An \alpha-additive spanner of an undirected graph G =(V, E)  is a subgraph H such that the distance between any two vertices in G is stretched by no more than an additive factor of \alpha. It is previously known that unweighted graphshave 2-, 4-, and 6-additive spanners containing (n^{3/2), }%DIFDELCMD < O%%% (n^{7/5), and O(n^{4/3}) edges, respectively. In this paper, we generalize these results to weighted graphs. We consider \alpha=}   2W , 4W , 6W, where W is the maximum edge weight in G. We first show that every n-node graph has a subsetwise 2W-spanner on O(n |S|^{1/2) edges where S \subseteq V and W is a constant. We then show that for every set P with |P| = p vertex demand pairs, there are pairwise 2W- and 4W-spanners on O(np^{1/3}) and O(np^{2/7}) edges respectively. We also show that for every set P, there is a 6W-spanner on O(np^{1/4}) edges where W is a constant. We then show that every graph has additive 2W- and 4W-spanners on O(n^{3/2}) and O(n^{7/5}) edges respectively. Finally, we show that every graph has an additive 6W-spanner on O(n^{4/3}) edges where W is a constant}  .
Aspanner of a graph G is a subgraph H ), }%DIFDELCMD < O%%% ), and O(n^{4/3}) edges, respectively. In this paper, we generalize these results to weighted graphs. We consider \alpha=} that approximately preserves its pairwise distances. Spanners are commonly applied to compress computation on metric spaces corresponding to weighted input graphs. Classic spanner constructions can seamlessly handle edge weights, so long as error is measuredmultiplicatively. In this work, we investigate whether one can similarly extend constructions of spanners withadditive error. These extensions are not immediate, due to a key lemma about the size of shortest path neighborhoods that fails for weighted graphs. Despite this, we recover a suitable amortized version, which lets us prove direct extensions of classic +2 and +4 unweighted spanners (both all-pairs and pairwise) to + 2W and + 4W weighted spanners, where W is the maximum edge weight ) edges where S \subseteq V and W is a constant. We then show that for every set P with |P| = p vertex demand pairs, there are pairwise 2W- and 4W-spanners on O(np^{1/3}) and O(np^{2/7}) edges respectively. We also show that for every set P, there is a 6W-spanner on O(np^{1/4}) edges where W is a constant. We then show that every graph has additive 2W- and 4W-spanners on O(n^{3/2}) and O(n^{7/5}) edges respectively. Finally, we show that every graph has an additive 6W-spanner on O(n^{4/3}) edges where W is a constant} . For a technical reason, the +6 unweighted spanner becomes a +8W weighted spanner; closing this error gap is an interesting remaining open problem .
R
An \alpha-additive spanner of an undirected graph G =(V, E)
A
A
spanner
A
of a graph G
D
such that the distance between any two vertices in G is stretched by no more than an additive factor of \alpha. It is previously known that unweighted graphshave 2-, 4-, and 6-additive spanners containing
D
(n^{3/2
D
(n^{7/5
A
that approximately preserves its pairwise distances. Spanners are commonly applied to compress computation on metric spaces corresponding to weighted input graphs. Classic spanner constructions can seamlessly handle edge weights, so long as error is measured
A
multiplicatively
A
. In this work, we investigate whether one can similarly extend constructions of spanners with
A
additive
A
error. These extensions are not immediate, due to a key lemma about the size of shortest path neighborhoods that fails for weighted graphs. Despite this, we recover a suitable amortized version, which lets us prove direct extensions of classic +2 and +4 unweighted spanners (both all-pairs and pairwise) to +
R
,
and +
R
, 6W,
weighted spanners,
D
in G. We first show that every n-node graph has a subsetwise 2W-spanner on O(n |S|^{1/2
A
. For a technical reason, the +6 unweighted spanner becomes a +8W weighted spanner; closing this error gap is an interesting remaining open problem