On Strong Diameter Padded Decompositions
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2019
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| _version_ | 1866917560501403648 |
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| author | Filtser, Arnold |
| author_facet | Filtser, Arnold |
| contents | Given a weighted graph $G=(V,E,w)$, a partition of $V$ is $Δ$-bounded if the diameter of each cluster is bounded by $Δ$. A distribution over $Δ$-bounded partitions is a $β$-padded decomposition if every ball of radius $γΔ$ is contained in a single cluster with probability at least $e^{-β\cdotγ}$. The weak diameter of a cluster $C$ is measured w.r.t. distances in $G$, while the strong diameter is measured w.r.t. distances in the induced graph $G[C]$. The decomposition is weak/strong according to the diameter guarantee.
Formerly, it was proven that $K_r$ minor free graphs admit weak decompositions with padding parameter $O(r)$, while for strong decompositions only $O(r^2)$ padding parameter was known. Furthermore, for the case of a graph $G$, for which the induced shortest path metric $d_G$ has doubling dimension $d$, a weak $O(d)$-padded decomposition was constructed, which is also known to be tight. For the case of strong diameter, nothing was known.
We construct strong $O(r)$-padded decompositions for $K_r$ minor free graphs, matching the state of the art for weak decompositions. Similarly, for graphs with doubling dimension $d$ we construct a strong $O(d)$-padded decomposition, which is also tight. We use this decomposition to construct strong $\left(O(d),\tilde{O}(d)\right)$ sparse cover scheme for such graphs. Our new decompositions and cover have implications to approximating unique games, the construction of light and sparse spanners, and for path reporting distance oracles. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1906_09783 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | On Strong Diameter Padded Decompositions Filtser, Arnold Data Structures and Algorithms Computational Geometry Given a weighted graph $G=(V,E,w)$, a partition of $V$ is $Δ$-bounded if the diameter of each cluster is bounded by $Δ$. A distribution over $Δ$-bounded partitions is a $β$-padded decomposition if every ball of radius $γΔ$ is contained in a single cluster with probability at least $e^{-β\cdotγ}$. The weak diameter of a cluster $C$ is measured w.r.t. distances in $G$, while the strong diameter is measured w.r.t. distances in the induced graph $G[C]$. The decomposition is weak/strong according to the diameter guarantee. Formerly, it was proven that $K_r$ minor free graphs admit weak decompositions with padding parameter $O(r)$, while for strong decompositions only $O(r^2)$ padding parameter was known. Furthermore, for the case of a graph $G$, for which the induced shortest path metric $d_G$ has doubling dimension $d$, a weak $O(d)$-padded decomposition was constructed, which is also known to be tight. For the case of strong diameter, nothing was known. We construct strong $O(r)$-padded decompositions for $K_r$ minor free graphs, matching the state of the art for weak decompositions. Similarly, for graphs with doubling dimension $d$ we construct a strong $O(d)$-padded decomposition, which is also tight. We use this decomposition to construct strong $\left(O(d),\tilde{O}(d)\right)$ sparse cover scheme for such graphs. Our new decompositions and cover have implications to approximating unique games, the construction of light and sparse spanners, and for path reporting distance oracles. |
| title | On Strong Diameter Padded Decompositions |
| topic | Data Structures and Algorithms Computational Geometry |
| url | https://arxiv.org/abs/1906.09783 |