15-Puzzle Problem Solving With The Artificial Bee Colony Algorithm Based On Pattern Database

Journal of Universal Computer Science, vol. 27, no. 6 (2021), 635-645
submitted: 27/2/2021, accepted: 31/5/2021, appeared: 28/6/2021 CC BY-ND 4.0
15-Puzzle Problem Solving with the Artificial Bee Colony
Algorithm Based on Pattern Database
Adem Tuncer
(Yalova University, Yalova, Turkey
https://orcid.org/0000-0001-7305-1886, adem.tuncer@yalova.edu.tr)
Abstract: The N-puzzle problem is one of the most classical problems in mathematics. Since the
number of states in the N-puzzle is equal to the factorial of the number of tiles, traditional
algorithms can only provide solutions for small-scale ones, such as 8-puzzle. Various uninformed
and informed search algorithms have been applied to solve the N-puzzle, and their performances
have been evaluated. Apart from traditional methods, artificial intelligence algorithms are also
used for solutions. This paper introduces a new approach based on a meta-heuristic algorithm
with a solving of the 15-puzzle problem. Generally, only Manhattan distance is used as the
heuristic function, while in this study, a linear conflict function is used to increase the
effectiveness of the heuristic function. Besides, the puzzle was divided into subsets named pattern
database, and solutions were obtained for the subsets separately with the artificial bee colony
(ABC) algorithm. The proposed approach reveals that the ABC algorithm is very successful in
solving the 15-puzzle problem.
Keywords: 15-puzzle, Artificial bee colony, Pattern database, Iterative deepening
Categories: I.2., I.2.0, I.2.1, I.2.8
DOI: 10.3897/jucs.65202
1 Introduction
The N-puzzle (e.g., 8-puzzle, 15-puzzle) has been usually used to test search techniques
because they can represent a particular area of problems [Drogoul and Dubreuil, 1993].
Research for the N-puzzle or sliding puzzle problem has generally focused on using
traditional search methods such as the breadth-first search and deriving various
heuristic methods for informed search methods such as the A-Star (A*) algorithm. The
N-puzzle consists of a square board with numbered tiles: e.g., a 3×3 board for the 8-
puzzle, a 4×4 board for the 15-puzzle, and a blank cell for sliding the tiles. In the puzzle,
a tile is slide to a blank cell at each step to reach a goal state from the initial state and
iteratively continued until the goal state is reached. The main purpose is to arrange the
tiles according to the goal using as few moves as possible. Tiles can only be moved
horizontally or vertically into blank cells.
Solving these sliding puzzles manually or automatically is so challenging. The
state-space for the 8-puzzle contains over 105
nodes, the 15-puzzle space about 1013
nodes, and the 24-puzzle almost 1025
nodes [Korf and Taylor, 1996], [Piltaver et al.
2012]. The uninformed search algorithms specifically used for 8-puzzle do not require
any domain-specific knowledge. On the other hand, one restriction of uninformed
search algorithms is that they expand and store all possible intermediate states [Wang

636 Tuncer A.: 15-Puzzle Problem Solving with the Artificial Bee Colony Algorithm...
and Li, 2017]. Considering the search area with trillion possible states, more than
uninformed (brute force, blind) searches are required.
Figure 1 shows an example with an arbitrary initial and goal state of 15-puzzle
denoting the tile by ti and the blank tile by t0, <t11, t15, t1, t4, t3, t0, t2, t6, t10, t13, t9, t7, t8,
t5, t4, t12> for initial state and <t1, t2, t3, t4, t5, t6, t7, t8, t9, t10, t11, t12, t13, t14, t15, t0,> for
goal state shown in the figure.
Figure 1: 15-puzzle a) arbitrary initial state, b) goal state
Some studies that use heuristic search algorithms such as A* [Hart et al. 1968] and
IDA* (Iterative Deepening A*) find optimal solutions to this problem. Solution
approaches to the N-puzzle problem are generally presented using different heuristic
functions with the A* algorithm in the literature. Many of the implementations use the
number of tiles that are not in place (misplaced tiles) or the distance of the tiles from
their current location to where they should be. Manhattan Distance (MD) is used for
distance measurement and is the sum of the absolute vertical and horizontal distance
between the current tile and the same tile in the goal location. A* and IDA* are always
capable of solving the N-puzzle problem when using an admissible heuristic function,
but time required increases exponentially when the N value rises [Drogoul and
Dubreuil, 1993]. Moreover, as the size of the puzzle increases, this makes it difficult to
solve with the A* algorithm.
[Reinefeld, 1993] indicated that the 8-puzzle can be solved in a maximum of 31
moves using IDA*. [Nayak, 2014] proposed a solution to the 8-puzzle problem using
A* and IDA* with various heuristic functions, aiming to minimize the memory
required. [Korf and Schultze, 2005] provided the first complete breadth-first search for
15-puzzle. 8-puzzle is easy to solve with A* because of its small search space, but a
15-puzzle is cannot [Kendall et al. 2008]. [Korf, 1985] found the optimal solution to
this problem using IDA*, eliminating the difficulties of the basic A* algorithm. [Bauer,
1994] used IDA* for the 15-puzzle with two heuristics combining Manhattan and pair
distance. [Piltaver et al. 2012] examined the effect of various features of the search tree
and the heuristic evaluation function on pathology and gain in the 8-puzzle. [Wang and
Li, 2017] proposed a DSolving algorithm that mainly imitates the directly solving
approach. The algorithm quickly moves them to their goal locations along the shortest
paths for different sizes of puzzles. The authors indicated that the most distinctive
feature of DSolving is that a sliding puzzle does not need to store intermediate states
passed.
Some studies proposed a pattern database heuristic based on dividing the problem
into sub-problems and producing solutions separately. [Korf and Felner, 2002]

