*2.2. Dijkstra Algorithm*

The most classic single source shortest path algorithm is Dijkstra Algorithm [6], which was proposed in 1956 and became well-known three years later. Dijkstra Algorithm can calculate the shortest path from one node to all nodes, ye<sup>t</sup> not all equal-cost multi-path shortest path.

Many modified algorithms based on Dijkstra Algorithm are proposed in [7–14]. An improved algorithm of Dijkstra Algorithm was proposed by Li [7]. Under the concept of precursor node, Li exploited the initial shortest path set calculated by Dijkstra Algorithm, to calculate most but not all of the shortest paths. The authors of [8] improved Dijkstra Algorithm for solving three issues, such as the ineffective mechanism to digraph. In addition, the work [9] proposed some modifications on Dijkstra Algorithm and made the number of iterations less than the number of the nodes. Work [10] proposed an optimized algorithm based on Dijkstra Algorithm to optimize logistics route for the supply chain. On the other hand, the study [11] modified Dijkstra Algorithm and the modified algorithm is very of efficiency for public transport route planning. Work [12] used Dijkstra Algorithm towards shortest path computation in navigation systems for making sensible decision and time saving decisions. By the way, the study [13] improved Dijkstra Algorithm to find the

maximum load path. Work [14] introduced an improved Dijkstra Algorithm for analyzing the property of 2D grid map and increased significantly the speed of Dijkstra Algorithm.

Referring to their proposed algorithms, we also improve Dijkstra Algorithm and propose an improved equal-cost multi-path shortest paths algorithm (ECMP-Dijkstra), which can calculate all equal-cost shortest paths from one node to all nodes, thus developing a cost-effective shortest path optimization algorithm between three points (i.e., S, D and L) with low time and space in hybrid SDN.

#### *2.3. The Minimum Vertex Cover Problem (MVCP)*

The traditional algorithm to solve the minimum vertex cover algorithm (MVCP) is 2-approximation [33]. This algorithm can find the set of vertex cover which is no more than twice of the optimal vertex cover, and the time complexity of the algorithm is O (E+V). More importantly, the results obtained by this algorithm are different each time, and thus may be inaccurate and not approximate solution. However, this algorithm has its advantages: every time a vertex is selected, and all the edges connected by the vertex are deleted.

The authors in [20–22] made much contribution to MVCP in theory and applications. The authors in [23,24] proposed parameterized algorithms for MVCP, and applied them in biochemistry. Work [25] proposed an improved greedy algorithm for minimum vertex cover problem, and the algorithm used the concept of degree (i.e., the number of links connected by a node) to carry out an order of degree and to select the node with the largest degree to add to the minimum vertex cover set until the degree of all nodes is 0 (i.e., the vertexes in the minimum vertex cover set has covered all the edges). Thus, the result is a very excellent approximate solution. However, the process of judging the degree of the algorithm is too complicated. Authors in [26] presented a greedy heuristic algorithm for MVCP to offer better results on dense graphs. The study [27] presented a breadth first search approach, which can ge<sup>t</sup> the exact result of MVCP for grid graphs. Work [28] proposed a near-optimal algorithm named MAMA to optimize the unweighted MVCP, and MAMA can return near optimal result in quick-time. Authors in [29] proposed a NHGA for MVCP to yield near-optimal solutions. In [30], authors studied an ameliorated genetic algorithm for the partial VCP to skip the local optimum by powerful vertex and adaptive mutation. All of their algorithm are based on the concept of degree.

Combining with the advantages of the above algorithms, we proposed an ameliorated restrictive minimum vertex cover algorithm (RMVCA) in hybrid SDN using the concept of degree to significantly simplify the process of degree judgment and to yield near-optimal result, thus, in the whole interception system, realizing the optimization of the deployment efficiency of IAPs and the improvement of intercept link coverage.

#### **3. ECMP-Dijkstra Algorithm and SDN Interception Models**
