Heuristic Function Influence to the Global Optimum Value in Shortest Path Problem

IOSR Journal of Computer Engineering (IOSR-JCE)
e-ISSN: 2278-0661,p-ISSN: 2278-8727, Volume 18, Issue 5, Ver. V (Sep. - Oct. 2016), PP 39-48
www.iosrjournals.org
DOI: 10.9790/0661-1805053948 www.iosrjournals.org 39 | Page
Heuristic Function Influence to the Global Optimum
Value in Shortest Path Problem
Andysah Putera Utama Siahaan
Faculty of Computer Science, Universitas Pembangunan Panca Budi
Jl. Jend. Gatot Subroto Km. 4,5 Sei Sikambing, 20122, Medan, Sumatera Utara, Indonesia
Abstract: Determination of the optimum route is often encountered in daily life. The purpose of the optimum
route itself is to find the best trajectory of the two pairs of vertices contained in a map or graph. The search
algorithm applied is A*. This algorithm has the evaluation function to assist the search. The function is called
heuristic. Two methods which have been introduced as a step to obtain the value of heuristic function are by
using Euclidean and Manhattan distance. Both of these methods create the optimum distance in shortest path
problem, but these functions gain the different results. This research performs the development of the heuristic
function using Euclidean, Manhattan, Euclidean Square and the author method (Andysah).
Keywords: Shortest Path, Artificial Intelligence, A*
I. Introduction
The optimum route is always required to travel in achieving a goal [7], but it needs a way how it is
reached. To determine the optimum route, there are advantages and disadvantages associated with the method
used where each method produces different output. The gap will affect the outcome of a process of finding the
optimum route. In the A* algorithm, heuristic function is a value that affects the optimum value; the route length
depends on the heuristic functions given [1][2].
In the previous research, the search for the optimum path can be done by two methods: conventional
methods and heuristic methods. The conventional method is a method that uses a simple calculation used to
resolve a few vertex alone. As for vertex more, more varied methods of heuristics for heuristic method
directional approach and conduct searches. Some heuristic methods that have been used to get the value of the
heuristic function are Euclidean and Manhattan distance. Euclidean will provide the results to the heuristic
function by measuring the direct distance between two vertices. It is done by drawing a straight line without a
hitch and then determine the distance between two vertices, while Manhattan, value heuristic function is
obtained by mapping the vertex to a coordinate plane two dimensions and then summing the difference in the
second coordinate of the vertex.
From some heuristic methods to calculate the optimum distance of a path, then there is a comparison
and calculations to determine which method is the best that works for the determination of the optimum route.
The author will discuss the methods used in the heuristic function to get the global optimum value. The author
will try to create a method to expect this method to have a better result than the method that has ever existed.
The method used is to determine the initial vertex to vertex distance and the purpose of the vertex after the
initial vertex to vertex of interest. This method will get the new distance measurement models that can be
applied to the heuristic function.
II. Related Work
In a heuristic search method, used a method used to evaluate the circumstances of individual problems
and determine how far they could be used to obtain the desired solution [4]. The heuristic function is different
for each destination, so this function is often used as an important parameter in resolving problems. The
heuristic function is implemented to find the distance between two vertices, which typically uses Euclidean and
Manhattan distance.In determining the value of the heuristic function will involve the coordinates x and y,
where x and y are the meeting point of the two-dimensional plane. Determining the value of x and y are
assumed to be taken of latitude and longitude which has been converted into two-dimensional coordinates.
Optimum Search Using Genetic Algorithm
This study describes the application of genetic algorithms to find the most optimum from the point of
origin to the point of destination. Genetic algorithm system that has been designed using chromosome
representation in the form of a bit string. Therefore this kind of mutation that is used is a bit mutation. The
system also uses several methods of selection, namely roulette wheel, elitism and a combined method of the
roulette wheel and elitism. There are two types of crossover used, one cut point and two cut point crossover [6].

