  In this paper we address the problem of computing a sparse subgraph of a
weighted directed graph such that the exact distances from a designated source
vertex to all other vertices are preserved under bounded weight increment.
Finding a small sized subgraph that preserves distances between any pair of
vertices is a well studied problem. Since in the real world any network is
prone to failures, it is natural to study the fault tolerant version of the
above problem. Unfortunately, it turns out that there may not always exist such
a sparse subgraph even under single edge failure [Demetrescu \emph{et al.}
'08]. However in real applications it is not always the case that a link (edge)
in a network becomes completely faulty. Instead, it can happen that some links
become more congested which can easily be captured by increasing weight on the
corresponding edges. Thus it makes sense to try to construct a sparse distance
preserving subgraph under the above weight increment model. To the best of our
knowledge this problem has not been studied so far. In this paper we show that
given any weighted directed graph with $n$ vertices and a source vertex, one
can construct a subgraph that contains at most $e \cdot (k-1)!2^kn$ many edges
such that it preserves distances between the source and all other vertices as
long as the total weight increment is bounded by $k$ and we are allowed to have
only integer valued (can be negative) weight on each edge and also weight of an
edge can only be increased by some positive integer. Next we show a lower bound
of $c\cdot 2^kn$, for some constant $c \ge 5/4$, on the size of the subgraph.
We also argue that restriction of integer valued weight and integer valued
weight increment are actually essential by showing that if we remove any one of
these two restrictions we may need to store $\Omega(n^2)$ edges to preserve
distances.
