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IOSR Journal of Mathematics (IOSR-JM)

e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 12, Issue 4 Ver. V (Jul. - Aug.2016), PP 41-45

www.iosrjournals.org

DOI: 10.9790/5728-1204054145 www.iosrjournals.org 41 | Page

Minimum Spanning Tree of City to City Road Network in Nigeria

Effanga, E.O. & Edeke, Uwe. E.

Department Of Statistics, University Of Calabar, Nigeria

Abstract: This paper deals with the construction of minimum spanning tree of city – to – city network in Nigeria.

Nodes in our network are the 36 state capitals in Nigeria and the FCT, and the arcs are the proposed major roads

that link the state capitals. The distances between states are computed using the Lad/Long converter software which

makes use of the latitude and longitude of each state capital. We employed Prim’s algorithm to determine the

minimum spanning tree with Yenagoa as the starting point. The result gives a total distance of 5, 128.5 km.

Keywords: Spanning tree, Network, Prim’s algorithm, Latitude and Longitude

I. Introduction

A lot of researches have been done on designing optimal network for purposes of distribution,

communication, machine scheduling, gas and pipelines, etc. The tool commonly used for the optimal design of such

networks is the Minimal Spanning Tree. A Minimal Spanning Tree problem is one of the most fundamental and

intensively studied problems in network optimization problem with many theoretical and practical applications

(Ahuja, et al, 1993), (Taha, 2006), (Winston, 2004), (Dippon, 1999), (Seth, 2002), (Nahla, 2011), (Mares, 2008).

(Rothfard, 1970) used minimum spanning tree to design optimal offshore natural gas pipeline systems

A number of algorithms exist for the determination of Minimum Spanning Tree; these include Kruskal

algorithm, Prim’s algorithm and Boruvka algorithm, (Agarval, 2010). The Prim’s algorithm was developed in 1957

by a computer scientist, Robert C. Prim and rediscovered by Edsger Diijkstra in the same year. For this reason the

algorithm is sometimes known as DJP algorithm (Wilkipedia, 2010). An optimal design of natural gas pipeline of

Amaco East Cross field gas pipeline project in Alberta – Canada with a total distance of 66km was reduced to

49.9km by using hamster program software. Steiner minimum spanning tree algorithm was later used to reduce the

distance to 48.84km (Dott, 1997).

A study on the optimal design of oil pipeline network for the South Gabon oil field having 33 nodes and

129 possible arcs reduces the total distance of 188.2 miles to 156.2 miles using Prim’s algorithm (Brimberg et al,

2003). (Donkor et al, 2011) used Prim’s algorithm to determine the Minimum Spanning Tree of length 712km of the

West African gas pipeline from Nigeria through Benin and Togo to Ghana.

In this paper, we construct a Minimum Spanning Tree covering 36 state capitals and FCT, namely:

Yenagoa, Port – Harcourt, Oweri, Asaba, Awka, Umuhia, Uyo, Calabar, Ababkiliki, Enugu, Lokoja, Lafia, Makurdi,

Jalingo, Yola, Maiduguri, Damaturu, Gombe, Bauchi, Jos, Kaduna, Minna, Ilorin, Oshogbo, Adoekiti, Akure, Benin

city, Ikeja, Abeokuta, Ibadan, Berrin - Kebbi, Sokoto, Gusau, Katsina, Kano, Dutse and Abuja.

II. Methodology

City – to – City road Network in Nigeria

A network is a collection of points (nodes) linked by arcs (branches). Let N be a set of finite number of

nodes, and A be a set of arcs linking the nodes. Then a network is defined by the pair (N, A). In a network, the arc

linking two distinct nodes i and j is denoted by (i, j). In every network, there is a flow of some type along its arcs.

For instance, in road network, the flows are the vehicle; in communication network, the flows are the messages

along the wires; in pipeline network, the flows are oil products, etc. If flows are allowed in only one direction along

an arc, the arc is said to be directed or oriented. If all the arcs in a network are directed, then the network is called a

directed network, otherwise it is undirected (Hillier & Lieberman, 2001).

A sequence of arcs linking two distinct nodes forms a path in a network. A path forms a loop or cycle in a

network if it connects a node to itself. Two nodes in a network are connected if there is at least one path linking

them. When all the nodes in a network are linked by at least one path, the network is said to be connected (Taha,

2006).In this paper, a city – to – city road network in Nigeria is constructed first. Then Prim’s algorithm is employed

to obtain a minimum spanning tree. Figure 1 shows City – to – City road network in Nigeria, while Fig. 2 shows

minimum spanning tree of the network.