637
Tuncer A.: 15-Puzzle Problem Solving with the Artificial Bee Colony Algorithm...
presented an optimal solution to the 15-puzzle problem using IDA* with pattern
database heuristics. Another study using the pattern database was carried out by [Felner
and Adler, 2005] for 24-puzzle.
When examined the solution studies with meta-heuristic methods, it is seen that
there are genetic algorithm-based studies for the 8-puzzle only [Shaban et al. 2010],
[Igwe et al. 2013]. However, the solutions of the 8-puzzle problem with the genetic
algorithm have not sufficiently been researched, and the experimental results have not
fully been revealed. A similar study to the N-puzzle problem was carried out by [Naz
et al. 2016]. In the study, the word tile puzzle was tried to be solved by moving a tile
up, down, right, or left to find the maximum number of words using the Bee Colony
Algorithm. The word can be two letters or longer. The complexity of the word tile
problem is less than the complexity of the N-puzzle problem due to the variable number
of words and the increased probability of the goal location of the letters for each
different word. Although the studies in the literature are limited, they have generally
been carried out with A* and its different versions. In this study, a meta-heuristic
algorithm approach has been proposed to the solution for the 15-puzzle problem, and
the artificial bee colony (ABC) algorithm, which is one of these approaches has been
used. The main contributions of this study are summarized as follows:
• To the best of our knowledge, although meta-heuristic methods such as genetic
algorithms are utilized for the 8-puzzle in the literature, studies for the 15-puzzle
do not exist.
• The use of the ABC algorithm in the solution of this problem is the first in the
literature.
• The study shows that meta-heuristic methods are also successful in solving 15-
puzzle.
• The algorithm's success in finding the solution has been increased by combining
the Pattern database, Linear conflict, and Manhattan distance.
2 Methodology
2.1 Principles of the ABC Algorithm
The ABC algorithm [Karaboga, 2005], one of the meta-heuristic optimization
algorithms, simulates the intelligent foraging behavior of honey bee swarms. Although
it has only a few control parameters when compared with other algorithms, it has been
successfully applied to solving different kinds of optimization problems. In the ABC
algorithm, the bees are grouped into three types, employed bees, onlooker bees, and
scout bees. While the first half of the colony consists of employed bees, the second half
consists of the onlooker bees. The employed bees are responsible for searching the food
source around the hive and evaluate its quality. They then share the information of the
food source with onlooker bees which wait in the dance area. The onlooker bees choose
good food sources based on this information. The choice probability of food sources is
directly proportional to their quality. The food source with better quality has a bigger
chance of being selected by the onlooker bees [Karaboga and Akay, 2009]. If the nectar
in a food source is depleted, the bee of that source becomes a scout bee, and the scout
bee searches for a new food source randomly. In the ABC algorithm, the position of a
food source corresponds to a candidate solution to the problem, and the nectar amount

