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**Published in:** LIPIcs, Volume 57, 24th Annual European Symposium on Algorithms (ESA 2016)

We revisit the complexity of online computation in the cell probe model. We consider a class of problems where we are first given a fixed pattern F of n symbols and then one symbol arrives at a time in a stream. After each symbol has arrived we must output some function of F and the n-length suffix of the arriving stream. Cell probe bounds of Omega(delta lg n/w) have previously been shown for both convolution and Hamming distance in this setting, where delta is the size of a symbol in bits and w in Omega(lg n) is the cell size in bits. However, when delta is a constant, as it is in many natural situations, the existing approaches no longer give us non-trivial bounds.
We introduce a lop-sided information transfer proof technique which enables us to prove meaningful lower bounds even for constant size input alphabets. Our new framework is capable of proving amortised cell probe lower bounds of Omega(lg^2 n/(w lg lg n)) time per arriving bit. We demonstrate this technique by showing a new lower bound for a problem known as pattern matching with address errors or the L_2-rearrangement distance problem. This gives the first non-trivial cell probe lower bound for any online problem on bit streams that still holds when the cell size is large.

Raphaël Clifford, Markus Jalsenius, and Benjamin Sach. Cell-Probe Lower Bounds for Bit Stream Computation. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 31:1-31:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{clifford_et_al:LIPIcs.ESA.2016.31, author = {Clifford, Rapha\"{e}l and Jalsenius, Markus and Sach, Benjamin}, title = {{Cell-Probe Lower Bounds for Bit Stream Computation}}, booktitle = {24th Annual European Symposium on Algorithms (ESA 2016)}, pages = {31:1--31:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-015-6}, ISSN = {1868-8969}, year = {2016}, volume = {57}, editor = {Sankowski, Piotr and Zaroliagis, Christos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.31}, URN = {urn:nbn:de:0030-drops-63827}, doi = {10.4230/LIPIcs.ESA.2016.31}, annote = {Keywords: Cell-probe lower bounds, algorithms, data streaming} }

Document

**Published in:** LIPIcs, Volume 20, 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)

We study the problem of parameterized matching in a stream where we want to output matches between a pattern of length m and the last m symbols of the stream before the next symbol arrives. Parameterized matching is a natural generalisation of exact matching where an arbitrary one-to-one relabelling of pattern symbols is allowed. We show how this problem can be solved in constant time per arriving stream symbol and sublinear, near optimal space with high probability. Our results are surprising and important: it has been shown that almost no streaming pattern matching problems can be solved (not even randomised) in less than Theta(m) space, with exact matching as the only known problem to have a sublinear, near optimal space solution. Here we demonstrate that a similar sublinear, near optimal space solution is achievable for an even more challenging problem.

Markus Jalsenius, Benny Porat, and Benjamin Sach. Parameterized Matching in the Streaming Model. In 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 20, pp. 400-411, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{jalsenius_et_al:LIPIcs.STACS.2013.400, author = {Jalsenius, Markus and Porat, Benny and Sach, Benjamin}, title = {{Parameterized Matching in the Streaming Model}}, booktitle = {30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)}, pages = {400--411}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-50-7}, ISSN = {1868-8969}, year = {2013}, volume = {20}, editor = {Portier, Natacha and Wilke, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2013.400}, URN = {urn:nbn:de:0030-drops-39513}, doi = {10.4230/LIPIcs.STACS.2013.400}, annote = {Keywords: Pattern matching, streaming algorithms, randomized algorithms} }

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**Published in:** LIPIcs, Volume 5, 27th International Symposium on Theoretical Aspects of Computer Science (2010)

The degree of a CSP instance is the maximum number of times that a variable may appear in the scope of constraints. We consider the approximate counting problem for Boolean CSPs with bounded-degree instances, for constraint languages containing the two unary constant relations $\{0\}$ and $\{1\}$. When the maximum degree is at least $25$ we obtain a complete classification of the complexity of this problem. It is exactly solvable in polynomial-time if every relation in the constraint language is affine. It is equivalent to the problem of approximately counting independent sets in bipartite graphs if every relation can be expressed as conjunctions of $\{0\}$, $\{1\}$ and binary implication. Otherwise, there is no FPRAS unless $\NPtime = \RPtime$. For lower degree bounds, additional cases arise in which the complexity is related to the complexity of approximately counting independent sets in hypergraphs.

Martin Dyer, Leslie Ann Goldberg, Markus Jalsenius, and David Richerby. The Complexity of Approximating Bounded-Degree Boolean #CSP. In 27th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 5, pp. 323-334, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{dyer_et_al:LIPIcs.STACS.2010.2466, author = {Dyer, Martin and Goldberg, Leslie Ann and Jalsenius, Markus and Richerby, David}, title = {{The Complexity of Approximating Bounded-Degree Boolean #CSP}}, booktitle = {27th International Symposium on Theoretical Aspects of Computer Science}, pages = {323--334}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-16-3}, ISSN = {1868-8969}, year = {2010}, volume = {5}, editor = {Marion, Jean-Yves and Schwentick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2010.2466}, URN = {urn:nbn:de:0030-drops-24669}, doi = {10.4230/LIPIcs.STACS.2010.2466}, annote = {Keywords: Boolean constraint satisfaction problem, generalized satisfiability, counting, approximation algorithms} }

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