Difference between revisions of "Open Problems:74"

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{{Header
 
{{Header
|title=Succinct Representation for Functions on Graphs
 
 
|source=baltimore16
 
|source=baltimore16
 
|who=Robert Krauthgamer
 
|who=Robert Krauthgamer
 
}}
 
}}
Given an undirected graph $G = (V,E_G)$ with weight function $w_G$, one may wish to design a data structure to store the value of the minimum $s-t$ cut, for any $s,t \in V$.
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Suppose we want to design a data structure that stores, for a given edge-weighted (undirected) graph $G = (V,E_G,w_G)$, the values of the minimum $st$-cuts for all $s,t \in V$.  
A naive method is to construct a table containing the value for each pair, requiring $O(|V|^2)$ space.
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A naive method is to construct a table containing the value for each pair, requiring $O(|V|^2)$ space (machine words).
Alternatively, one may construct a Gomory-Hu tree {{cite|GomoryHu-61}}.
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Alternatively, one may construct a Gomory–Hu tree {{cite|GomoryH-61}}.
This is an undirected tree $T = (V,E_T)$ with weight function $w_T$, such that the minimum $s-t$ cut values from $G$ are preserved.
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This is a tree $T = (V,E_T,w_T)$, in which the minimum $st$-cut values are equal to those in $G$.
 
Since $T$ is a tree, it requires only $O(|V|)$ space.
 
Since $T$ is a tree, it requires only $O(|V|)$ space.
  
For this problem, it turns out the most space-efficient data structure is a graph which preserves the value of the function we wish to query.
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Thus for this problem, a very space-efficient data structure (perhaps even the best one) is itself a graph $G'$, and it encodes the desired values in a natural manner, just compute the same function (min $st$-cut) on $G'$.  
But is this the case for all such functions on graphs, or is there a case where another (potentially complicated) data structure could out-perform a graph?
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But is this the case for all such functions on graphs, or is there a (natural) case where a potentially complicated data structure outperforms a graphical encoding? The question applies both to exact and approximate computations of the function.
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'''Some relevant examples:'''
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* A randomized data structure for $(1+\varepsilon)$-approximating any cut value (or Rayleigh quotient) in $G$, which is more space-efficient than the known graphical representation, can be found in {{cite|AndoniCKQWZ-16}}.
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* For a given graph $G$ with $k$ terminals, the exact values of all the minimum cuts between subsets of terminals can be stored simply as a list of $2^k$ numbers. The best graphical representation known is of size roughly $2^{2^k}$ {{cite|HagerupKNR-98|KhanR-14}}.
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* A deterministic data structure for $(1+\varepsilon)$-approximating any multicommodity flow on a set of $k$ given terminals in $G$. There is no known graphical representation for this, whose size depends on $k$ and $\varepsilon$, but not on $G$ {{cite|AndoniGK-14}}.

Latest revision as of 18:43, 18 January 2016

Suggested by Robert Krauthgamer
Source Baltimore 2016
Short link https://sublinear.info/74

Suppose we want to design a data structure that stores, for a given edge-weighted (undirected) graph $G = (V,E_G,w_G)$, the values of the minimum $st$-cuts for all $s,t \in V$. A naive method is to construct a table containing the value for each pair, requiring $O(|V|^2)$ space (machine words). Alternatively, one may construct a Gomory–Hu tree [GomoryH-61]. This is a tree $T = (V,E_T,w_T)$, in which the minimum $st$-cut values are equal to those in $G$. Since $T$ is a tree, it requires only $O(|V|)$ space.

Thus for this problem, a very space-efficient data structure (perhaps even the best one) is itself a graph $G'$, and it encodes the desired values in a natural manner, just compute the same function (min $st$-cut) on $G'$. But is this the case for all such functions on graphs, or is there a (natural) case where a potentially complicated data structure outperforms a graphical encoding? The question applies both to exact and approximate computations of the function.

Some relevant examples:

  • A randomized data structure for $(1+\varepsilon)$-approximating any cut value (or Rayleigh quotient) in $G$, which is more space-efficient than the known graphical representation, can be found in [AndoniCKQWZ-16].
  • For a given graph $G$ with $k$ terminals, the exact values of all the minimum cuts between subsets of terminals can be stored simply as a list of $2^k$ numbers. The best graphical representation known is of size roughly $2^{2^k}$ [HagerupKNR-98,KhanR-14].
  • A deterministic data structure for $(1+\varepsilon)$-approximating any multicommodity flow on a set of $k$ given terminals in $G$. There is no known graphical representation for this, whose size depends on $k$ and $\varepsilon$, but not on $G$ [AndoniGK-14].