
Simple dynamic algorithms for Maximal Independent Set and other problems
We study three fundamental graph problems in the dynamic setting, namely...
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Conditional Lower Bounds for Variants of Dynamic LIS
In this note, we consider the complexity of maintaining the longest incr...
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Faster Dynamic Matrix Inverse for Faster LPs
Motivated by recent Linear Programming solvers, we design dynamic data s...
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CliqueBased Lower Bounds for Parsing TreeAdjoining Grammars
Treeadjoining grammars are a generalization of contextfree grammars th...
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Error correction in fast matrix multiplication and inverse
We present new algorithms to detect and correct errors in the product of...
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The I/O complexity of hybrid algorithms for square matrix multiplication
Asymptotically tight lower bounds are derived for the I/O complexity of ...
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Subcubic Certificates for CFL Reachability
Many problems in interprocedural program analysis can be modeled as the ...
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Dynamic Matrix Inverse: Improved Algorithms and Matching Conditional Lower Bounds
The dynamic matrix inverse problem is to maintain the inverse of a matrix undergoing element and column updates. It is the main subroutine behind the best algorithms for many dynamic problems whose complexity is not yet wellunderstood, such as maintaining the largest eigenvalue, rank and determinant of a matrix and maintaining reachability, distances, maximum matching size, and kpaths/cycles in a graph. Understanding the complexity of dynamic matrix inverse is a key to understand these problems. In this paper, we present (i) improved algorithms for dynamic matrix inverse and their extensions to some incremental/lookahead variants, and (ii) variants of the Online MatrixVector conjecture [Henzinger et al. STOC'15] that, if true, imply that these algorithms are tight. Our algorithms automatically lead to faster dynamic algorithms for the aforementioned problems, some of which are also tight under our conjectures, e.g. reachability and maximum matching size (closing the gaps for these two problems was in fact asked by Abboud and V. Williams [FOCS'14]). Prior best bounds for most of these problems date back to more than a decade ago [Sankowski FOCS'04, COCOON'05, SODA'07; Kavitha FSTTCS'08; Mucha and Sankowski Algorithmica'10; Bosek et al. FOCS'14]. Our improvements stem mostly from the ability to use fast matrix multiplication "one more time", to maintain a certain transformation matrix which could be maintained only combinatorially previously (i.e. without fast matrix multiplication). Oddly, unlike other dynamic problems where this approach, once successful, could be repeated several times ("bootstrapping"), our conjectures imply that this is not the case for dynamic matrix inverse and some related problems. However, when a small additional "lookahead" information is provided we can perform such repetition to drive the bounds down further.
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