrives,
the algorithm is given the connecting costs of i to each
facility, and is expected to react.
We consider the graph formulation described ear-
lier. Let r be a root node; each facility is represented
as a facility node, and has an edge to r, associated
with its opening cost. Upon the arrival of a new client
i, the algorithm creates a client node for it and adds an
edge from this node to each facility node, associated
with the given connecting cost. Let G = (V,E) be this
graph.
Each edge added to E is given a fraction, set inti-
tially to 0. The algorithm does not allow these frac-
tions to decrease over time. These form a fractional
solution for ONMFL. The maximum flow between
node u and node v in G is the smallest total weight
of edges which if removed would disconnect u from
v. These edges form a minimum cut between u and v
in G. Let c
e
and f
e
be the cost and fraction of edge
e, respectively. A path is purchased if and only if its
An Algorithmic Approach to Online Multi-Facility Location Problems
31
edges are purchased.
Random Process. The algorithm makes its ran-
dom choices, based on α, the minimum among
2
d
log(kn + 1)
e
independently chosen random vari-
ables, distributed uniformly in the interval [0,1].
Next, we describe how the algorithm reacts upon the
arrival of a new client.
Input: G = (V,E) and client node i D
Output: Set of edges purchased
Make a copy G
0
of G;
As long as there are < k disjoint paths purchased
between r and i in G, do the following:
1. While the maximum flow between r and i in G
0
is
less than 1, construct a minimum cut Q between r
and i in G
0
; for each edge e Q, make the follow-
ing fraction increase:
f
e
= f
e
· (1 + 1/c
e
) +
1
|
Q
|
· c
e
2. Purchase each edge e with f
e
> α.
3. If there is no purchased path between r and i in
G
0
, find a minimum-cost such path and purchase
it.
4. Refer to all facilities whose corresponding edges
were purchased as open; delete from G
0
the pur-
chased edges between i and each open facility.
2.3 Competitive Analysis
The algorithm buys edges in the second and third
steps. In the second step, its choices are made based
on the random process, whereas in the third step, its
choices are made to guarantee a feasible solution. We
measure the expected cost of each separately. Let Opt
be the cost of the optimal offline solution and let f rac
be the cost of the fractional solution constructed by
the algorithm in the first step.
Choices Based on Random Process: Let S
0
be the
set of edges purchased in the second step of the al-
gorithm and let C
S
0
be its expected cost. These edges
are purchased by the algorithm based on the random
process described earlier. Let us fix an l : 1 l
2
d
log(kn + 1)
e
and an edge e. We denote by X
e,l
the
indicator variable of the event that e is chosen by the
algorithm based on the random choice of l.
C
S
0
=
eS
0
2
d
log(kn+1)
e
l=1
c
e
· Exp [X
e,l
]
= 2
d
log(kn + 1)
e
eS
0
c
e
f
e
(1)
Notice that
eS
0
c
e
f
e
is upper bounded by the cost
of the fractional solution. The latter can be measured
against the optimal offline solution, as follows. The
idea here is that every time the algorithm performs a
fraction increase, it does not exceed 2. Moreover, the
total number of fraction increases can be measured in
terms of the cost of the optimal offline solution.
The fraction increase contributed by each edge e
in a minimum cut Q is
f
e
c
e
+
1
|
Q
|
·c
e
. The algorithm
would make a fraction increase only if the maximum
flow is less than 1. This means we have that
eQ
f
e
<
1 before a fraction increase. Therefore, each fraction
increase does not exceed:
eQ
c
e
·
f
e
c
e
+
1
|
Q
|
· c
e
< 2 (2)
As long as the algorithm hasn’t purchased at least
k disjoint paths between r and a given client i, it enters
the loop that starts by constructing a maximum flow
on the graph G
0
. Notice how G
0
shrinks over time, as
the algorithm purchases the paths.
Lemma 1. Whenever the algorithm makes a fraction
increase, G
0
contains at least one path that is also in
the optimal offline solution.
Proof. To see why this holds, fix any time t before a
fraction increase. Let s < k be the number of disjoint
paths purchased by the algorithm at time t. G
0
at time
t must contain at least one optimal path since G
0
is
constructed by removing s (less than k) feasible paths
from G and the optimal offline solution contains at
least k disjoint paths in G.
