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Nutan Limaye

Publications and source records attributed to Nutan Limaye.

23 records · Page 2Linked to original sources
arXiv2012

Streaming algorithms for recognizing nearly well-parenthesized expressions

We study the streaming complexity of the membership problem of 1-turn-Dyck2 and Dyck2 when there are a few errors in the input string. 1-turn-Dyck2 with errors: We prove that there exists a randomized one-pass algorithm that given x checks whether there exists a string x' in 1-turn-Dyck2 such that x is obtained by flipping at most $k$ locations of x' using: - O(k log n) space, O(k log n) randomness, and poly(k log n) time per item and with error at most 1/poly(n). - O(k^{1+epsilon} + log n) space for every 0 <= epsilon <= 1, O(log n) randomness, O(polylog(n) + poly(k)) time per item, with error at most 1/8. Here, we also prove that any randomized one-pass algorithm that makes error at most k/n requires at least Omega(k log(n/k)) space to accept strings which are exactly k-away from strings in 1-turn-Dyck2 and to reject strings which are exactly (k+2)-away from strings in 1-turn-Dyck2. Since 1-turn-Dyck2 and the Hamming Distance problem are closely related we also obtain new upper and lower bounds for this problem. Dyck2 with errors: We prove that there exists a randomized one-pass algorithm that given x checks whether there exists a string x' in Dyck2 such that x is obtained from x' by changing (in some restricted manner) at most k positions using: - O(k log n + sqrt(n log n)) space, O(k log n) randomness, poly(k log n) time per element and with error at most 1/poly(n). - O(k^(1+epsilon)+ sqrt(n log n)) space for every 0 <= epsilon <= 1, O(log n) randomness, O(polylog(n) + poly(k)) time per element, with error at most 1/8.

cs.CC↗
arXiv2011

Streaming algorithms for language recognition problems

We study the complexity of the following problems in the streaming model. Membership testing for \DLIN We show that every language in \DLIN\ can be recognised by a randomized one-pass $O(\log n)$ space algorithm with inverse polynomial one-sided error, and by a deterministic p-pass $O(n/p)$ space algorithm. We show that these algorithms are optimal. Membership testing for \LL$(k)$ For languages generated by \LL$(k)$ grammars with a bound of $r$ on the number of nonterminals at any stage in the left-most derivation, we show that membership can be tested by a randomized one-pass $O(r\log n)$ space algorithm with inverse polynomial (in $n$) one-sided error. Membership testing for \DCFL We show that randomized algorithms as efficient as the ones described above for \DLIN\ and $\LL(k)$ (which are subclasses of \DCFL) cannot exist for all of \DCFL: there is a language in \VPL\ (a subclass of \DCFL) for which any randomized p-pass algorithm with error bounded by $ε< 1/2$ must use $Ω(n/p)$ space. Degree sequence problem We study the problem of determining, given a sequence $d_1, d_2,..., d_n$ and a graph $G$, whether the degree sequence of $G$ is precisely $d_1, d_2,..., d_n$. We give a randomized one-pass $O(\log n)$ space algorithm with inverse polynomial one-sided error probability. We show that our algorithms are optimal. Our randomized algorithms are based on the recent work of Magniez et al. \cite{MMN09}; our lower bounds are obtained by considering related communication complexity problems.

cs.DS↗
arXiv2009

A Log-space Algorithm for Canonization of Planar Graphs

Graph Isomorphism is the prime example of a computational problem with a wide difference between the best known lower and upper bounds on its complexity. We bridge this gap for a natural and important special case, planar graph isomorphism, by presenting an upper bound that matches the known logspace hardness [Lindell'92]. In fact, we show the formally stronger result that planar graph canonization is in logspace. This improves the previously known upper bound of AC1 [MillerReif'91]. Our algorithm first constructs the biconnected component tree of a connected planar graph and then refines each biconnected component into a triconnected component tree. The next step is to logspace reduce the biconnected planar graph isomorphism and canonization problems to those for 3-connected planar graphs, which are known to be in logspace by [DattaLimayeNimbhorkar'08]. This is achieved by using the above decomposition, and by making significant modifications to Lindell's algorithm for tree canonization, along with changes in the space complexity analysis. The reduction from the connected case to the biconnected case requires further new ideas, including a non-trivial case analysis and a group theoretic lemma to bound the number of automorphisms of a colored 3-connected planar graph. This lemma is crucial for the reduction to work in logspace.

cs.CC↗
arXiv2008

Longest paths in Planar DAGs in Unambiguous Logspace

We show via two different algorithms that finding the length of the longest path in planar directed acyclic graph (DAG) is in unambiguous logspace UL, and also in the complement class co-UL. The result extends to toroidal DAGs as well.

cs.CC↗