Thesis for the degree Master of Science Mimicking Networks and Succinct Representations of Terminal Cuts
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چکیده
Given a large edge-weighted network G with k vertices designated as terminals, we wish to compress it and store, using little memory, the value of the minimum cut (or equivalently, maximum flow) between every bipartition of terminals. One appealing methodology to implement a compression of G is to construct a mimicking network : a small network G′ with the same k terminals, in which the minimum cut value between every bipartition of terminals is the same as in G. This notion was introduced by Hagerup, Katajainen, Nishimura, and Ragde [JCSS ’98], who proved that such G′ of size at most 2 k always exists. Obviously, by having access to the smaller network G′, certain computations involving cuts can be carried out much more efficiently. We provide several new bounds, which narrow the previously gap between known upper and lower bounds from doubly-exponential to only singly-exponential, both for planar and for general graphs. Our first and main result is that every k-terminal planar network admits a mimicking network G′ of size O(k2), which is moreover a minor of G. On the other hand, some planar networks G require |E(G′)| ≥ Ω(k). For general networks, we show that certain bipartite graphs only admit mimicking networks of size |V (G′)| ≥ 2, and moreover, every data structure that stores the minimum cut value between all bipartitions of the terminals must use 2 machine words, which proves that the trivial upper bound is tight.
منابع مشابه
Mimicking Networks and Succinct Representations of Terminal Cuts
Given a large edge-weighted network G with k terminal vertices, we wish to compress it and store, using little memory, the value of the minimum cut (or equivalently, maximum flow) between every bipartition of terminals. One appealing methodology to implement a compression of G is to construct a mimicking network : a small network G′ with the same k terminals, in which the minimum cut value betw...
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تاریخ انتشار 2013