Tree A tree is any subset of a network not containing a loop. Given a set of nodes N, a tree can be grown by

linking any two nodes by an arc, and subsequently adding new arc in such a way that it links a node already linked

to other nodes to a new node not previously linked to any other node. When nodes are linked this way, the problem

of creating a chain is avoided and the number of nodes linked will be 1 greater than the number of arcs. (Taha, 2002)

A tree is spanning tree if all the nodes in a network are linked and are connected. Every spanning tree has exactly (m

– 1) arcs in a network of m nodes, since this is the minimum number of arcs needed to have a connected network

and the maximum number possible without having a chain.

The Minimal Spanning Tree problem

A spanning tree is a group of (m – 1) arcs that links all the m nodes of the network and contains no chain. A

spanning tree of minimum length in a network is a minimum spanning tree. The minimal spanning tree problem

found its applications in the creation of a network of paved roads that links several rural towns, where the road

between two towns may pass through one or more other towns.

The Minimal Spanning Tree algorithm (The Prim’s Algorithm)

The steps are as follows:

Step 0: set C = ø and

m} , 2, {1, C

Minimum Spanning Tree Of City To City Road Network In Nigeria

DOI: 10.9790/5728-1204054145 www.iosrjournals.org 42 | Page

Step 1: Start with any node, say node i, in

0

C

and connect it to node j that is closest to node i.

Set

j} {i,\m} , 2, {1, C and j} {i, C

.

Step 2: Now choose node l in C’ that is closest to some node k in C. Then connects node k to node l. Set C = {i, j,

k} and C’ = C’\C

Step 3: Repeat step 2 until C’ = ø. Ties for the closest node and arc to be included in the minimum spanning tree

may be broken arbitrarily. (Hillier and Lieberman, 2001).

Method of Data collection

The data used for this study is generated using Lad/Long converter software which makes use of latitude and

longitude of cities in Nigeria to determine the distances between cities. Table 1 shows latitude and longitude of each

state capital, while Table 2 shows city – to – city distances.

City-to-City Road Network in Nigeria.

