Prim 's Minimum Spanning Tree Research Articles

A Novel Method for Network Design and Optimization of District Energy Systems: Considering Network Topology Planning and Pipe Diameter

This paper proposes a new network topology design method that considers all the road nodes, energy stations and load centers to ensure the distribution of pipes along the road. The traditional graph theory and Prim Minimum Spanning Tree (MST) are used to simplify the map and minimize the length of the pipeline. After analyzing the limitations of the traditional network topology model, Point-to-Point (PTP), we present a new model, Energy Station-to-Load Point (ESLP). The model is optimized by minimum cost, not the shortest path. Finally, Pipe Diameter Grading (PDG) is proposed based on ESLP by solving for the pipe diameter that gives the minimum cost under different load demands in the process of optimization. The network design method is effectively applied in a case, and the results show that the path of the optimized plan is 1.88% longer than that of the pre-optimized plan, but the cost is 2.38% lower. The sensitivity analysis shows that the cost of pipeline construction, project life and electricity price all have an impact on the optimization results, and the cost of pipeline construction is the most significant. The difference between the different classifications of pipelines affects whether PDG is effective or not.

Board Game Supporting Learning Prim’s Algorithm and Dijkstra’s Algorithm

The concept of minimum spanning tree algorithms in data structure is difficult for students to learn and to imagine without practice. Usually, learners need to diagram the spanning trees with pen to realize how the minimum spanning tree algorithm works. In this paper, the authors introduce a competitive board game to motivate students to learn the concept of minimum spanning tree algorithms. They discuss the reasons why it is beneficial to combine graph theories and board game for the Dijkstra and Prim minimum spanning tree theories. In the experimental results, this paper demonstrates the board game and examines the learning feedback for the mentioned two graph theories. Advantages summarizing the benefits of combining the graph theories with board game are discussed.

Computation of Minimal Uniform Transmission Range in Ad Hoc Wireless Networks

Power conservation is a critical issue for ad hoc wireless networks. The main objective of the paper is to find the minimum uniform transmission range of an ad hoc wireless network, where each node uses the same transmission power, while maintaining network connectivity. Three different algorithms, Prim's Minimum Spanning Tree (MST), its extension with Fibonacci heap implementation, and an area-based binary search are developed to solve the problem. Their performance is compared by simulation study together with Kruskal's MST, a known solution proposed by Ramanathan and Rosales-Hain for topology control by transmission power adjustment, and an edge-based binary search used by the same study in order to find the per-node-minimality after Kruskal's algorithm is applied. Our results show that Prim's MST outperforms both Kruskal's MST and the two binary searches. The performance between Prim's MST implemented with binary heap and Fibonacci heap is fairly close, with the Fibonacci implementation slightly outperforming the other.

A Unified Algorithm for Designing Multidrop Teleprocessing Networks

The problem of designing minimum-cost multidrop lines which connect remote terminals to a concentrator or a central data-processing computer is studied. In some cases, optimal solutions can be obtained by using either linear integer programming or a branch-bound method. These approaches are not practical, since they lack flexibility and require an enormous amount of computer time for most practical problems. As a consequence, heuristic algorithms have been developed by various authors. In this paper, we point out that all of these algorithms fall into the class of minimum spanning tree (MST) problems, constrained by traffic or response time requirements. The difference between them is mainly the sequential order with which a branch or a line is selected into the tree. Without the constraints, all algorithms converge to a MST. With the constraints, they form different subtrees. Most of the algorithms can be unified into a modified Kruskal's MST algorithm. In the modified algorithm, a weight is associated with each terminal. Let w i be the weight associated with terminal i , and d_{i}{j} be the cost for the line directed from terminal i to terminal j . When the algorithm fetches the cost for the line, it replaces it with d_{i}{j} - w_{i} . In some cases, w i 's need to be readjusted in the middle of the algorithm. The difference between all existing heuristic algorithms is in the way w i 's are defined. If w i is zero for all i , the algorithm reduces to the unmodified Kruskal's algorithm; if w i is set to zero whenever a line incident to terminal i is selected as a tree branch, the algorithm reduces to Prim's MST algorithm. An extension of the algorithm to the solution of an associated problem of partitioning the terminals with respect to a predetermined set of concentrators, multiplexers, terminal interface processors, or central computers is also derived. The efficiency of an algorithm depends greatly on how it is implemented. The computational complexity of the unified algorithm is in the order of N^{2} log N for the most general case, where N is the number of terminals. By using good heuristics, it reduces to K_{1}N log N + K_{2}N , where K 1 and K 2 are constants, for many practical applications. The algorithm has been applied to large networks with over 1000 terminals, yielding excellent results and using only 15 seconds of computer time on a CDC 6600 computer. Designs obtained by using different w i 's are compared.