Heuristic Function Influence to the Global Optimum Value in Shortest Path Problem
Optimum Route Finders On Map to Increase Efficiency Travel Time Road Users By Method A* and BFS
This study develops an application for the determination of the optimum route using a digital map
where in this application there is congestion factor, road condition, and mileage. To search routes used method A
* and Best First Search, which uses a heuristic function to 'direct' the search on the map which is represented in
the concept graph [5].
Shortest Variations Determination Function Using Heuristic Algorithm A* In Mobile Devices
This study also aims to build an application on mobile devices, especially cell phones that can provide
the shortest route from a starting location to a destination location. It also compares the execution time of the
algorithm and the number of nodes are examined for each heuristic function in the search for the shortest route
by using algorithms A*.
III. Proposed Work
A. Graph Design
The graph that will be used in this study was obtained by taking a cut from the fields of the digital map
obtained with the help of Google Maps rather take the city of Medan Baru subdistrict of Medan. The streets on
the digital map will be formed into a graph by hand by way of drawing a straight line on its lines (remapping).
Fig. 1Pieces of the field map obtained from Google Maps
Once a map is obtained as shown in Figure 1, then it can be that each line has a junction and will be
placed vertex. From the point of intersection of these lines will be established weighted graph. The goal is that
defined distances between vertices. This distance will then be processed as a determinant of the direction of the
road users who want to move from the starting point to the end point. Branching on the map is so complex that
there are many routes to be passed from the destination point to the end point. After the mapping phase is
completed, then the vertex will be formed at each branch and at the end of a dead end road. Then the vertex
indexed for easy programming.
Fig. 2A Graph that has been formed

Figure 2 is a map that has been converted into a graph, but the graph has yet to do the best distance
calculation of specific points. The next step is to give weight to contain the shortest distance (straight line) from
the two points and the names on the vertex. The number of vertices in the graph reaches 120 pieces so it is a
fairly complex graph for examination. It must be emphasized, rather than relying on the truth of little or number
of items involved, but depending on the programming algorithms.
Fig. 3Final result of the mapping graph
Figure 3 is already a graph, and more precisely as a map as well as have a real comparison on each of
the vertices. However, to do a search by using the A* algorithm takes a heuristic function that supports the
movement of the election.
B. Heuristic Function
The heuristic function will determine the cost to get to the goal by using the Euclidean distance,
Manhattan, Euclidean Square and a new method. In the discussion, will the tests on methods of heuristic
functions on some models of cases, which method is superior to other methods, and how it functions heuristic
influence the outcome of the global optimum.
Euclidean Distance and Euclidean Squares
Heuristic value for h (n) is obtained by the calculation using the Euclidean distance is the distance that
determines the direct relationship freeway from two points while Euclidean Squared takes the value of the sum
of squares difference of x and y coordinates.
Known:
- Vertex coordinate 12: 287, 424
Asked:
- Euclidean Distance
- Euclidean Squares
Answer:
- x1 : 287
- y1 : 424
- x2 : 669
- y2 : 648
- ED2 : (x2 - x1)2
+ (y2 - y1)2
: (669 - 287)2
+ (648 - 424)2
: 3822 + 2242
: 145.924 + 50.176
: 196.100
- ED : √196.100
: 442,8318
: 443

Manhattan Distance
The value of h (n) will be obtained from the calculation of the difference of the coordinates of the
vertices start, and end vertices are represented in the form of 2D coordinates.
Known:
Asked:
- Manhattan Distance
Answer:
- x1 : 287
- y1 : 424
- x2 : 669
- y2 : 648
- MD : (x2 - x1) + (y2 - y1)
: (669 - 287) + (648 - 424)
: 382 + 224
: 606
Fig. 4 Euclidean and Manhattan calculation
The result of the calculation of Euclidean and Manhattan earlier showed the same results as in figure
3.5. The position of the origin point is at the lower left of the image. The values of x1, y1, x2 and y2 taken by
the shift of the position of the origin point. To determine the point coordinates, use the help of AutoCAD
software that was used to draw a map or maps. Moreover, having done the math, Euclidean calculation results
have a smaller value compared with the results obtained from the calculation of Manhattan for calculating
Euclidean diagonal from the difference between x and y coordinates.
C. Andysah Method
The method used in this study is Euclidean, Manhattan, and Euclidean Squares. The third method of
generating optimum values vary. As a complement to the study, the authors created a new method that will be
used as a comparison to the three methods that have been done, and it is hoped this method has better results
than previous methods. The new method is called Andysah where the calculation result of this formula will be
given to the heuristic function. The purpose of this method is to direct the movement towards the destination
point by selecting the vertex is better under the terms are given to this method. Selection of the best vertex
raised by way of calculating three-point coordinates, i.e., the coordinates of origin, and the next destination.