of a food source represents the quality, known as fitness, of the related solution [Ozger
et al. 2019]. The ABC generates a random population as initial of SN food sources by
Eq. (1) at the first step.
𝑥!" = 𝑥"
#!$
+ 𝑟𝑎𝑛𝑑(0,1).𝑥"
#%&
− 𝑥"
#!$
0 (1)
where i ∈ {1,2,…,SN}, j ∈ {1,2,…,D} and rand(0,1) generates a uniformly distributed
random number in the range of (0,1). xi which is a D-dimensional vector represents ith
candidate solution. xj
max
and xj
min
refer lower and upper bounds of the jth variable,
respectively.
Then, the fitness value of each solution xi represented by fiti is calculated by Eq.
(2).
𝑓𝑖𝑡! =
1
1 + 𝑓!
(2)
where fi represents the objective function value of the solution xi.
The employed bee searches for a new food source, vij, around the current source by
Eq. (3).
𝑣!" = 𝑥!" + ø!"(𝑥!" − 𝑥'") (3)
where k ∈ {1,2,…,SN}. k is a randomly selected candidate solution (i ≠ j), and øij is a
random real number in the range of [-1,1]. Then, the objective function value of the
source vi is calculated and compared with the source xi. If the objective function value
of vi is better than that of xi, vi is replaced with xi, otherwise, xi is retained. The greedy
selection process is usually used for comparison.
In the standard ABC algorithm, only one parameter of each source of xi (randomly
chosen parameter, j) is modified. This section has been modified with an approach
proposed by [Karaboğa & Akay, 2011], so the performance of the algorithm has been
increased. In the modified ABC algorithm shown in Eq. (4), there is a probability that
each j parameter changes according to a predetermined control parameter, C. Thus, it
is possible to change more than one parameter in a food source. If no parameter changes
for all j values, change a random parameter in the solution by Eq. (3). Thus, it is
guaranteed to change at least one parameter in no case parameter is changed depending
on the control parameter.
𝑣!" = 9
𝑥!" + ø!".𝑥!" − 𝑥'"0, 𝑖𝑓𝑅" < 𝐶
𝑥!", 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
(4)
where Rj is a uniformly distributed random number in the range of [0,1] and C is a
control parameter in the range of [0,1] which controls the number of parameters to be
modified of the ABC algorithm.
After the employed bee phase is completed, the onlooker bee phase starts. The
onlooker bee chooses a food source depending on the probability value, pi, related to
its nectar amount, using Eq. (5).

639
𝑝!" =
𝑓𝑖𝑡!
∑ 𝑓𝑖𝑡!
()
"*+
(5)
As with the employed bee, the onlooker bee also searches for a new food source
around the existing food source. The objective function value of the new source is
calculated, a greedy selection then is applied to choose between an existing source and
a new source.
A parameter named limit value is defined to indicate that the food source can no
longer be updated after a predetermined number of cycles for abandonment. This value
shows how many times the food source visited has not been improved. If the food
source reaches the limit value, this food source is abandoned, and then the scout bee
searches for a random new food source using the Eq. (1). In the standard ABC
algorithm, only one scout bee is allowed per cycle. These steps in the algorithm are
repeated until a predetermined maximum number of cycles (MCN). Detailed
pseudocode of the modified ABC algorithm [Karaboga and Akay, 2009] is given in the
Algorithm 1.
Algorithm 1. Fundamental steps of the modified ABC algorithm
Initialize the population of solutions xij, i = 1,2,…,SN, j = 1,2,…,D
Evaluate the fitness fiti of the population
set cycle to 1
Repeat
For each employed bee {
Produce new solution vi by (4) and calculate fiti
Apply the greedy selection between xi and vi
}
For each onlooker bee {
Calculate the probability values pi for the solution xi by (5)
Select a solution xi depending on pi
Produce new solution vi and calculate fiti
Apply the greedy selection process between xi and vi
}
If there is an abandoned solution for the scout
Then replace it with a new produced solution by (1)
Memorize the best solution obtained so far
cycle = cycle + 1
Until cycle = MCN
2.2 Solving 15-Puzzle Problem
Each food source in the ABC algorithm has a vector of directions for sliding actions
labeled with ‘1’, ‘2’, ‘3’, and ‘4’ representing a move of the blank one place left, right,
up, down, respectively. The standard ABC algorithm has been developed to solve
continuous optimization problems. For combinatorial problems, it is necessary to
convert the real numbers to integers. In this regard, Eq. (1) has been updated to generate
integer numbers. A food source example is shown in Figure 2. The initial population of