Finally, the algorithm would have an edge e from
the optimal offline solution in every minimum cut Q
of G
0
, since Q must contain an edge from each path,
by definition. Based on the equation for the fraction
increase, after O(log |Q|) fraction increases, the frac-
tion f
e
of e becomes 1, and no further increases can
be made, as e will not be in any future minimum cut.
The size of any minimum cut is upper bounded by
m, the number of facilities or the maximum available
paths between r and client i. We can now bound the
fractional solution:
f rac O(log m · Opt) (3)
Combining Equations 1, 2, and 3 imply an upper
bound on the expected cost C
S
0
of the edges bought in
the second step of the algorithm:
C
S
0
O(log(kn)logm · Opt) (4)
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32
Choices to Guarantee Feasible Solution: Let S
00
be the set of edges purchased in the third step of the al-
gorithm and let C
S
00
be its expected cost. These edges
are purchased by the algorithm only if a path has not
been bought by the random process in the second step.
Every time the algorithm purchases a path in this step,
its cost does not exceed Opt since the algorithm buys
the minimum-cost path in G
0
, and as we showed ear-
lier in Lemma 1, G
0
contains at least one path that is
also in the optimal solution.
(one client, one path) We start with calculating the
expected cost incurred by a single client for pur-
chasing a single path. Fix a client i. Let Q
j+1
be
a minimum cut of G
0
constructed after the algo-
rithm has purchased j < k disjoint paths between
r and i and has completed the first step. The prob-
ability of purchasing the ( j + 1)
th
path for a single
1 l 2
d
log(kn + 1)
e
is:
eQ
j+1
(1 f
e
) e
eQ
j+1
f
e
1/e
Notice that the last inequality holds because the
algorithm ensures that
eQ
j+1
f
e
1 at the end
of the first step (Max-flow min-cut theorem). The
expected cost of purchasing the ( j + 1)
th
path for
all 1 l 2
d
log(kn + 1)
e
, is less than 1/(kn)
2
·
Opt.
(one client, k paths) The expected cost of purchas-
ing all k paths is the sum of the expected costs for
each path and is less than k · 1/(kn)
2
· Opt.
(total cost of all clients) The total expected cost
incurred by all n
0
clients that arrived is less than:
n
0
· k · 1/(kn)
2
· Opt n · k · 1/(kn)
2
· Opt
= 1/kn · Opt
.
Therefore, the expected cost C
S
00
of the edges
bought in the third step of the algorithm is:
C
S
00
1/kn · Opt (5)
Equations 4 and 5 yield to the following theorem.
Theorem 1. There is an online O(log(kn)logm)-
competitive randomized algorithm for the Online
Non-metric Multi-Facility Location, where m is the
number of facilities, n is the number of clients, and k
is the number of required connections.
3 ONLINE METRIC
MULTI-FACILITY LOCATION
In this section, we start with an overview of works
related to OMMFL and some preliminaries. Then we
present an online randomized algorithm for OMMFL
and analyze its competitive ratio.
While this problem has been intensively studied
in the offline setting (Byrka et al., 2010; Yan and
Chrobak, 2011), we are not aware of any online al-
gorithm for it.
3.1 Preliminaries & Related Work
OMMFL is a generalization of metric Online Facility
Location (OFL) (Fotakis, 2007; Fotakis, 2008; Mey-
erson, 2001) with k = 1. Meyerson (Meyerson, 2001)
introduced metric OFL and proposed an O(logn)-
competitive randomized algorithm, where n is the
number of clients. Fotakis (Fotakis, 2008) showed
that no randomized online algorithm can achieve
a competitive ratio better that (
logn
loglog n
) against
an oblivious adversary and gave a deterministic al-
gorithm with asymptotically matching O(
logn
loglog n
)-
competitive ratio. In another work later, he proposed
a primal-dual deterministic algorithm with O(log n)-
competitive ratio, that was simpler to formulate, ana-
lyze, and implement (Fotakis, 2007). The competitive
ratio we achieve for OMMFL is based on running the
deterministic algorithm of Fotakis (Fotakis, 2008) for
metric OFL.
The lower bound on the competitive ratio of
metric OFL by Fotakis (Fotakis, 2008) implies an
(
logn
loglog n
) lower bound on the competitive ratio of
OMMFL.
3.2 Online Algorithm
Let A
OFL
be any online (deterministic or random-
ized) algorithm for metric Online Facility Location
(OFL), with competitive ratio r. Given an instance
I of OMMFL with positive integer k. Client i arrives.
