Results 1 
8 of
8
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 ..."
Abstract

Cited by 26 (8 self)
 Add to MetaCart
(Show Context)
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 ..."
Abstract

Cited by 22 (4 self)
 Add to MetaCart
(Show Context)
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
Lecture 8: Undirected Connectivity is in logspace
, 2005
"... In the second half of today’s lecture, we will discuss a deterministic logspace algorithm for undirected connectivity, a recent and beautiful result due to Omer Reingold [Rei]. In fact, Reingold’s algorithm is one of the reasons this course is being offered this quarter. 8.1 Undirected ST Connectiv ..."
Abstract
 Add to MetaCart
In the second half of today’s lecture, we will discuss a deterministic logspace algorithm for undirected connectivity, a recent and beautiful result due to Omer Reingold [Rei]. In fact, Reingold’s algorithm is one of the reasons this course is being offered this quarter. 8.1 Undirected ST Connectivity The undirected st connectivity problem is the problem of finding if there exists a path between two specified vertices in a given undirected graph. More formally, the problem is as follows: Input: A undirected graph G = (V, E) and two vertices s, t ∈ V (s denotes source and t target). Problem: Are s and t connected? I.e., does there exist a path in G from the source s to the target t? USTCONN = {〈G, s, t 〉  G – undirected graph, s, t ∈ V (G); s and t are connected in G}. Clearly, any of the standard search algorithms (depthfirstsearch, breadthfirstsearch etc.) solve USTCONN in linear time. Thus, the time complexity of USTCONN is wellunderstood. What we would be interested in today’s lecture is the same complexity of
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 ..."
Abstract
 Add to MetaCart
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
1 Credits These are the lecture notes for the course CS369E: Expanders in Computer Science taught at
"... ..."