Fig. 5Andysah method
Figure 5 shows how to perform calculations on Andysah method. To determine the functionality of the
heuristic A to C obtained through an intermediary B is H = BC + AC, while AB is not used. To find a long-BC
and AC can use Euclidean Distance (equation 2.3). Flowchart can be found in Appendix E. It is formed with the
formulation as follows:
Ha-b-c = BC + AC
Ha-b-c = xc– xb ^2 + yc– yb ^2 +
xc– xa ^2 + yc– ya ^2
Keterangan:
- Ha-b-c : total cost a to c via b
- xa : coordinate x on a
- xb : coordinate x on b
- ya : coordinate y on a
- yb : coordinate y on b
Fig. 6 Sample of Andysah method
In Figure 6, it is known that the coordinates of the point A (0, 20), B (40, 120), C (100, 70), D (70, 0)
and E (250, 45). The purpose of this method to get a path from vertex A to vertex E. The position of the current
search is in vertex A, while towards the vertex E 3 pieces (B, C, D) track selection. The method given in this
heuristic function will calculate the smallest cost in the third vertex selection. The value of the smallest cost will
be elected as the track be passed after the vertex A.
The calculation:
Coordinate:
A(0, 20), B(40, 120), C(100, 70), D(70, 0) and E(250, 45)
Vertex A => B
BE = sqrt((250-40)2
- (45-120)2
)
= 222,991
AE = sqrt((250-0)2
- (45-20)2
)
= 251,2469
T = BE + AE
= 222, 991 + 251,2469

= 474,2379
Vertex A => C
CE = sqrt((250-100)2
- (45-70)2
)
= 154,0292
AE = sqrt((250-0)2
- (45-20)2
)
= 251,2469
T = CE + AE
= 154,0292 + 251,2469
= 405,2761
Vertex A => D
DE = sqrt((250-70)2
- (45-0)2
)
= 185,5398
AE = sqrt((250-0)2
- (45-20)2
)
= 251,2469
T = DE + AE
= 185,5398 + 251,2469
= 436,7867
If compared to the value obtained between AB, AC, and AD, the smallest value that is owned by AC is
405.2761 after going through vertex A, the next track will pass through the vertex C.
IV. Evaluation
A. Observation
The following test will try taking a sample calculation of the graph. Here will be displayed 25 pieces
scattered vertex at particular coordinates. By using the A* algorithm, the process looks for the best path from
vertex "G" to "S". There is some way from the vertex G towards S. The heuristic function takes a role in
choosing the best route. The following calculation tests the heuristic functions by using the same graph model.
Here are the coordinates for the vertex position information that is spread in the form of 2D coordinates and the
number of branching vertices.
Graph = { 'A' : [100, 100, ['B', 'F']],
'B' : [100, 250, ['A', 'C', 'G']],
'C' : [100, 400, ['B', 'H', 'D']],
'D' : [100, 550, ['I', 'C', 'E']],
'E' : [100, 700, ['D', 'J']],
'F' : [300, 100, ['A', 'G', 'K']],
'G' : [300, 250, ['B', 'H', 'F', 'L']],
'H' : [300, 400, ['C', 'G', 'I', 'M']],
'I' : [300, 550, ['D', 'H', 'N', 'J']],
'J' : [300, 700, ['E', 'I', 'O']],
'K' : [500, 100, ['F', 'L', 'P']],
'L' : [500, 250, ['G', 'K', 'Q', 'M']],
'M' : [500, 400, ['H', 'L', 'R', 'N']],
'N' : [500, 550, ['I', 'M', 'S', 'O']],
'O' : [500, 700, ['J', 'N', 'T']],
'P' : [700, 100, ['K', 'Q', 'U']],
'Q' : [700, 250, ['L', 'P', 'V', 'R']],
'R' : [700, 400, ['M', 'Q', 'W', 'S']],
'S' : [700, 550, ['N', 'R', 'X', 'T']],
'T' : [700, 700, ['O', 'S', 'Y']],
'U' : [900, 100, ['P', 'V']],
'V' : [900, 250, ['U', 'Q', 'W']],
'W' : [900, 400, ['V', 'R', 'X']],
'X' : [900, 550, ['W', 'S', 'Y']],
'Y' : [900, 700, ['X', 'T']]}

