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Undirected STConnectivity in LogSpace
, 2004
"... We present a deterministic, logspace algorithm that solves stconnectivity in undirected graphs. The previous bound on the space complexity of undirected stconnectivity was log 4/3 (·) obtained by Armoni, TaShma, Wigderson and Zhou [ATSWZ00]. As undirected stconnectivity is complete for the clas ..."
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Cited by 167 (3 self)
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We present a deterministic, logspace algorithm that solves stconnectivity in undirected graphs. The previous bound on the space complexity of undirected stconnectivity was log 4/3 (·) obtained by Armoni, TaShma, Wigderson and Zhou [ATSWZ00]. As undirected stconnectivity is complete for the class of problems solvable by symmetric, nondeterministic, logspace computations (the class SL), this algorithm implies that SL = L (where L is the class of problems solvable by deterministic logspace computations). Our algorithm also implies logspace constructible universaltraversal sequences for graphs with restricted labelling and logspace constructible universalexploration sequences for general graphs.
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 47 (5 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
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 31 (8 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].
Realizable paths and the NL vs L problem
 Electronic Colloquium on Computational Complexity (ECCC
"... A celebrated theorem of Savitch [Sav70] states that NSPACE(S) ⊆ DSPACE(S2). In particular, Savitch gave a deterministic algorithm to solve STCONNECTIVITY (an NLcomplete problem) using O(log2n) space, implying NL ⊆ DSPACE(log2n). While Savitch’s theorem itself has not been improved in the last fo ..."
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A celebrated theorem of Savitch [Sav70] states that NSPACE(S) ⊆ DSPACE(S2). In particular, Savitch gave a deterministic algorithm to solve STCONNECTIVITY (an NLcomplete problem) using O(log2n) space, implying NL ⊆ DSPACE(log2n). While Savitch’s theorem itself has not been improved in the last four decades, studying the space complexity of several special cases of STCONNECTIVITY has provided new insights into the spacebounded complexity classes. In this paper, we introduce new kind of graph connectivity problems which we call graph realizability problems. All of our graph realizability problems are generalizations of UNDIRECTED STCONNECTIVITY. STREALIZABILITY, the most general graph realizability problem, is LogCFLcomplete. We define the corresponding complexity classes that lie between L and LogCFL and study their relationships. As special cases of our graph realizability problems we define two natural problems, BALANCED STCONNECTIVITY and POSITIVE BALANCED STCONNECTIVITY, that lie between L and NL. We present a deterministic O(lognloglogn) space algorithm for BALANCED STCONNECTIVITY. More generally we prove that SGSLogCFL, a generalization of BALANCED STCONNECTIVITY, is contained in DSPACE(lognloglogn). To achieve this goal we generalize several concepts (such as graph squaring and transitive closure) and algorithms (such as parallel algorithms) known in the context of
1 Credits These are the lecture notes for the course CS369E: Expanders in Computer Science taught at
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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 ..."
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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 ..."
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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