Our algorithm needs to connect i to k different open
facilities.
1. The algorithm starts by running A
OFL
on instance
I, where k = 1. This results in opening some fa-
cilities and connecting i to one open facility.
2. If i is the first client, we open the cheapest k 1 fa-
cilities other than the one i is connected to. From
this point on, there are at least k open facilities.
Until A
OFL
opens these facilities itself, these re-
main closed with respect to A
OFL
.
3. The algorithm will then connect i to any other k
1 open facilities.
An Algorithmic Approach to Online Multi-Facility Location Problems
33
3.3 Competitive Analysis
Let I be an instance of OMMFL with positive integer
k. Let I
0
be the same instance as I except for k = 1.
Let Opt and Opt
0
be the cost of the optimal solution
for I and that for I
0
, respectively. Let C and C
0
be the
cost of our algorithm for I and that of A
OFL
for I
0
,
respectively. We denote by C
f ac
and C
con
the costs in-
curred by our algorithm to open facilities and to con-
nect clients, respectively. We denote by C
0
f ac
and C
0
con
the costs incurred by A
OFL
to open facilities and to
connect clients, respectively.
The algorithm opens the cheapest k 1 facilities
other than the ones opened by A
OFL
. Let f
max
be the
maximum facility cost and f
min
the minimum facility
cost. Thus, we have that:
C
f ac
C
0
f ac
+ f
max
· (k 1) (6)
Given any client i. Apart from its connecting cost
c
i
incurred by A
OFL
, our algorithm connects i to k 1
other facilities, each resulting in a connecting cost at
most
c
max
c
min
· c
i
, where c
max
and c
min
are the maximum
and minimum connecting costs, respectively. This
implies an overall connecting cost:
C
con
C
0
con
· (1 +
c
max
c
min
(k 1)) (7)
We now add the two equations above and do some
algebraic manipulations by using:
C
0
con
C
0
C
0
f ac
C
0
C
0
c
min
C
0
f
min
This yields to:
C C
0
· (2 +
f
max
f
min
(k 1) +
c
max
c
min
(k 1)) (8)
Recall that A
OFL
is r-competitive and so C
0
r ·
Opt
0
. Since Opt
0
Opt, the theorem below follows.
Theorem 2. Given an online (deterministic or ran-
domized) r-competitive algorithm for metric Online
Facility Location. Then there is an online
O(max{
f
max
f
min
,
c
max
c
min
} · k · r)-competitive algorithm for
the Online Metric Multi-Facility Location, where k is
the number of required connections; c
max
and c
min
are
the maximum and minimum connecting costs, respec-
tively; f
max
and f
min
are the maximum and minimum
facility costs, respectively.
Running the deterministic algorithm of Fo-
takis (Fotakis, 2008) for metric OFL, with O(
logn
loglog n
)-
competitive ratio, results in the following.
Corollary 1. There is an online O(max{
f
max
f
min
,
c
max
c
min
} ·
k ·
logn
loglog n
)-competitive deterministic algorithm for the
Online Metric Multi-Facility Location, where n is the
number of clients; k is the number of required connec-
tions; c
max
and c
min
are the maximum and minimum
connecting costs, respectively; f
max
and f
min
are the
maximum and minimum facility costs, respectively.
4 CONCLUDING REMARKS &
FUTURE WORK
In this paper, we have assumed there is a unique pos-
itive integer k for all clients. In many application
scenarios, it is likely that clients have different num-
ber of required connections. A slight modification
in our algorithms would yield to O(log(k
max
n)logm)
and O(max{
f
max
f
min
,
c
max
c
min
} · k
max
·
logn
loglog n
) competitive ra-
tios for ONMFL and OMMFL, respectively, where
k
max
is the maximum required connections. One re-
search direction is to target better competitive ratios
for these variants.
This brings us to the next question, for Online
Metric Multi-Facility Location (OMMFL), whether it
is possible to get rid of the parameters c
max
, c
min
, f
max
,
and f
min
from the competitive ratio or achieve lower
bounds in terms of these parameters. To achieve the
former, one may want to attempt a primal-dual ap-
proach, for instance, by trying to extend the algorithm
of Fotakis (Fotakis, 2007) for metric Online Facility
Location.
Finally, in our model here, demands and their ar-
rival order are given by an oblivious adversary. One
may want to consider other types of adversary for
these sequences, for instance, by exploring various
probability distributions.
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