Fig.1

Table 1: Latitudes and Longitudes of State capitals in Nigeria

S/N

STATE CAPITALS

LATITUDE

LONGITUDE

S/N

STATE CAPITALS

LATITUDE

LONGITUDE

1

Ikeja

6.5833oN

3.3333oE

23

Lafia

8.4917oN

8.5167oE

2

Abeokuta

7.1608oN

3.3483oE

24

Jos

9.9333oN

8.8833oE

3

Ibadan

7.3964oN

3.9167oE

25

Bauchi

10.5000oN

10.0000oE

4

Oshogbo

7.7660oN

4.5667oE

26

Gombe

10.2500oN

11.1667oE

5

Ado – Ekiti

7.6211oN

5.2214oE

27

Jalingo

8.9000oN

11.3667oE

6

Akure

7.2500oN

5.1950oE

28

Yola

9.2300oN

12.4600oE

7

Benin City

6.3176oN

5.6145oE

29

Maiduguri

11.8333oN

13.1500oE

8

Asaba

6.1978oN

6.7285oE

30

Damaturu

11.7444oN

11.9611oE

9

Awka

5.0000oN

7.8333oE

31

Dutse

11.7011oN

9.3419oE

10

Enugu

6.4527oN

7.5103oE

32

Kano

12.0000oN

8.5167oE

11

Abakaliki

6.3333oN

8.1000oE

33

Katsina

12.2500oN

7.5000oE

12

Umuahia

5.5333oN

7.4833oE

34

Gusau

12.1500oN

6.6667oE

13

Calabar

4.7500oN

8.3250oE

35

Sokoto

13.0667oN

5.2333oE

14

Uyo

5.0500oN

7.9333oE

36

Berrin – Kebbi

11.5000oN

4.0000oE

15

Owerri

5.4850oN

7.0350oE

37

Kaduna

10.5167oN

7.4333oE

16

Port – Harcourt

6.3176oN

7.0000oE

17

Yenagoa

4.7500oN

6.3333oE

18

Ilorin

8.5000oN

4.5500oE

19

Minna

9.6139oN

6.5569oE

20

Abuja

9.0667oN

7.4833oE

21

Lokoja

7.8167oN

6.7500oE

22

Makurdi

7.7306oN

8.5361oE

Minimum Spanning Tree Of City To City Road Network In Nigeria

DOI: 10.9790/5728-1204054145 www.iosrjournals.org 43 | Page

Table 2: City – to – City Road Network showing Distances in KM

CITY

YENAGOA

PH

OWERRI

ASABA

AWKA

UMUAHIA

UYO

CALABAR

ABAKALIKI

ENUGU

LOKOJA

ABUJA

LAFIA

MAKURDI

YENAGOA

80.3

136.6

PH

80.3

81.3

108.6

OWERRI

86.2

ASABA

180.7

AWKA

180.7

70.8

165.3

UMUAHIA

70.8

73.2

UYO

108.6

73.2

44.9

CALABAR

44.9

115.7

309.9

ABAKALIKI

115.7

85.5

ENUGU

165.3

85.5

85.5

173.2

181.6

LOKOJA

173.2

160.6

196.9

ABUJA

160.6

130.2

LAFIA

130.2

84.6

MAKURDI

309.9

181.6

196.9

84.6

JALINGO

337.3

YOLA

MAIDUGURY

DAMATURU

GOMBE

BAUCHI

JOS

165.2

KADUNA

161.2

MINNA

118.4

ILORIN

OSHOGBO

ADOEKITI

381.8

AKURE

BENIN CITY

123.8

208.4

IKEJA

ABEOKUTA

IBADAN

BERIN KEBI

SOKOTO

GUSAU

KATSINA

KANO

DUTSE

CITY

JALINGO

YOLA

MAIDUGURY

DAMATURU

GOMBE

BAUCHI

JOS

KADUNA

MINNA

ILORIN

OSHOGBO

ADOEKITI

AKURE

YENAGOA

PH

OWERRI

ASABA

AWKA

UMUAHIA

UYO

CALABAR

ABAKALIKI

ENUGU

LOKOJA

381.8

ABUJA

161.2

118.4

LAFIA

165.2

MAKURDI

337.3

JALINGO

125.5

YOLA

125.5

298.9

MAIDUGURY

298.9

129.7

DAMATURU

129.7

187.3

GOMBE

130.5

BAUCHI

130.5

137.4

JOS

137.4

171.3

KADUNA

171.3

138.7

MINNA

138.7

252.6

ILORIN

252.6

81.5

OSHOGBO

81.5

73.9

89.9

ADOEKITI

73.9

AKURE

89.9

203.8

BENIN CITY

113.5

IKEJA

ABEOKUTA

IBADAN

141.1

82.6

BERIN KEBI

349.2

338.8

SOKOTO

GUSAU

199.8

KATSINA

KANO

232.2

282.7

DUTSE

285

Minimum Spanning Tree Of City To City Road Network In Nigeria

DOI: 10.9790/5728-1204054145 www.iosrjournals.org 44 | Page

CITY

BENIN CITY

IKEJA

ABEOKUTA

IBADAN

BERIN KEBI

SOKOTO

GUSAU

KATSINA

KANO

DUTSE

YENAGOA

PH

OWERRI

ASABA

123.8

AWKA

UMUAHIA

UYO

CALABAR

ABAKALIKI

ENUGU

LOKOJA

208.4

ABUJA

LAFIA

MAKURDI

JALINGO

YOLA

MAIDUGURY

DAMATURU

285

GOMBE

BAUCHI

232.2

JOS

KADUNA

199.8

282.7

MINNA

349.2

ILORIN

141.1

338.8

OSHOGBO

82.6

ADOEKITI

AKURE

113.5

BENIN CITY

253.6

IKEJA

253.6

64.2

110.9

ABEOKUTA

67.9

64.2

IBADAN

110.9

67.9

BERIN KEBI

219.6

SOKOTO

219.6

185.8

361.9

GUSAU

185.8

91.2

KATSINA

361.9

91.2

113.9

209.3

KANO

113.9

95.6

DUTSE

209.3

95.6

III. Result

The result of applying the Prim’s algorithm to the network in figure 1 is as shown in figure 2. From the

Minimum Spanning Tree in fig. 2 a total distance of 5,128.5km is obtained.

Minimum Spanning Tree Network of city to City Distance in Nigeria.

Fig.2

IV. Conclusion

The optimal road network so constructed can be used for various purposes in Nigeria, for example,

designing of telecommunication network, transportation network, network of high – voltage electrical power

transmission lines, network of gas and petroleum pipelines, etc. Although most of these networks have been in

existence in Nigeria, the network design in this work would be needful in other applications yet to be considered.

Minimum Spanning Tree Of City To City Road Network In Nigeria

DOI: 10.9790/5728-1204054145 www.iosrjournals.org 45 | Page

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