Performance of A Star with Dynamic Programming Algorithms in

Determining the Shortest Route

Henny Yusnita

, Muhammad Zarlis

and Sawaluddin

Faculty of Computer Science and Information Technology, Universitas Sumatera Utara, Indonesia

Faculty of Math and Natural Science, Universitas Sumatera Utara

Keywords: Shortest path, dynamic programming, shortest route.

Abstract: A Star's algorithms tend to perform better when combined with heuristics. By implementing the heuristic

function-based Best First Search (BFS) search mechanism, the system will find the shortest path from the

origin to the destination nodes. In the Dynamic Programming algorithm, the search for the shortest route

was based on interconnected stages by loading the route to be searched. In this study, a combination of the

A Star algorithm and Dynamic Programming methods was used. Testing of A Star Algorithm with Dynamic

Programming in the search for the shortest route from Pasar Petisah to Pasar Lau Cih was chosen. The

results of the study found that the A Star - Dynamic Programming algorithm produced the shortest distance

of 14.12 Km.

1 INTRODUCTION

The A Star algorithm is a development of the

Dijkstra algorithm. To get the best results, a route

search design by analyzing inputs and testing

potential routes is carried out. In general, this

algorithm uses traversal graphs and possible routes

when searching for a route around a node. The basic

characteristic of the algorithm used is the application

of the Open List. The Open List is a node that has

probably been skipped, has never been traversed, or

is not a dead-end route. The Close List is an

impassable node, such as a dead end or has been

passed. Thus, the Close List function aims to re-

analyze nodes that have been passed previously.

Therefore, the route search algorithm becomes faster

and does not repeat itself indefinitely. The closest

route search algorithm will stop when the destination

node has been found or the node in the Open List no

longer exists. Implementation of Close List and

Open List in finding the shortest route with a small

number of nodes becomes less productive, which

leads to the use of more complex algorithms (Taufiq,

2015).

Sharma & Pal (2015) compared the closest route

search using the A Star algorithm and Dijkstra's

algorithm. This study shows that the A Star

algorithm performs better than the Dijkstra

algorithm in all cases, where weights and distances

are not limited. The A Star algorithm is proven to

provide the best search results compared to others.

Further research entitled Guided Hybrid A Star Path

Planning Algorithm for Valet Parking Applications

was piloted by Sedighi (2019). This research

presents an innovative and computationally efficient

approach to combining the search engine and the

famous Hybrid A Star with visibility diagrams in the

search for the shortest possible non-holonomic path

in a hybrid (continuous discrete) atmosphere for

valet parking. In the study, the Visibility Diagram

was applied to introduce a cost saving function as

well as for the Hybrid A Star algorithm.

Furthermore, the shortest path found is used to

provide the correct waypoint so that Hybrid A Star

can plan the optimal route by taking into account

non-holonomic constraints. This method has been

researched extensively with results 40% faster than

the Hybrid A-star algorithm or an average of 20%.

Karova & Penev's (2016) conducted a

comparative study between A Star Algorithm and

backtracking algorithms to find the shortest route of

a maze. The results reveal that the backtracking

algorithm can be used effectively in small mazes

with few constraints. Likewise, it can also trace the

shortest route for a shorter time. For larger size and

number of obstacles in the maze, the algorithm

produces the best results. In several trials, this

536

Yusnita, H., Zarlis, M. and Sawaluddin, .

Performance of A Star with Dynamic Programming Algorithms in Determining the Shortest Route.

DOI: 10.5220/0010335700003051

In Proceedings of the International Conference on Culture Heritage, Education, Sustainable Tourism, and Innovation Technologies (CESIT 2020), pages 536-542

ISBN: 978-989-758-501-2

algorithm produced a shorter route in a shorter time

compared to the other two algorithms. The genetic

algorithm can be applied in some cases, where the

computation time required is shorter compared to the

shorter route. Obstacle removal will be simulated

during the real-time search process starting from

10% to 30%. With such an algorithm, the robot will

be able to move in a dynamically changing

environment.

In his research entitled Improving A Star

Hierarchy Algorithm for Optimal Parking Path Plan

from Large Parking Spaces, Cheng & Yan (2014)

revealed that optimal parking lane planning in a

large parking area. The optimal evaluation index is

determined in a short period of time and then

increases. Given the situation, the corresponding

evaluation function is well developed by the A Star

plus algorithm, which results in better, more

efficient and more accurate searches. The evaluation

index in the parking lane planning from the parking

lot is the optimal time rather than the optimal

distance. The algorithm must be able to avoid

congested roads at the same time. The experimental

results show that the built algorithm can effectively

plot different optimal path information with

additional attributive data parameters.

