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The Complexity of Planarity Testing
, 2000
"... We clarify the computational complexity of planarity testing, by showing that planarity testing is hard for L, and lies in SL. This nearly settles the question, since it is widely conjectured that L = SL [25]. The upper bound of SL matches the lower bound of L in the context of (nonuniform) circ ..."
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Cited by 23 (7 self)
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We clarify the computational complexity of planarity testing, by showing that planarity testing is hard for L, and lies in SL. This nearly settles the question, since it is widely conjectured that L = SL [25]. The upper bound of SL matches the lower bound of L in the context of (nonuniform) circuit complexity, since L/poly is equal to SL/poly. Similarly, we show that a planar embedding, when one exists, can be found in FL SL . Previously, these problems were known to reside in the complexity class AC 1 , via a O(log n) time CRCW PRAM algorithm [22], although planarity checking for degreethree graphs had been shown to be in SL [23, 20].
Estimating PageRank on Graph Streams
"... This study focuses on computations on large graphs (e.g., the webgraph) where the edges of the graph are presented as a stream. The objective in the streaming model is to use small amount of memory (preferably sublinear in the number of nodes n) and a few passes. In the streaming model, we show ho ..."
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Cited by 21 (4 self)
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This study focuses on computations on large graphs (e.g., the webgraph) where the edges of the graph are presented as a stream. The objective in the streaming model is to use small amount of memory (preferably sublinear in the number of nodes n) and a few passes. In the streaming model, we show how to perform several graph computations including estimating the probability distribution after a random walk of length l, mixing time, and the conductance. We estimate the mixing time M of a random walk in Õ(nα+Mα √ q q Mn M n+) space and Õ( α α) passes. Furthermore, the relation between mixing time and conductance gives us an estimate for the conductance of the graph. By applying our algorithm for computing probability distribution on the webgraph, we can estimate the PageRank p of any node up to an additive error of √ ɛp in Õ( q M α) passes and Õ(min(nα + 1
Notes on SpaceBounded Complexity
, 2009
"... of the syllabus. The problem called PATH in Siper’s book is called STCONN here. L.T. ..."
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of the syllabus. The problem called PATH in Siper’s book is called STCONN here. L.T.
The Computational Complexity of Randomness
, 2013
"... This dissertation explores the multifaceted interplay between efficient computation andprobability distributions. We organize the aspects of this interplay according to whether the randomness occurs primarily at the level of the problem or the level of the algorithm, and orthogonally according to wh ..."
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This dissertation explores the multifaceted interplay between efficient computation andprobability distributions. We organize the aspects of this interplay according to whether the randomness occurs primarily at the level of the problem or the level of the algorithm, and orthogonally according to whether the output is random or the input is random. Part I concerns settings where the problem’s output is random. A sampling problem associates to each input x a probability distribution D(x), and the goal is to output a sample from D(x) (or at least get statistically close) when given x. Although sampling algorithms are fundamental tools in statistical physics, combinatorial optimization, and cryptography, and algorithms for a wide variety of sampling problems have been discovered, there has been comparatively little research viewing sampling throughthelens ofcomputational complexity. We contribute to the understanding of the power and limitations of efficient sampling by proving a time hierarchy theorem which shows, roughly, that “a little more time gives a lot more power to sampling algorithms.” Part II concerns settings where the algorithm’s output is random. Even when the specificationofacomputational problem involves no randomness, onecanstill consider randomized