A common solution, and the one I did, is to:

• sort by W

• subtract B from A

• for each node, find the distance between it and the next, and between its right and left neighbour, and store this in two sorted lists

• have a dsu that stores int min_odd, min_even, min_internal; storing the min value of A at odd positions, even positions, and nodes that can be skipped

• iterating through values of E in increasing:

• for every distance of the first type with value less than E, unite(), maintining the min_*

• for every distance of the second type, min the node's value into min_internal of its component

the answers are now sum(min(min_even,min_internal)), which can be maintained through the operations, totaling O(nlogn) for sorting and O(n a(n)) for dsu.