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Research Article Impact Factor: 4.226 ISSN: 2319-507X

Diptee Warhade, IJPRET, 2015; Volume 3 (9): 938-945 IJPRET

Available Online at www.ijpret.com

938

INTERNATIONAL JOURNAL OF PURE AND

APPLIED RESEARCH IN ENGINEERING AND

TECHNOLOGY A PATH FOR HORIZING YOUR INNOVATIVE WORK

A STUDY ON SHORTEST PATH ALGORITHM BASED ON DIJKSTRA’S AND BELLMAN

FORD ALGORITHM

DIPTEE WARHADE, DR. A. S. ALVIDepartment Of Computer Science and Engineering, Prof. Ram Meghe Institute of Technology and Research, Amravati (India)

Accepted Date: 05/03/2015; Published Date: 01/05/2015

Abstract: The shortest path is an important research topic in graph theory. The shortest path

algorithm in the graph theory includes both the one between the particular node pair and

the one among all the nodes. Aiming at the shortcomings of traditional algorithm, this paper

has proposed an optimization method which has mainly improved the nodes selection of the

shortest path and data storage structure and organization. Through comparison and

analysis, the improved algorithm has been obtained, which has reduced the storage space,

improved the operational efficiency and has a better applicability in the shortest path

calculation.

Keywords: Bellman Ford, Dijkstra’s Algorithm

Corresponding Author: MISS. DIPTEE WARHADE

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INTRODUCTION

The shortest path is an important research topic in graph theory. The shortest path algorithm in

the graph theory includes both the one between the particular node pair and the one among all

the nodes [1]. The former can be used to rationalize the transportation decision-makinganalysis, that Dijkstra algorithm is a classical algorithm for solving such problems, while the

latter is for the selection of a reasonable distribution center. A well-known shortest path

algorithm is Dijkstra's, also called "label algorithm". We need to solve many shortest path

problems [2].

For example, in the process of production, to complete production tasks quickly and with

efficacy, we should find the shortest path to complete each production task; in the process of

management, to make large gains with minimal cost, we should develop rational plans; in the

existing transport network, to transport large quantity of goods with minimal costs, we should

arrange for reasonable transport path. All these questions can be summed up as the “shortest

path problem” [3].

The Bellman-Ford algorithm uses relaxation to find single source shortest paths on directed

graphs. And it is also contain negative edges. The algorithm will also detect if there are any

negative weight cycles (such that there is no solution). If talking about distances on a map,

there is no any negative distance. The basic structure of bellman-ford algorithm is similar to

Dijkstra algorithm. It relaxes all the edges, and does this |V| - 1 time, where |V| is the number

of vertices in the graph [2]. The cost of a path is the sum of edge weights in the path [4].

I.

Literature Review

A.

Description of Algorithm:

Dijkstra’s algorithm: Identify a shortest distance from the source point to the other target

nodes, and then use the path length to find the shortest path iteratively, that is, the basic idea

of Dijkstra algorithm. However, with the development of the computer, the scale of the

problems is increasing continuously, and meanwhile the use of traditional Dijkstra has

increased the space and time complexity [1]. Before going into details of the pseudo-code of

the algorithm it is important to know how the algorithm works. Dijkstra’s algorithm works by

solving the sub problem k, which compute the shortest path from source to vertices among the

k closest vertices to the source [5].

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Dijkstra (G,s)

For each vertex v in graph G

{

D[s] = 0

D[v] = ∞

}

Initialize visited vertices S in graph G

S = null

Initialize Queue Q as set of all nodes in graph G

Q = all vertices V

While Q ≠ ∅

{

u=mind (Graph G, distance d)

Visited vertices S = S+u

for each vertex v in neighbor[u]

{

If d[v] > d[u] + w(u,v)

Then d[v] = d[u] + w (u,v)

}

Return d

}[4]

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1) Why It Doesn’t Work for Negative-Weight Edges

Dijkstra’s algorithm is based on the greedy method. It adds vertices by increasing distance.

If a node with a negative incident edge were to be added late to the cloud, it could mess up

distances for vertices already in the cloud.

Bellman –Ford algorithm: It Works even with negative-weight edges must assume directed

edges (for otherwise we would have negative-weight cycles).Negative edge weights are found

in various applications of graphs, hence the usefulness of this algorithm. If a graph contains a

"negative cycle" (i.e. a cycle whose edges sum to a negative value) that is reachable from the

source, then there is no cheapest path: any path can be made cheaper by one more walk

around the negative cycle. In such a case, the Bellman –Ford algorithm can detect negative

cycles and report their existence.

