A generalized construction produces (2 floor(k/2)-1, 2 ceil(k/2) W1 + max(0, 2(ceil(k/2)-2)) W2)-emulators of size O~(n^{1+1/k}) that improve on prior spanners and emulators for small distances.
Compact routing schemes
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Weighted Emulators with Local Heaviest Edges Stretch for Undirected Graphs
A generalized construction produces (2 floor(k/2)-1, 2 ceil(k/2) W1 + max(0, 2(ceil(k/2)-2)) W2)-emulators of size O~(n^{1+1/k}) that improve on prior spanners and emulators for small distances.