Fig. 7 Graph model
Figure 7 describes the results forming the vertex position by using data on the graph above information.
The formation of this graph deliberately made irregular by the number five columns and five rows to see the
path that will be skipped program using four methods described previously. Can be seen the problems that occur
is how to get the optimum out of the vertex G to the vertex S. In eyesight, the possibility of the path there are
five, namely G-L-Q-R-S, G-L-M-R-S, G-H-M-R-S, G-H-M-N-S, and G-H-I-N-S. The fifth possibility is to have
the same total cost is 700. The laying of point coordinates with regularly aims to obtain a search pattern
(searching pattern) of the four methods used.
Fig. 8 Euclidean result
The result shown in Figure 8 states that by using the Euclidean distance in the method of use of
heuristic functions, the A * algorithm will approach the first vertically and horizontally eventually approach
after the vertex vertically movement has traversed one by one. The distance G to H closer than the distance G to
L and while the cost (GH + HS) <(GL + LS), thus making G rather than L H to move to the next vertex.
Moreover, when the process is in the first position, will move horizontally in the direction N due to the costs
involved (IN + NS) <(IJ + JS). At the moment the process is in the S search will stop because the vertex of
interest has been found.
Fig. 9 Manhattan result

The result shown in Figure 9 stated that by using the method of use of Manhattan distance heuristic
function, the A * algorithm have the same result by using the Euclidean distance journeys will be made
vertically advance and eventually approached horizontally as vertically vertex movement has traversed one by
one. Results from Manhattan distance calculation produces a value not very different from the Euclidean
distance. Can be seen (GH + HS) <(GL + LS), so that G prefers H instead of L to move to the next vertex.
Moreover, when the process is in the first position, will move horizontally in the direction N due to the costs
involved (IN + NS) <(IJ + JS). At the moment the process is in the S search will stop because the vertex of
interest has been found.
Fig. 10 Euclidean Square result
The results shown in Figure 10 has a clear difference compared with the results obtained in figure 4.4
and 4.5. The use of the Euclidean distance squared has a different approach than the previous method. The trip
will be made in a zig-zag movement starting horizontal and then vertical movement. The results of the
calculation of the Euclidean distance squared calculation produces much different from the Euclidean distance
and Manhattan. Can be seen (GH + HS)> (GL + LS), so that G prefers L than H to move to the next vertex.
When the process is at stake L, the thing done is the opposite, namely (MN + NS)> (MR + RS), so I would
prefer R than N. And finally, when the process is in the S search will stop because the vertex of interest have
been found.
Fig. 11Andysah result
The results shown in Figure 11 is inversely proportional to the results obtained in Figure 4.6. The use
of this method is not much different from the Euclidean distance squared. The trip will be made in a zig-zag
starting the vertical movement and then the horizontal movement. The results of calculations using this method
produce a different calculation so as to have different results. Can be seen (GH + HS) <(GL + LS), so that G
prefers H instead of L to move to the next vertex. When the process is at stake H, the thing done is the opposite,
namely (HM + MS) <(HI + IS), so that H would prefer M than I. And finally, when the process is in the S
search will stop because the vertex of interest have been found.