Dynamic Programming is a method for finding

the shortest route by dividing the search area and

stages. In this case, the route obtained from each

region is considered a series of fastest paths that

support each other. Dynamic Programming can

identify and solve multiple problems, including

work scheduling, optimal operation division,

shortest route, and workforce size. When getting

optimal results, the Dynamic Programming

algorithm applies the principle of optimality by

dividing the problem into several parts called stages.

The set is then resolved with optimization until all

cases are resolved. The best decision is a decision

body made of all stages, also called optimal

decisions (Fawwaz, 2019).

This study aims to determine, analyze, and study

how the Dynamic Programming algorithm works

integrated with the A Star algorithm to identify the

shortest route on a city graph with certain

limitations. Solve the shortest route problem using

advanced recursive dynamic programming. The

route discussed in this study is the route from Pasar

Petisah to Pasar Lau Cih. However, it does not

discuss road congestion, road sections, road

conditions and road priorities. The routes covered in

the analysis are land routes with good conditions

that can be traversed by motorized vehicles.

The AI applied in this research is the Dynamic

Programming algorithm to find a path when

attacking opposing participants in a game. Police

figures are tested in a game. From 10 experiments,

he obtained 90% success in finding the shortest and

optimal path with the Dynamic Programming

algorithm (Fawwaz, 2019).

Dynamic Programming (DP) is an optimization

algorithm that can be applied in everyday life. The

DP algorithm is carried out in several stages so that

problems can be perceived through a series of

interdependent decisions. In solving the problem, the

DP algorithm carries out two approaches, namely

Forward and Backward so that the problem can be

resolved as a whole by using these two procedures.

Even though the final optimal value is the same, the

achievement of the optimal value at each stage

between the Forward and Backward approaches is

different. Differences in absolute optimal values are

obtained under the same conditions if resolved using

Forward and Backward strategies (Kamal, 2017).

In this study, the proposed strategy provides a

popular file-to-cache combination that maximizes

the likelihood of optimal cache being found with

pseudo-polynomial time complexity. For these

purposes, the resource algorithm is known as the

0/1-Knapsack problem assuming that files are

cached without the partition being used (Ayenew,

2018).

The A Star algorithm is a classic, efficient

algorithm for searching the shortest path on a graph.

The productivity of the algorithm depends on an

evaluation function that estimates the heuristic value

of the shortest path from the starting point to the

destination. If you know the coordinates of the

vertices (x, y), the heuristic value of the shortest path

becomes the distance. This study presents the Index-

Based A Star algorithm which aims to solve the

shortest path problem on directed and weighted

graphs with unknown vertices of coordinates. The

development is done by using three indexes for each

node, namely the earliest, second, and newest arrival

index. Thus, the lower limit and upper limit of the

shortest distance from the starting point to the target

are based on these three indices. This algorithm

makes use of the earliest arrival index to define the

evaluation function of the A Star algorithm and

applies three indexes to unused nodes to improve

algorithm performance. Compared with the A Star

algorithm, the additional time and space complexity

of the experimental results proves the effectiveness

of the proposed design (Lie, 2018).

The investigation aims to find out how the A Star

algorithm determines the shortest route during traffic

Performance of A Star with Dynamic Programming Algorithms in Determining the Shortest Route

537

jams. The results of the experiment provide a more

realistic picture of the A Star algorithm process in

determining the closest path (Syukriyah & Solihin,

2016).

1.1 A Star Algorithm

Nils Nilsson developed the A star algorithm in 1964

based on Dijkstra's algorithm with the name A1

algorithm. In 1967, Bertram Raphael refined the first

algorithm to become the A2 algorithm. However, in

1968, Peter E. managed to prove the supremacy of

the A2 algorithm over the previous version. Thus,

the A2 algorithm is claimed to be the most optimal

algorithm in finding the closest route and is renamed

to A Star. Based on time, the A Star algorithm is

better than Dijkstra's algorithm which uses a

heuristic search function. The A Star algorithm is the

most frequently used algorithm in route finding and

graph exploration, which plots the most efficient

paths between points called nodes. This algorithm

looks for a path or route very efficiently where this

algorithm is also developed from the basic method

of Best First Search (Syukriyah, 2016).

The main principle of the A Star algorithm is to

find the shortest path from the starting point to the

destination node which underlies the shortest

distance and lowest cost. This algorithm process

begins by placing A at the starting point and entering

all neighboring nodes, without the obstacle attribute

with A into the Open List. Then, the system

identifies the smallest H node value in the Open

List, and then moves A to the smallest H node value.

The nodes before A are stored as the primary of A

and are then classified into a Closed List. If another

node has been relocated adjacent to A but is not a

member of the Open List, that node will be

categorized as Open List. After that, compare the

existing G value with the previous one. If the

previous G value was smaller, then A will return to

the starting position. Node is designed to enter the

Close List. These steps can be rerun until a solution

is found or no other nodes are available in the Open

List. Figure 1 illustrates how the concept of the A

Star algorithm works.

