Editing Open Problems:14
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− | {{ | + | {{DISPLAYTITLE:Problem 14: Graph Distances}} |
− | | | + | {{Infobox |
− | | | + | |label1 = Proposed by |
+ | |data1 = Andrew McGregor | ||
+ | |label2 = Source | ||
+ | |data2 = [[Workshops:Kanpur_2006|Kanpur 2006]] | ||
+ | |label3 = Short link | ||
+ | |data3 = http://sublinear.info/14 | ||
}} | }} | ||
Given a stream of edges defining a graph $G$, how well can we estimate $d_G(u,v)$, the length of the shortest path between two nodes $u$ and $v$? Progress that has been made on this problem is based on constructing ''spanners'' {{cite|FeigenbaumKMSZ-05|FeigenbaumKMSZ-05a|ElkinZ-06|Baswana-06|Elkin-06}} where subgraph $H$ of $G$ is an $(\alpha,\beta)$-spanner | Given a stream of edges defining a graph $G$, how well can we estimate $d_G(u,v)$, the length of the shortest path between two nodes $u$ and $v$? Progress that has been made on this problem is based on constructing ''spanners'' {{cite|FeigenbaumKMSZ-05|FeigenbaumKMSZ-05a|ElkinZ-06|Baswana-06|Elkin-06}} where subgraph $H$ of $G$ is an $(\alpha,\beta)$-spanner | ||
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Clearly, an $(\alpha,\beta)$-spanner gives an $\alpha+\beta/d_G(u,v)$ approximation to $d_G(u,v)$. | Clearly, an $(\alpha,\beta)$-spanner gives an $\alpha+\beta/d_G(u,v)$ approximation to $d_G(u,v)$. | ||
Since a spanner is constructed independently of $u$ and $v$ it is perhaps surprising that this approach gives nearly optimal results for approximating $d_G(u,v)$ in a single pass {{cite|FeigenbaumKMSZ-05}}. It is unclear whether there is a better approach for multiple pass algorithms. Clearly, $d_G(u,v)$ can be computed exactly in $d_G(u,v)$ passes but for $d_G(u,v)$ large this is infeasible. Can we do better? For example, how well can $d_G(u,v)$ be approximated in $O(\log n)$ passes? What if the edges arrived in random order? | Since a spanner is constructed independently of $u$ and $v$ it is perhaps surprising that this approach gives nearly optimal results for approximating $d_G(u,v)$ in a single pass {{cite|FeigenbaumKMSZ-05}}. It is unclear whether there is a better approach for multiple pass algorithms. Clearly, $d_G(u,v)$ can be computed exactly in $d_G(u,v)$ passes but for $d_G(u,v)$ large this is infeasible. Can we do better? For example, how well can $d_G(u,v)$ be approximated in $O(\log n)$ passes? What if the edges arrived in random order? | ||
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