Initialize graph

Bellman-ford (vertices V, edges E, source vertex srv)

For each v in V

{

If v = srvthan d[v] =0

Else d[v] = infinite

Cost[v] = null

}

Relaxation of edges

For i form 1 to IVI-1

{

For each [u,v] with w in E

{

If d[u]+w

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{

If d[v]=d[u]+w

Cost[v] =u

}

}

}[4]

Checking negative cycle

For each e(u ,v) with weight winE

{

If d[u]+w,d[v]

{

Print “graph has negative cycle”

}

}[4]

II.

Comparisons Between Dijkstra's and Bellman Fords Algorithm

Both of these functions solve the single source shortest path problem. The primary difference in

the function of the two algorithms is that Dijkstra's algorithm cannont handle negative edge

weights. Bellman-Ford's algorithm can handle some edges with negative weight. It must be

remembered, however, that if there is a negative cycle there is no shortest path.

For Single source shortest paths:

Dijkstra Algorithm - No negative weight allowed - O(E+Vlg(V))

Bellman ford Algorithm - Negative weight is allowed. But if a negative cycle is present Bellman

ford will detect the -ve cycle - O(VE)

Directed Acyclic Graph - as name suggests it works only for DAG - O(V+E)

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All pairs shortest paths:

Dijkstra Algorithm - No negative weight allowed - O(VE + V^2lg(V))

Bellman ford Algorithm - O(V^2E)

Matrix chain multiplication method -complexity same as Bellman ford algorithm.

III.

Application of Algorithm

1) Application of Bellman Ford’s Algorithm

A distributed variant of the Bellman Ford algorithm is used in distance vector

Routing protocols.

Saving network resource

Routing Information Protocol (RIP)

Two metric routing problem

2) Application of Dijkstra’s Algorithm

Robot path planning.

Logistics Distribution Lines.

Link-state routing protocols.

OSPF(Open Shortest Path First).

IS-IS (Intermediate System to Intermediate System).

IV. Disadvantage of Algorithm

1) Bellman Ford’s Algorithm

The main disadvantage of the Bellman –Ford algorithm in RIP is that it doesn’t take weightings

into consideration. Another disadvantage of the Bellman -Ford algorithm is due to the slow

updates passed from one RIP router to the next. This results in a slow response to changes in

the network topology, which in turn results in more attempts to use routes that are down,

which wastes time and network resources.

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2) Dijkstra’s Algorithm

The major disadvantage of the algorithm is the fact that it does a blind search there by

consuming a lot of time waste of necessary resources.

It cannot handle negative edges. This leads to acyclic graphs and most often cannot obtain the

right shortest path.

V. Similarity and Dissimilarity of Dijkstra Algorithm and Bellman Ford Algorithm

Bellman Ford and Dijkstra both algorithms are used to find the shortest path of a network

where the value of the shortest path may vary according to the situation of the network. We

can solve cost and n length problems using this both algorithm. Dijkstra algorithm is faster than

bellman ford algorithm. Bellman Ford algorithm is not suggested for larger networks. And in

GIS, there are lots of data so we cannot suggest this algorithm. Bellman ford algorithm work in

negative weighted graph and also detect negative cycle [4].

VI.

CONCLUSIONS

In this paper, after studying these two algorithms we concluded that Dijkstra’s algorithm is

faster and better than Bellman Ford algorithm. Thus, Bellman –Ford is usually used only when

there are negative edge weights. The small difference in these two algorithms is that Dijkstra's

algorithm cannot handle negative edge weights. Bellman-Ford's algorithm can handle some

edges with negative weight. It must be remembered, however, that if there is a negative cycle

there is no shortest path.

REFERENCES

1. Jinhao Lu and Chi Dong, Research of Shortest Path Algorithm Based on the Data Structure,

3rd International Conference on Software Engineering and Service Science (ICSESS), 2012 IEEE,

22-24 June 2012, pp. 108 - 110

2.

Y. Cao, the shortest path algorithm in data structures, Journal of Yibin University, Vol. 6,

2007, pp.82 -84.

3.

Q. Sun, J. H. Shen, J. Z. Gu, An improved Dijkstra algorithm, [J]. Computer Engineering andApplications, vol. 3, 2002, pp.99-101.

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4.

Thippeswamy. K, Hanumanthappa. J, and Dr. Manjaiah D. H., A Study on Contrast and

Comparison between Bellman-Ford algorithm and Dijkstra’s algorithm In: National Conference

on wireless Networks-09(NCOWN-2010), Dec 2014.

5.

Goyal Abhishek, Mogha Prateek, Luthra Rishabh, and Ms. Sangwan Neeti, Path finding: A* orDijkstra's?, IJITE Vol.02 Issue-01, (January, 2014) ISSN: 2321 –1776, pp. 1 –15

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