Constraint Satisfaction Through
GBP-Guided Deliberate Bit Flipping
Item Type Article
Authors Bahrami, Mohsen; Vasić, Bane
Citation Bahrami M., Vasić B. (2019) Constraint Satisfaction Through GBP-
Guided Deliberate Bit Flipping. In: Ćirić M., Droste M., Pin JÉ.
(eds) Algebraic Informatics. CAI 2019. Lecture Notes in Computer
Science, vol 11545. Springer, Cham
DOI 10.1007/978-3-030-21363-3_3
Publisher SPRINGER INTERNATIONAL PUBLISHING AG
Journal ALGEBRAIC INFORMATICS, CAI 2019
Rights © Springer Nature Switzerland AG 2019.
Download date 09/08/2022 13:13:38
Item License http://rightsstatements.org/vocab/InC/1.0/
Version Final accepted manuscript
Link to Item http://hdl.handle.net/10150/634958
Constraint Satisfaction through GBP-Guided
Deliberate Bit Flipping
Mohsen Bahrami and Bane Vasi´c
Electrical and Computer Engineering, University of Arizona, Tucson, AZ 85721, USA
bahrami@email.arizona.edu and vasic@ece.arizona.edu
Abstract. In this paper, we consider the problem of transmitting bi-
nary messages over data-dependent two-dimensional channels. We pro-
pose a deliberate bit flipping coding scheme that removes channel harmful
configurations prior to transmission. In this method, user messages are
encoded with an error correction code, and therefore the number of bit
flips should be kept small not to overburden the decoder. We formulate
the problem of minimizing the number of bit flips as a binary constraint
satisfaction problem, and devise a generalized belief propagation guided
method to find approximate solutions. Applied to a data-dependent bi-
nary channel with the set of 2-D isolated bit configurations as its harmful
configurations, we evaluated the performance of our proposed method in
terms of uncorrectable bit-error rate.
Keywords: Probabilistic inference · Graphical models · Generalized be-
lief propagation.
1 Introduction
Many of probabilistic inference problems can be reformulated as the computa-
tion of marginal probabilities of a joint probability distribution over the set of
solutions of a constraint satisfaction problem (CSP) [1, 2]. A CSP consists of a
number of variables and a number of constraints, where each constraint specifies
admissible values of a subset of variables. A solution to a CSP is an assign-
ment of variables satisfying all the constraints. Message passing algorithms have
been successfully used for solving hard CSPs [3]. Traditional low-complexity ap-
proximate algorithms for solving these problems are based on belief propagation
(BP) [4, 5] which operate on factor graphs. BP, as an algorithm to compute
marginals over a factor graph, has its roots in the broad class of Bayesian infer-
ence problems [6]. It is well known that the BP algorithm gives exact inference
only on cycle-free graphs (trees). It has been also observed that in some ap-
plications BP surprisingly can provide close approximations to exact marginals
on loopy graphs. However, an understanding of the behavior of BP in the lat-
ter case is far from complete. Moreover, it is known that BP does not perform
well on graphs which contain a large number of short cycles. A new class of
message-passing algorithm called generalized belief propagation (GBP) is intro-
duced in [7] to solve the problem of computing marginal probability distributions
2 Mohsen Bahrami and Bane Vasi´c
on factor graphs with short cycles. The algorithm relies on the extension of clus-
ter variation method [8, 9], which is called the region graph method. The GBP
algorithm provides approximate marginals by minimizing the Gibbs free energy
using region graph method. In GBP, messages are sent among clusters of vari-
ables nodes instead of the node-to-node message passing fashion in BP and SP.
More recently GBP has been shown empirically to have good performance, in
either accuracy or convergence properties, for certain applications [10, 11].
In this paper, we consider the problem of transmitting a binary message
over a data-dependent communication channel and recovering it back at the
receiver side. This problem is one of the most fundamental problems in com-
munication theory, and can be considered as an instance of a CSP. Shannon in
his seminal work [12] introduced two coding schemes for reliable transmission
of information over a noisy channel, namely error correction coding and con-
strained coding. The first method protects user messages against random errors,
which are independent of input data, by introducing redundancy in the messages
prior to transmission. On the other hand, a constrained coding method assumes
that channel solely introduces errors in response to specific patterns in input
messages, and removing these problematic patterns makes the channel noiseless.
Recent advances in emerging data storage technologies like magnetic record-
ing systems [13, 14], optical recording devices [15] and flash memory drives [16]
necessitate to study two-dimensional coding (2-D) techniques for reliable stor-
age of information. In these systems, user information bits are arranged into
2-D arrays for storing over the recording channel, and occurrences of specific
patterns in input arrays are the significant cause of errors during read-back
process. These systems require the use of some form of error-correction coding
in addition to constrained coding of the input data or symbol sequences. It is
therefore natural to investigate the interplay between these two forms of coding
and the possibilities for efficiently combining their functions into a single coding
operation. For this purpose, we introduce a generic 2-D channel with a set of
harmful configurations to model patterning effects on an information bit from
its neighboring bits in a 2-D channel input array. In this model, information
bits contained in the harmful configurations are more vulnerable to errors than
the other bits. Different 2-D constrained coding methods have been proposed
to remedy the patterning effects in data-dependent 2-D channels, e.g., [17–22].
