The Application of Minimum Spanning Tree Algorithm in the Water Supply Network

Abstract

The system of urban water supply network is the important lifeline project of the city. With the continuous development of social economy, people are no longer satisfied with water supply requirements, but to put forward higher requirements for the safety, reliability and economy of the water supply. Based on actual demands to solve the economic problems of water supply network to ensure the lowest costs in the laying the pipelines. First, establishing a mathematical model of water supply network, so we can use the knowledge of graph theory to solve this problem; from the above that, the minimum spanning tree was needed to establish to ensure that costs are the lowest in the case of pipeline connectivity. Then using the Kruskal algorithm to generate minimum spanning tree; finally, an example was analyzed to verify its practicality, and the algorithm solved the problem of water supply network in laying pipelines successfully. Introduction Water is the source of life, and is closely related to human survival. Water supply network is a water distribution system created by people, which is a vital part of water supply system. Water supply system in the order usually consists of water intake structures, water treatment structures, water supply pumping stations, adjustment structures, drainage pipes and water supply pipe network. The water supply network mainly refers to the urban water supply pipe network system, which is an important material base to protect the city people's life and develop production and construction [1]. Urban water supply pipe network system can be regarded as an important lifeline of urban engineering. The traditional view is that the water supply network's mission is to provide sufficient amount of water, the residents have enough water to use. However, with the continuous development of social economy, people put forward higher requirements for the safety, reliability and economy of the water supply. Specifically, in the process of water supply, the quality of supplied water is healthy or not, such as water pipe corrosion or other factors lead to water quality problems; after an earthquake or major disaster, the water supply network is reliable or not, urban water supply network can be normal without the occurrence of secondary disasters; pipe network not only to have above two characteristics, economic issues is the focus. In the case of the entire water supply network connectivity, we must to ensure the lowest costs that aim to produce hedge-fund-like returns at lower cost. So in recent years, research on the economic aspect of water supply network is increasingly attracted people's attention. Water supply system is an important infrastructure of the city, and it also is an important part of urban lifeline project, which plays an irreplaceable role in protecting economic development, ensuring social production and meeting human life [2]. The Establishment of the Mathematical Model for Water Supply Network In laying city network, the street interchanges must be considered because the pipelines must along the street to lay. In case of that water supply network can connect all users to make it with the lowest costs. To achieve this goal, you must consider how to select and handle these interchanges. This problem is a serious problem. The solution of this problem can provide a standard for the International Industrial Informatics and Computer Engineering Conference (IIICEC 2015) © 2015. The authors Published by Atlantis Press 52 laying of water supply network to ensure best design effect. According to graph theory, the water supply network can be viewed as a graph. So the contents of the water supply network need to be translated into the language of graph theory to help solve the problem of water supply network costs. The language from pipelines to graph is described as follows [3]: a) The water supply center and users in the planning area are referred to as nodes, the intersection of the street known as the intersections. The nodes and intersections are regarded as the vertices of graph. So the issue can be converted to the shortest path between each vertex, and each vertex must be connected indirectly or not indirectly. b) The routes that may be laying between nodes and intersections can be considered as edges of the graph. c) The sum of construction costs and operating costs of each line is regarded as the weights of edges. The sum of weights is the minimum that is the lowest costs, that is the purpose of the design you want to achieve. Through the above three steps, the water supply network can form a graph, this graph includes the vertices, edges and weights. Using G (V, E, W) to represent, V represents the set of vertices in the graph; E represents the set of edges in the graph; W represents the set of weights of each edge in the graph. Setting T is a spanning tree of diagram of G, then: W(T)=∑ Wuv euv∈T (1) Among them, W(T) is the sum of weights in the tree of T; euv is the any edge in the tree of T; Wuv is the weights of euv. The purpose of design is to require the minimum values of W(T). Only this way can ensure the lowest costs of laying pipelines. In summary, the problem of the minimum costs of the water supply network may be as a problem of seeking minimum spanning tree in the graph. The minimum spanning tree must exist. According to the actual situation, each node will certainly connected when laying water supply network, so there will be a minimum spanning tree certainly. There are a variety of algorithms to generate minimum spanning tree, such as Prim algorithm, Kruskal algorithm and simple algorithm and so on [4]. The Basic Concept of Algorithm Kruskal algorithm chooses the right edge according to the ascending order of weights to construct a minimum spanning tree. Kruskal algorithm, also known as avoidance circle method, starting from the shortest side, the edge attached to the tree does not form a loop, then the edge can be added to the tree, otherwise examine the next edge [5]. Specific steps are as follows: (1) Firstly, all vertices in connectivity network need to be added to minimum spanning tree to Start Arranging according to the ascending order of weights Setting d(vj)=min{d(u), d(v)}