B. Andysah Complexity
To determine the efficiency of the Andysah method on A* algorithm, it must perform the calculation
and analysis of algorithms. The complexity of the algorithm A* depends on the method applied to the heuristic
function. In the worst case, the number of nodes that can be developed exponentially in the solution, but its
complexity can be polynomial if the heuristic function meets the following conditions:
| h(x) - h*(x) | = O(log h*(x))
Where h* is heuristic an optimal or the cost of which is certain to get to the destination point. With the
use of heuristic function H (n), the A * algorithm can focus the search on the vertex closer to the purpose of the
search. Then the search will be dismissed if the vertex of interest is encountered. This method can minimize the
number of vertices to be examined and the time it takes to get on line directly proportional to the number of
vertices is checked, then the search time can be optimized.
C. Circumstances Worth
Circumstances Worth going if it is found if there are two or more nodes examined has a price f (n) the
same. It often happens because f (n) depend on two functions, namely the function g (n) and h (n). It is very
likely to occur between nodes located far apart, and that the most likely node located near the goal while the
other node is located away from the destination node.
10 G
9 1
8
7
6
5
4
3 2
2 S
1
1 2 3 4 5 6 7 8 9 10
Fig. 12Circumstances Worth
In Figure 12 it can be seen that S is the vertex of the initial (start), G is a vertex end (goal), "1" and "2"
is the vertex position that will be compared to the value heuristic functions. From the evaluation function f (n) =
g (n) + h (n) :
vertex “1” : f(n) = (8 - 2) + (9 - 2) + (9 - 8) + (10 - 9)
= 6 + 7 + 1 + 1
= 15
vertex “2” : f(n) = (6 - 2) + (3 - 2) + (9 - 6) + (10 - 3)
= 4 + 1 + 3 + 7
= 15
Because the A * algorithm gives priority based on the price of f(n), then if the state is worth going on,
there is more than one node with the same priority. The result is that these nodes will be checked first, which is
probably the nodes located far apart the destination node. It causes the performance of the A* algorithm
decreased.
V. Conclusion
Based on the discussion and evaluation of the previous chapters and the existing theory, it can be
concluded that the function heuristic affects the global optimum results on the search path because different
methods were applied to the heuristic function, will produce global optimum results are different, too. The
overall method of Euclidean, Manhattan, and Andysah be the best method to use on a heuristic function for
optimum yield more lanes than the Euclidean Squares. The Euclidean method generates the shortest route.

While Manhattan is rated second. The results produced by the Manhattan have differences that are not too far
away, even in certain cases the result is the same as the Euclidean. Regarding running time, Andysah method
has better results than the Euclidean and squared Manhattan.Euclidean generate good execution speed by the
way just raised slightly vertices in the search process, but these produced no better than Euclidean, Manhattan,
and Andysah.
References
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[2]. G. Gerla, "Effectiveness and Multivalued Logics," The Journal of Symbolic Logic, vol. 71, pp. 137-162, 2006.
[3]. H. J. Cho and C. L. Lan, "Hybrid Shortest Path Algorithm For Vehicle Navigation," Journals Springer Science, vol. 10, no. 49, pp.
234-247, 2008.
[4]. F. Lam and A. Newman, "Travelling Salesman Path Problems," Journal Math Program, vol. 10, no. 113, pp. 39-59, 2005.
[5]. R. Adipranata, "Aplikasi Pencari Rute Optimum Pada Peta Guna Meningkatkan Efisiensi Waktu Tempuh Pengguna Jalan Dengan
Metode A* dan Best First Search," Jurnal Teknik Informatika, vol. 8, no. 2, pp. 100-108, 2007.
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pp. 78-83, 2002.
[7]. A. P. U. Siahaan, "Adjustable Knapsack in Travelling Salesman Problem Using Genetic Process," International Journal of Science
& Technoledge, vol. 4, no. 9, pp. 46-55, 2016.

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