candidate solutions (foods) is generally generated randomly in the standard ABC
algorithm. But some of the randomly generated candidate solutions may include
infeasible actions for the puzzle problem like the movements that are opposite each
other are likely to be generated back-to-back. For example, a down movement after an
up movement means that no action will be taken. The initial population might be
generated by preventing back-to-back opposite movements to overcome this problem.
However, the same problem might come up in the new food sources that will be
produced from the old food sources. In this case, it may not be sufficient to produce a
feasible initial population. Furthermore, initialization with feasible solutions requires
more time. This process may reduce the search capability of the algorithm and may
increase the time to find an optimal solution [Karaboga and Akay, 2011].
Figure 2: Example of the food source in the ABC algorithm
It is obvious that generation candidate solutions considering the constraints will
add an additional cost to the algorithm. Therefore, candidate solutions were generated
without considering the constraints in the study. The movements that do not comply
with the constraints in the food sources have been ignored and removed from the food
source. Thus, a constrained optimization problem has been replaced by an
unconstrained optimization problem ignoring the constraint with movements.
Examples of possible constraints for movements in candidate solutions are shown in
Figure 3.
Figure 3: Constraints. a) back-to-back opposite moves, b) moves that exceed the
boundary of the puzzle, c) moves that exceed the boundary of the puzzle between
back-to-back opposite moves
The heuristic function based on MD used in the study has been enhanced by linear
conflict (LC). The linear conflict heuristic was the first described in [Hansson et al.
1992]. Two tiles ti and tj are in a linear conflict if they are in the same row or column,
their goal positions are both in that row or column, both tiles are different positions than
they should be. MD and LC calculations are shown in Eqs. (6) and (7), respectively.

641
For tiles with conflict, the cost of linear conflict heuristics will be at least 2 more than
MD. The objective function used as the sum of the MD and LC is shown in Eq. (8).
𝑀𝐷 = |𝑥+ − 𝑥,| + |𝑦+ − 𝑦,| (6)
𝐿𝐶 = L
1, 𝑙𝑖𝑛𝑒𝑎𝑟 𝑐𝑜𝑛𝑓𝑙𝑖𝑐𝑡 𝑒𝑥𝑖𝑠𝑡𝑠
0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
(7)
𝑓! = P 𝑀𝐷(𝑡!) + 2 × 𝐿𝐶(𝑡!)
)-+
!*+
(8)
where N represents the size of the puzzle and ti is the i. tile. In the row or column
conflict, extra movement is required for the movement of the tiles and this cost is added
to the MD.
2.3 Pattern Database
A pattern is the partial specification of an N-puzzle and a method [Culberson and
Schaeffer, 1998] based on dividing the N-puzzle into subsets and solving each subset
separately, ignoring the other tiles. This database has been referred to as the fringe.
Studies [Korf and Felner, 2002], [Felner and Adler, 2005], [Culberson and Schaeffer,
1998] show that it is more effective to solve the puzzle by dividing it into subsets
instead of solving the entire puzzle.
Figure 4: The fringe pattern for the 15-puzzle
Figure 4 shows a 15-puzzle divided into two subsets. As can be seen in the figure,
the tiles of <t1, t2, t3, t4, t5, t9, t13> are placed where they should be, and the locations
of the other tiles are ignored. Then the rest tiles are located. In the study, a pattern
database has been used as in the figure to increase the ABC algorithm’s effectiveness
in solving the puzzle.
The efficiency of the ABC algorithm's search strategy has been increased by
including an iterative deepening search approach. The iterative deepening search,
which is of the uninformed search strategies, is a method with an increasing depth limit
until a solution is found. In the study, the steps in the ABC algorithm to approach the
solution iteratively have been updated with inspiration from the iterative deepening
search. The algorithm firstly tries to find a solution by generating food resources

according to the predetermined number of moves within a depth of 1. Only if it does
not find the goal state, the algorithm then initializes the food sources and starts
searching within a depth of 2. The value of depth is incremented iteratively until the
predefined depth value. The initial state of the puzzle at each depth is the best state
obtained at the previous depth. Thus, the algorithm starts a little closer to the solution
at each depth and the time to reach the solution is shortened. This approach improves
the heuristic search with the strength of iterative deepening.
3 Experiments
In the study, the ABC algorithm was applied by combining the advantages of
Manhattan distance and linear conflict heuristics, and pattern database. The algorithm
was run on 25 randomly generated solvable instances of the 15-Puzzle. The parameters
of the ABC algorithm are as follows: the number of foods is 80, the limit number is
100, and the max cycle is 100 for each depth. The food length is determined
dynamically, different at each depth. As the goal is approached more at each depth, the
food length, which is initially determined as large, is gradually reduced. In this study,
food length is taken as an initial value of 30. Although a fixed-length value for food
length has been determined, this value becomes less in terms of the total number of
movements when consecutive undesirable moves in the sources are ignored.
Table 1 summarizes the results, averaged over 10 independent runs varying the
initial states of the puzzles. In the table, the best, mean, and worst number of moves
obtained using the ABC algorithm are given. In addition, the number of moves in the
solution of each instance with the IDA* has been also added to the table. However,
with IDA*, it is seen that the number of moves in solutions is less. The reason for this
is that the approach has a deterministic feature. The number of moves in the meta-
heuristic algorithm is greater than IDA* but the difference is negligible. Heuristic
algorithms are expected to produce acceptable results in a reasonable time, rather than
always the optimum results by nature. Considering the number of moves in the
experimental results, it is clearly seen that the ABC algorithm produces acceptable
results.
In Figure 5, the convergence graphic of the ABC algorithm running based on
iterative deepening is given for the first instance in the Table 1. Convergence graphs
also for the other instances occur similarly. As can be seen from the figure, there are
two distinct convergences. The first convergence graphic shows the convergence in the
solution of the first subset of the puzzle divided into two subsets, and the other graphic
shows the convergence in the solution of the second. It is seen from the graphic; the
first subset of the puzzle was solved at the 3rd depth and the second at the 2nd depth.