Figure 1: A Star Algorithm.

The Star algorithm has two main functions when

determining the best solution. The first function is as

g (x) which functions to calculate the total cost

(distance or time) required from origin to destination

nodes. The second function is as h (x) to estimate the

total cost estimated from origin node to destination

node.

G(n) =

√

𝑋𝑛



 𝑌𝑛



(1)

H(n) = |X(target) – X(n)| + |Y(target) – Y(n)| (2)

F(n) = G(n)+F(n) (3)

1.2 Dynamic Programming

Dynamic Programming was first introduced by

Richard Bellman in the 1940's. This describes the

problem-solving process, where the desired solution

is the most optimal, often referred to as the global

optimal (Ayenew, 2018). The optimal substructure

means that the optimal solution for the previous sub-

problem can be used to find the optimal solution for

the problem at hand. In the graph, for example, the

shortest path from origin to destination node can be

found by first calculating the shortest subpath from

all neighboring origin nodes to the destination node

and then using the shortest subpath to obtain the

shortest complete path. There are three stages in

problem solving with an optimal substructure,

namely:

1. Break the problem into smaller sub-problems,

2. Find the optimal solution to the sub-problem

using these three steps recursively,

3. Use the optimal solution for the sub-problem

to build the optimal solution for the problem

at hand.

The problem-solving process is continued by

dividing it into smaller parts until a simple case can

CESIT 2020 - International Conference on Culture Heritage, Education, Sustainable Tourism, and Innovation Technologies

538

be solved quickly. Dynamic Programming is a

method of solving problems by breaking down

solutions into groups (stages). Problem solving can

be observed from a series of interrelated decisions.

Dynamic Programming emerged because of its

ability to analyze and document the results at each

stage using multiple tables for more detailed

solutions. The characteristics of problem solving

with Dynamic Programming can be described as

follows:

1. Several possible options,

2. Solutions at each stage are built from the

results of the completion of the previous stage,

3. The algorithm takes optimization

requirements and limits and limits the number

of options to be considered at each stage.

In Dynamic Programming, optimal decisions are

made on the principle of optimality. If the total

solution is optimal, then the solution part to the k-

stage must also be optimal. (Plancher, 2017).

In the principle of optimality, if you work from

stage k to stage k + 1 we can use the optimal results

from stage k without having to return to the initial

stage where the cost at stage k + 1 is the cost

generated in stage k plus the cost from stage k to

stage k + 1 as shown in Figure. 2.

Figure 2: Principle of Optimality (Munir, 2013).

Dynamic Programming has the following

characteristics:

1. The problem is divided into stages where only

one decision is taken per set.

2. Each stage consists of a corresponding state

and its state is a number of possible inputs at

that stage.

3. The solution of each stage is declared with a

status corresponding to the status of the next

step in the next step.

4. The cost at each stage increases regularly as

the number of stages increases.

5. The value at each stage depends on the current

value.

6. The solution at each stage does not depend on

the solution made in the previous stage.

7. The recursive relation marks the solution of

each state at stage k, which provides the best

solution for each state in the previous stage.

There are 2 (two) approaches in Dynamic

Programming, namely:

1. The forward or up-down approach,

2. Backward or bottom-up approach.

For example x1, x2,…, xn represent the decision

variables that must be made respectively for stages

1, 2, ..., n, then:

1. Dynamic Programming Forward moves from

stage 1, continues to stage 2, 3, and so on until

stage n where the sequence of decision

variables is x1, x2, ..., xn.

2. Dynamic Programming backward moves from

stage n, continues backward to stages n - 1, n -

2 and so on until stage 1. The set of decision

variables is xn, xn-1, ..., x1. stage k + 1 =

(costs generated in stage k) + (costs from

stage k to stage k + 1), k = 1, 2, ..., n - 1.

While the principle of optimality in backward

dynamic programming is the cost at stage k = (costs

generated in stage k + 1) + (costs from stage k + 1 to

stage k), k = n, n - 1, .., 1.

Following are the steps to develop the Dynamic

Programming algorithm:

a) Perform the structural characteristics of the

optimal solution,

b) Recursively determine the optimal solution

value,

c) Calculate the optimal solution value for forward

or backward,

d) Develop optimal solutions.

As for the principle, Dynamic Programming is

based on multi-stage graph. Each node in the graph

represents a state, while V1, V2, ... represent a stage.

Figure 3 shows a multi-stage process.

Figure 3: Multi-Stage Graph.

Several problems in the Multi-Layout Graph related

to Dynamic Programming are categorized as

follows:

Performance of A Star with Dynamic Programming Algorithms in Determining the Shortest Route

539

1. Stage (k) is the process of selecting the

destination node as shown in Figure 3; there

are five stages.

