## 2 Related Work  
To assign multiple controllers to control SFDs is a vital issue in a large-scale SDN to provide good network management, which has been well studied.  
First, many studies have been done on static controller assignment in pure SDN, which fails to consider the dynamic network environment, such as network traffic. The average latency and the worst latency of control messages were used to indicate the performance of the network in [15]. In [16], spectral clustering was used to partition a large network into several small domains, each of which is under the control of one or a group of SDN controllers. The K-Critical was proposed in [17] to decide the controller assignment based on a robust control tree. In [18], a controller assignment scheme was proposed based on community detection algorithms, which limited the spread of maximum number of label hops and maximum difference of nodes number between control domains to meet the load balancing and reliability requirements. In [19], a framework for Pareto-based optimal controller placement, was proposed to get optimal placements with respect to different performance metrics.  
However, these static assignment schemes between the switches and the controllers result in long and varied controller response times [20]. To solve this problem, several methods for dynamic SDN controller assignment were proposed, which considered different factors. Firstly, the average controller response time is one of the most important factors. The study in [21] proposed dynamic SDN controller assignment scheme based on the stable matching transfers to minimize the average response time. The work in [22] leveraged a learning automaton (LA) approach to solve the controller placement problem to minimize the overall propagation latency. Secondly, the cost of maintenance on the cluster of controllers is considered in some works. For example, [PERSON] et al. [23] tried to minimize the total cost caused by response time and maintenance on the cluster of controllers. Thirdly, the load balance of controllers can be considered in this problem, which can affect the latency of response. ElastiCon [24], an elastic distributed controller architecture, was proposed to address the load imbalance problem, in which the controller pool is dynamically grown or shrunk according to traffic conditions. Furthermore, some works considered the energy consumption of SDN controller. For example, [PERSON] et al. [25] proposed a controller assignment scheme to save the energy to switch off the maximum number of controller-switch links subject to the following constraints: delay, link and controller load. Besides, some works considered other factors, such as the recovery of failure and path selection. For example, the work in [26] considered the control-plane recovery performance to assign controllers to switches and choose the routing of controller-switch control paths. The study in [27] proposed a low-complexity master assignment (LCMA) algorithm for the controller problem to minimize the average flow setup latency with consideration of the number of assigned master controllers along a path.  
Considering the gradual migration of SDNs, hybrid SDNs are inevitable and important states, which have some special characteristics compared with pure SDNs and legacy networks. Most of previous studies on the topic of hybrid SDNs focused on traffic engineering and SDN migration schemes [28; 29]. Nevertheless, as one kind of SDNs, the controller assignment is also critical to hybrid SDNs, which has almost been ignored in the previous studies. A few studies noticed the importance of controllers in hybrid SDNs. In [30], the controller placement problem for hybrid SDN/legacy networks over a period of migration time is firstly introduced and formulated. However, the research on the controller assignment is still very scarce.