643
No Initial state
Number of moves
IDA*
ABC
Best Mean Worst
1 1 5 2 7 10 14 11 6 15 12 9 3 13 0 8 4 34 37 42 45
2 5 6 10 7 1 3 11 8 13 4 15 9 14 0 2 12 38 43 47 51
3 1 11 6 2 10 13 15 5 3 12 0 4 9 7 14 8 40 46 54 62
4 6 5 2 7 13 0 10 12 4 1 3 14 9 11 15 8 44 49 59 68
5 4 3 10 7 6 0 1 2 12 15 5 14 9 13 8 11 44 52 60 68
6 4 10 3 2 1 0 7 8 9 6 13 15 14 12 11 5 44 51 58 69
7 3 4 11 2 9 1 14 15 7 6 0 8 5 13 12 10 44 51 60 72
8 3 10 2 5 15 6 13 4 0 11 1 7 9 12 8 14 46 52 68 79
9 9 4 0 3 14 7 5 12 15 2 13 6 10 1 8 11 46 54 66 79
10 7 1 12 10 6 11 15 4 0 2 5 14 3 13 8 9 48 59 71 85
11 1 13 5 7 14 9 10 12 11 8 2 15 6 0 4 3 48 62 78 88
12 13 9 5 12 10 2 4 11 3 8 0 7 1 14 6 15 48 64 77 86
13 2 13 6 1 14 5 11 0 12 4 8 10 9 3 15 7 50 66 77 89
14 11 3 12 9 2 8 10 14 0 7 15 13 1 6 5 4 50 68 78 92
15 7 6 15 12 14 1 13 3 0 9 8 4 2 11 5 10 50 68 81 94
16 5 8 13 15 14 0 1 7 4 6 10 2 11 9 12 3 52 59 77 92
17 12 2 5 11 10 0 1 6 3 14 8 9 7 4 13 15 52 62 78 91
18 13 3 2 8 12 0 5 1 11 6 9 15 4 14 7 10 52 63 76 89
19 7 13 1 4 9 12 8 5 15 14 0 6 11 2 3 10 52 59 78 91
20 8 11 12 10 2 0 15 1 14 6 4 3 7 9 5 13 54 61 81 93
21 6 8 12 13 7 2 5 14 9 3 1 15 11 0 10 4 54 65 77 94
22 9 12 2 5 11 1 10 14 0 4 3 8 6 15 7 13 54 67 80 92
23 10 12 11 7 8 9 14 5 3 13 4 1 6 0 2 15 56 69 80 96
24 3 10 14 5 1 12 11 8 15 7 9 6 2 0 13 4 56 71 81 95
25 9 3 12 5 4 14 6 11 8 7 15 13 10 0 2 1 56 71 81 92
Table 1: Solutions for 25 randomly generated 15-Puzzle instances using ABC
algorithm
Figure 5: Convergence graphic for ABC algorithm based on iterative deepening and
pattern database

4 Conclusion
A new approach based on the meta-heuristic method using the ABC algorithm with a
solving of the 15-puzzle problem has been proposed in the study. The efficiency of the
Manhattan distance heuristic was improved by combining it with the linear conflict
function. The pattern database method was implemented for reaching the solution
faster. Besides, the efficiency of the ABC algorithm's search strategy was increased by
including an iterative deepening search approach. It is clear that the ABC algorithm
produces acceptable results given the number of moves in the experimental results,
which proves that the ABC algorithm is an effective method for solving the 15-puzzle
problem. It is obvious that the problem cannot be solved with traditional algorithms as
the dimensions of the N-puzzle grow. In this respect, meta-heuristic algorithms such as
the ABC algorithm can be used effectively in the solution of the N-puzzle problem.
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