Let $G = (V, E)$ be a graph with edge weights $w: E \rightarrow \mathbb{R} \cup \{\infty\}$. Let $P := \{(a_i, b_i, w_i)\}$ be a set of tuples of nodes $a_i, b_i \in V$ with shortest distance $w_i$ from $a_i$ to $b_i$.
We want to update the weights of a subset of edges $E' \subseteq E$, and then efficiently update the $w_i$ in $P$. We assume that $E'$ is small compared to $E$, but not constant-sized.
How do we update the set $P$? Precomputations of some data structure based on the initial $G$, $w$ and $P$ are allowed. Are there some existing publications on this or similar problems?
