Finding the shortest paths by node combination

By repeatedly combining the source node's nearest neighbor, we propose a node combination (NC) method to implement the Dijkstra's algorithm. The NC algorithm finds the shortest paths with three simple iterative steps: find the nearest neighbor of the source node, combine that node with the source node, and modify the weights on edges that connect to the nearest neighbor. The NC algorithm is more comprehensible and convenient for programming as there is no need to maintain a set with the nodes' distances. Experimental evaluations on various networks reveal that the NC algorithm is as efficient as Dijkstra's algorithm. As the whole process of the NC algorithm can be implemented with vectors, we also show how to find the shortest paths on a weight matrix. } is the set of nodes, E = {e ij j if there is a link from v i to v j } is the set of edges and W = {w ij j 1 6 i, j 6 N} is the weight matrix for E. Given two nodes v s , v t of G, the shortest path problem can be defined as how to find a path with the minimum sum of the weights on the edges in a v s , v t-path. Generally, v s and v t are called source node and sink node, respectively. The shortest path problem is one of the most fundamental network optimization problems with widespread applications [1–4]. Among the various shortest path algorithms developed [5–12], Dijkstra's algorithm is probably the most well-known. It maintains a set S of solved nodes, comprising those nodes whose final shortest path distance from the source v s has determined, and labels d(i), storing the upper bound of the shortest path distance from v s to v i. The algorithm repeatedly selects the node v k 2 VnS with the minimum d(i), adds v k to S, and updates d(i) for nodes that are incident to v k Step 1. Select a node v k from Q such that dðv k Þ ¼ min v j 2Q dðv j Þ, if d(v k) = 1, stop, otherwise go to Step 2. Step 3. for every v j 2 Q, update d(v j) = min{d(v j), d(v k) + w kj }. Go to Step 1. In practice, Dijkstra's algorithm relies heavily on the strategies used to select the next minimum labeled …