2. The status associated with each stage is a node

in the graph

1.3 Research Design

The research design is to analyze the Dynamic

Programming method - A Star to find the shortest

route. The research design is presented in Figure 4.

Figure 4: Research Design.

Initially the user enters the route as data input

and the initial process is carried out by the route

processing software. After the processing of the

route is complete, the search results performed using

the A Star algorithm produce the shortest route

distance. Combination with Dynamic Programming

algorithm is then performed.

1.4 Problem Identification

Based on the previous explanation, the problem lies

with the A Star algorithm which finds the shortest

possible route according to the standard rules

regardless of the whole problem. This algorithm

works thoroughly against all available alternatives.

So as to produce solutions and processing time for

relatively small problems, which are less than

optimal. Like the A Star algorithm in finding a

solution for each iteration, it forms two arrays: Open

and Close List as temporary data stores that require

memory resources.

2 RESEARCH METHOD

This study discusses the shortest route search

method with the Dynamic Programming algorithm

on the A Star algorithm on a dataset in the form of a

route between the Petisah market to the Lau Cih

market. There are several possible routes to the

intended destination, namely through the Captain

Patimura-Jamin Ginting road, through the Captain

Patimura road - Dr Mansyur-Setia Budi Street and

Captain Patimura Street - Jamin Ginting Street -

Harmonika-Setia Budi Street. Following are the

steps to analyze the shortest route search using

Dynamic Programming on the A Star algorithm:

1. Perform a search for the shortest distance

using the A Star algorithm.

2. Perform a search for the closest distance with

the A Star-Dynamic Programming algorithm.

3. Perform an analysis to evaluate which of the

two methods above gives the best results.



3 RESULTS AND DISCUSSIONS

3.1 Results

The optimal route search test results for each

algorithm can be seen in table 1, table 2 and table 3.

1. Testing Results with A Star Algorithm

Table 1: Test Results of A Star Algorithm.

o Route Distance (Km)

1 Route -1 15.2

2 Route -2 14

3 Route -3 15

4 Route -4 14

Average 14.57

2. Testing Algorithm with Dynamic

Programming.

Table 2: Results of Testing Dynamic Programming

Algorithms.

o Route Distance (Km)

1Route-1 14.7

2 Route -2 14

ut Routes

Preprocessing Route Data

Algorithm Analysis

A Star Method

Research Conclusions

Search The Nearest Routes

Dynamic Programming

CESIT 2020 - International Conference on Culture Heritage, Education, Sustainable Tourism, and Innovation Technologies

540

3 Route -3 14.2

4 Route -4 14

Average 14.22

3. Testing algorithm with Dynamic

Programming - A Star.

Table 3: Test Results of Dynamic Programming - A Star

Algorithm.

o Route Distance (Km)

1 Route -1 14.5

2 Route -2 13.3

3 Route -3 14.7

4 Route -4 14

Average 14.12

3.2 Discussions

In the Table 1, the search process for the shortest

distance from the separating market to the Lau Cih

market was carried out with four routes using the A

star algorithm, the test process for each route is

carried out 10 times so that the test results are

obtained with an average of 14.57 km. In Table 2,

the search process with Dynamic Programming

algorithm was also carried out by testing each route

10 times with the four routes so that an average of

14.22 km was obtained. Table 3, In the Dynamic

Programming - A Star algorithm, the testing process

is carried out on each route 10 times on the four

routes, so that an average of 14.12 km is obtained.

From the test results of each algorithm above

through the route through Pasar Petisah to Pasar Lau

Cih Medan with the testing process carried out 10

times on each route, it is found that Dynamic

Programming - A Star is the most optimal route with

an average distance of 14.12 Km. as shown in Table

Table 4: Route Optimum Distance of Each Algorithm.

No Algorithm Distance

(Km)

1 A Sta

14.57

2 D

namic Pro

rammin

14.22

3 Dynamic Programming - A

Sta

14.12

From Table 4, the optimal distance for each

algorithm can be plotted as in the graph in Figure 5.

Figure 5: Optimum Distance of Each Algorithm.

From the graph above, it can be seen that

Dynamic Programming - A Star algorithm is the

most optimal route with an average distance of 14.12

4 CONCLUSIONS

By using these three algorithms, it can be concluded

that the shortest route from Petisah market to Lau

Cih Medan market is as follows; the optimal route

distance between the two zones is 14.12 km. The

results of the three searches show that the closest

distance is almost the same, but not significantly

different. This is because the route you are looking

for is a road with low complications or a limited

number of nodes. The analysis of the three methods

above requires further research that needs to be done

by further researchers by adding more nodes and the

weight of the congestion and difficulties when

crossing the route.

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