## 2 Related Work  
The number of flow entries in an SDN switch is limited for several reasons. Firstly, there are limitations of hardware. Flow entries are stored in a switch, which is commonly implemented with Ternary Content Addressable Memory (TCAM) technology [14]. However, TCAM is power-hungry, expensive, and available in limited space or capacity that can only accommodate between 750 and 20,000 flow entries [15]. Secondly, the number of flow entries can affect the performance of a switch. Managing and querying flow entries require computational resources. Having too many flow entries can lead to increased processing time and querying time, thereby affecting the speed of packet forwarding. Additionally, when events such as topology changes, network reconfigurations [16], and the creation of rerouting rules [17] occur, the routing update operation must be completed within 25 milliseconds to meet the stringent QoS requirements of real-time applications [18]. Fast routing updates rely on the number of flow entries. Due to these reasons, network designers need to configure flow entries reasonably based on actual needs and hardware capabilities. There are some studies on multi-topology construction methods. They can be divided into three categories. The first category aims at fast recovery from link/node failures. [PERSON] [2] proposed an approach called FAMTAR. It uses the optimal path between two endpoints in the physical topology and automatically finds new paths in case of congestion by setting the cost of the congested link to a high value, forming logical topologies. [PERSON] [3] described a network design principle that emphasized redundancy and resilience through multi-topology routing. This approach ensures that even if one link or node fails, there remains at least one available route between any two nodes. Similarly, [PERSON] [4] proposed a method based on multi-topology routing for pre-planning for geographically correlated failures. It constructs logical topologies with a network-wide coverage approach. [PERSON] [5] implemented an IP fast reroute method based on multi-topology routing for SDN. It constructs a multi-topology on the data plane of SDN and relies on the existing failure detection mechanisms of the controller. These methods do not take into account the distribution of data flows across different logical topologies, which makes them fail to support simultaneous transmission across multiple paths. Another category focuses on traffic engineering, which aims to maximize link utilization or maximize throughput. [PERSON] [6] proposed a method called AMPLE. It computes the link weights in different sub-topologies to maximize intra-domain path diversity across a multi-topology construction. Based on AMPLE, [PERSON] [7] introduced a QoS-aware cost function for topology state monitoring. Additionally, the method establishes a policy-based routing scheme to select the optimal logical topology, ensuring QoS for new users while enhancing overall network performance. [PERSON] [8] constructed a multi-topology based on minimum set cover to enhance path diversity and introduced novel offline optimization features using a dynamic cost function. [PERSON] [9] proposed a multi-topology construction method that removes each link from exactly one logical topology, continuing until either all links are isolated within a single topology or the removal of a link would disconnect the topology. To prevent congestion, the algorithm aims to balance the number of links removed from each logical topology. Although these methods construct multi-topologies with good path diversity and reasonably distribute data flows across different logical topologies, they fail to ensure balanced bandwidth from the source node to each destination node. The third category aims to provide different QoS levels to various flows.