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Minimum Spanning Tree of City to City Road Network in Nigeria

DOI:10.9790/5728-1204054145
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Effanga Effanga
Effanga Effanga
Effanga Effanga
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Uwe Edeke at University of Calabar
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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
STATE CAPITALS
LATITUDE
LONGITUDE
1
Ikeja
6.5833oN
3.3333oE
Lafia
8.4917oN
8.5167oE
2
Abeokuta
7.1608oN
3.3483oE
Jos
9.9333oN
8.8833oE
3
Ibadan
7.3964oN
3.9167oE
Bauchi
10.5000oN
10.0000oE
4
Oshogbo
7.7660oN
4.5667oE
Gombe
10.2500oN
11.1667oE
5
Ado – Ekiti
7.6211oN
5.2214oE
Jalingo
8.9000oN
11.3667oE
6
Akure
7.2500oN
5.1950oE
Yola
9.2300oN
12.4600oE
7
Benin City
6.3176oN
5.6145oE
Maiduguri
11.8333oN
13.1500oE
8
Asaba
6.1978oN
6.7285oE
Damaturu
11.7444oN
11.9611oE
9
Awka
5.0000oN
7.8333oE
Dutse
11.7011oN
9.3419oE
10
Enugu
6.4527oN
7.5103oE
Kano
12.0000oN
8.5167oE
11
Abakaliki
6.3333oN
8.1000oE
Katsina
12.2500oN
7.5000oE
12
Umuahia
5.5333oN
7.4833oE
Gusau
12.1500oN
6.6667oE
13
Calabar
4.7500oN
8.3250oE
Sokoto
13.0667oN
5.2333oE
14
Uyo
5.0500oN
7.9333oE
Berrin – Kebbi
11.5000oN
4.0000oE
15
Owerri
5.4850oN
7.0350oE
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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... Minimum Spanning Tree A tree is a subset of a network not containing loops. Minimum spanning tree is the tree which connects all nodes of the network with a minimum length/travel time of links (Effanga and Edeke, 2016). It gives the most feasible tree for an economy when considering cost of construction & maintenance (Kumar et al., 2014). ...
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Dott,D.R.(1997).OptimalNetworkDesignforNaturalGaPipelines.UniversityofCalgary,Alberta.http://dspace.ucalgary.ca/bitstream/1880/26 992/1/31387Dott.pdf. [7]. Mares, M. (2008). The Saga of Minimum Spanning Trees. Department of Applied Mathematics, Charles University, Czech Republic. http://www.sciencedirect.com. [8]. Nahla, M. A. I. (2011). Optimal Spanning Trees. African Institute for Mathematical Science.http://www/users.aims.ac.za/nahla/nahla.pdf [9].
The cost of local Telecommunication Network. A comparison of Minimum Spanning Tree and the HAI model. National Economic Research Associates, Enthropy Estimation Workshop Optimal connection for theWest African Gas Pipeline Project
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Dippon, M. C. and Kenneth, E. T. (1999). The cost of local Telecommunication Network. A comparison of Minimum Spanning Tree and the HAI model. National Economic Research Associates, Enthropy Estimation Workshop. [5]. Donkor, E. K., Amponsah, S. K. and Darkwa, F. K. (2010). Optimal connection for theWest African Gas Pipeline Project. http://Maxellsci.com/print/rjaset/v3-67-73pdf. [6].