The goal of most of these methods is to achieve tighter bounds on the Shannon
noiseless channel capacity of constraint. However, these schemes are non-linear
in nature, and their encoder/decoder has a memory. Therefore, combinations of
these methods with an error-correction coding scheme are challenging, and even
a small number of bit errors can result multiple errors and severely degrade the
performance of an error correction decoder. As an alternative coding scheme to
address the non-linear effects of conventional 2-D constrained coding schemes,
we present a deliberate bit flipping (DBF) coding scheme for data-dependent
2-D channels, where passing through channel specific patterns in inputs are the
main cause of errors. The user message is first encoded by an error correction
code, and is arranged into a 2-D array as an input to the channel. The idea is to
Constraint Satisfaction through GBP-Guided Deliberate Bit Flipping 3
completely eliminate a constrained encoder and, instead, to remove the harmful
configurations by deliberately flipping the selected bits prior to transmission.
The DBF method relies on the error correction capability of the error correction
code (ECC) being used so that it should be able to correct both deliberate errors
and channel errors. Therefore, it is crucial to keep the number of flipped bits
small in order not to overburden the error correction decoder.
The problem of minimizing the number of deliberate bit flips for removing a
set of configurations from a 2-D array is an instance of a CSP, where variables
are arranged into a 2-D array, and constraints are defined locally over a set of
neighboring variables. Assignments to variables are chosen from encoded mes-
sages of information bits (the codewords of ECC being used), and a constraint is
violated if the realization of the neighboring variables involved in the constraint
belongs to the given set of configurations. An initial realization of variables may
violate some of constraints, and the goal is to change values of minimum number
of variables to make all the constraints satisfied. This is equivalent to removing
the forbidden configurations entirely from the 2-D array by flipping minimum
number of bits. Using a factor graph representation, we devise a constrained com-
binatorial formulation for minimizing the number of bit flips in the DBF scheme
for removing a given set of configurations. We find an approximate solution by
reformulating the minimization problem as a 2-D maximum a posteriori (MAP)
problem using a probabilistic graphical model. In this framework, patterns which
do not contain harmful configurations are assumed to be uniformly distributed,
and each pattern containing a harmful configuration has zero probability. The
GBP algorithm, as a MAP inference method, is used to find the approximate
solution for the 2-D MAP problem. Applied to a data-dependent 2-D channel
with 2-D isolated bit patterns as the set of harmful patterns for the channel, we
have shown the performance of DBF method in terms of uncorrectable bit-error
rate.
The organization of the paper is as follows. Section 2 introduces the data-
dependent 2-D channel model. The DBF coding scheme is presented in Section 3.
Section 4 explains the probabilistic formulation devised for minimizing the num-
ber of bit flips in DBF coding scheme. Numerical results are given in Section 5.
2 Channel Model
In this section, we present a data-dependent 2-D communication channel which
transmits binary rectangular patterns and produces as an output a binary pat-
tern. Passing through the channel, information bits belong to a predefined set of
configurations are more prone to errors than the other bits. The channel is char-
acterized by this set of binary configurations, which is called the set of harmful
configurations and is denoted by F.
The set of channel input patterns and the set of channel output patterns are
denoted by X and Y. An input pattern x = [x
i,j
] is chosen uniformly and ran-
domly from X , and is transmitted through the channel. A pattern y = [y
i,j
] ∈ Y
is observed through the channel. The input pattern x can be considered as a
4 Mohsen Bahrami and Bane Vasi´c
(a) (b)
Fig. 1. Fig. shows (a) Q
+
(i, j) and (b) P
i,j
over the lattice Z
2
for the case of cross-
shaped polyomino.
square binary tiling of a rectangle, where each information bit x
i,j
on the 2-
D input pattern represents a colored tile (0 (1) refers to a white (black) tile).
The channel is data-dependent, and for each tile x
i,j
, error is characterized by a
Bernoulli random variable which depends on the realization of polyominoes hav-
ing intersection with this tile. A polyomino of order k is constructed by joining
k square tiles. Here we consider cross-shaped polyominoes of order 5 which are
defined over the 2-D lattice Z
2
as the following
Q
+
(i, j) = {(i, j − 1), (i − 1, j), (i, j), (i, j + 1), (i + 1, j)} . (1)
The set of cross-shaped polyominoes that have intersection with tile x
i,j
over an
m × n rectangle is identified by
P
i,j
=
[
(i
0
,j
0
)∈Q
+
(i,j)
Q
+
(i
0
, j
0
). (2)
Fig. 1 shows Q
+
(i, j) and P
i,j
on a 2-D lattice Z
2
.
The received tile y
i,j
is characterized by
y
i,j
= x
i,j
⊕ z
i,j
, (3)
where z
i,j
is a Bernoulli random variable which depends on the realization of
P
i,j
, x
P
i,j
, and is defined by
z
i,j
∼
(
Bern(α
b
), x
P
i,j
∈ F,
Bern(α
g
), x
P
i,j
6∈ F.
(4)
Passing through the channel, colors of input tiles belong to F invert with prob-
ability α
b
, while colors of other tiles invert with probability α
g
. Since patterns
belong to the set F are the main source of errors for this communication channel,
we have α
b
α
g
.
The introduced channel has two states where in each state acts as a binary
symmetric channel with a different cross-over probability, and can be considered
