## 1 Introduction  
Wireless sensor technologies have been used to collect data using inexpensive, battery-powered (hence, limited energy), tiny sensors. A wireless sensor network is comprised of thousands of sensor nodes, which are distributed in the harsh area and one or more sink nodes. Deployed sensors collect, process and distribute data through the network. Generally, when sensors are spread out, they start to sense the information and forward it to the sink node using multi-hop paths. Wireless sensor networks (WSNs) use broadcast, multicast, unicast and convergecast messages, just like other networks. The broadcast method is used to distribute a message from one node to all the other nodes in the networks. Multicast is used to distribute a message from one node to a group of nodes, and unicast is used to send a message from one node to another. Convergecast is the process in which every node sends a message to the sink node. In the convergecast process, a sink node gets the collected information from each sensor, and the collected information can be analyzed and integrated as useful information. One of the challenging problems is to reduce the number of transmissions to preserve the limited energy in WSNs. Thus, the convergecast problem is to find the minimum scheduling time to send a message from every node to the sink node. We are interested in the convergecast problem in WSNs, because most of the WSN applications use the many-to-one communication pattern; in other words, many or all sensors send data to the sink node. WSN applications require periodic data aggregation and quick delivery to conserve energy. This can be accomplished by minimizing the number of transmissions [1]. Fast and accurate data delivery also guarantees the improvement of the network performance [2].  
We are motivated to find the minimum scheduling time using the optimal solution of a linear topology by [PERSON] _et al._[3]. Finding the minimum scheduling time for convergecast is proven to be NP-hard in general graphs [4]. Other researchers have also studied the problem of finding the minimum number of time slots for a general topology in WSNs [5; 6; 7; 8; 9; 10; 11; 12], but they have barely succeeded, because of the high data rate originating from multiple hops to deliver a message to the sink node. However, [PERSON] _et al._[4] proved that the minimum scheduling time for the optimal transmission can be calculated if the deployed sensor nodes are organized in either line- or tree-based topologies. Using the optimal solution in linear topology guarantees the minimum time slots if we construct several minimum spanning trees from the general topology, even though it takes longer hops to reach the sink node. Therefore, in this paper, we study the problem of finding the minimum scheduling time for convergecast in WSNs by developing a heuristic algorithm that applies the optimum algorithm for line topology and the minimum weight spanning tree to general graphs.  
The convergecast problem has been widely researched over a couple of decades. We discuss more details in Section 2. The rest of the paper is organized as follows. In Section 3, we formulate the convergecast problem using graph theory, propose a heuristic algorithm in Section 4, show the result of the simulation in Section 5 and conclude in Section 6.