About “Improved Dijkstra Algorithm” Paper

Question

I want to ask a few questions about the algorithm in this paper.

https://pdfs.semanticscholar.org/9208/65aee7b072c054afa628ba413adda6f3e962.pdf

You may wonder why I don't ask the authors directly, actually I have, but unfortunately, I've got no response and I couldn't find direct emails to the authors.

You can see the core formula (8) of the algorithm.

• d(n) = weight value from starting point to current node
• r(n) = constraint function
• ɷ = weighted value denoting the impact factor
• θn = angle between vector that consists of nodes from starting point to current node1 and vector that consist of nodes from starting point to end point[2]

The steps said that to obtain the best nodes, all subsequent nodes vp (open list) will be compared to satisfy the constraint function above.

What I don't understand is:

1. Regarding the θn vectors, I don't really understand says, we have a graph with nodes 1,2,3,4,5,6,7,8,9,10. Suppose we start at node 1 and end at node 10, currently we have traversed node 1,2,4,5 (current node= 5). So for 1 it's supposed to be closed list (List of traversed optimal node) and the vector is v= {1,2,4,5}. Then, for [2], it's supposed to be v= {1,2,4,5,..,10}. How is that possible? For what I know, to find vector angle both vectors must have the same amount of nodes and it's weird since nodes between 5 and 10 haven't traversed yet, so it's not possible to use it. So, from my thinking, the vectors is supposed to be v1 = {1,5} and v[2] = {1,10}, but I wonder if that's even right.
2. About the step 3 of the algorithm, I wonder about the choosing of the best node? I can't understand the meaning of satisfying here since I can't find what condition does the compared nodes need, so it could satisfy the constraint function. Is D(n) must be the smallest or biggest among the compared list or what?

I hope anyone could answer this, and I hope this reach the author too, thanks.