# Tight Cell-Probe Bounds for Online Hamming Distance Computation

@inproceedings{Clifford2013TightCB,
title={Tight Cell-Probe Bounds for Online Hamming Distance Computation},
author={Rapha{\"e}l Clifford and Markus Jalsenius and Benjamin Sach},
booktitle={SODA},
year={2013}
}
• Published in SODA 8 July 2012
• Computer Science
We show tight bounds for online Hamming distance computation in the cell-probe model with word size w. The task is to output the Hamming distance between a fixed string of length n and the last n symbols of a stream. We give a lower bound of Ω(δ/w log n) time on average per output, where δ is the number of bits needed to represent an input symbol. We argue that this bound is tight within the model. The lower bound holds under randomisation and amortisation.
7 Citations

## Figures from this paper

### Cell-probe bounds for online edit distance and other pattern matching problems

• Computer Science
SODA
• 2015
Cell-probe bounds for the computation of edit distance, Hamming distance, convolution and longest common subsequence in a stream are given and an exponential gap between the known cell- Probe and RAM model complexities is established.

### Cell-Probe Lower Bounds for Bit Stream Computation

• Computer Science, Mathematics
ESA
• 2016
A lop-sided information transfer proof technique which enables us to prove meaningful lower bounds even for constant size input alphabets and 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.

### Time Bounds for Streaming Problems

• Computer Science, Mathematics
Theory Comput.
• 2019
Tight cell-probe bounds for the time to compute convolution, multiplication and Hamming distance in a stream are given within the cell probe model, a particularly strong computational model that subsumes the popular word RAM model.

### The complexity of computation in bit streams

• Computer Science, Mathematics
ArXiv
• 2015
This work uses a new lop-sided information transfer proof technique which enables them to prove meaningful lower bounds even for constant size input alphabets and proves an amortised cell probe lower bound of $\Omega(\lg^2 n/(w\cdot \lg \ lg n))$ time per arriving bit for an online version of a well studied problem known as pattern matching with address errors.

### Time Lower Bounds for Nonadaptive Turnstile Streaming Algorithms

• Computer Science
STOC
• 2015
This work proves the first non-trivial update time lower bounds for both randomized and deterministic turnstile streaming algorithms, which hold when the algorithms are non-adaptive.

### New Unconditional Hardness Results for Dynamic and Online Problems

• Computer Science, Mathematics
2015 IEEE 56th Annual Symposium on Foundations of Computer Science
• 2015
Improved unconditional lower bounds for matrix-vector multiplication and a version of dynamic set disjointness known as Patrascu's Multiphase Problem are given by studying the cell probe complexity of two conjectured to be hard problems of particular importance.

### Cell-probe lower bounds for dynamic problems via a new communication model

A new communication model is developed to prove a data structure lower bound for the dynamic interval union problem, and the sparse set disjointness protocol of Håstad and Wigderson is used to speed up a reduction from a new kind of nondeterministic communication games, for which lower bounds are proved.

## References

SHOWING 1-10 OF 28 REFERENCES

### Better Gap-Hamming Lower Bounds via Better Round Elimination

• Computer Science
APPROX-RANDOM
• 2010
This paper shows that every k-round bounded-error communication protocol for the Gap Hamming Distance problem sends a message of at least Ω(n/(k2 log k) bits, which implies strong space lower bounds on algorithms for a number of data stream computations, such as approximating the number of distinct elements in a stream.

### An optimal lower bound on the communication complexity of gap-hamming-distance

• Computer Science, Mathematics
STOC '11
• 2011
We prove an optimal Ω(n) lower bound on the randomized communication complexity of the much-studied Gap-Hamming-Distance problem. As a consequence, we obtain essentially optimal multi-pass space

### The One-Way Communication Complexity of Hamming Distance

• Computer Science, Mathematics
Theory Comput.
• 2008
This note gives a simple proof of a linear lower bound for the randomized one-way communication complexity of the Hamming distance problem using a simple reduction from the indexing problem and avoids the VC-dimension arguments used in the previous paper.

### Lower Bounds for Online Integer Multiplication and Convolution in the Cell-Probe Model

• Mathematics, Computer Science
ICALP
• 2011
Time lower bounds for both online integer multiplication and convolution in the cell-probe model with word size w are shown and an Ω(δ/w log n) lower bound for the time required per new number in the stream is shown.

### The cell probe complexity of dynamic range counting

This paper develops a new technique for proving lower bounds on the update time and query time of dynamic data structures in the cell probe model and proves the highest lower bound to date for any explicit problem, namely a lower bound of tq=Ω((lg n/lg(wtu))2).

### Logarithmic Lower Bounds in the Cell-Probe Model

• Computer Science, Mathematics
SIAM J. Comput.
• 2006
A new technique for proving cell-probe lower bounds on dynamic data structures is developed, which enables an amortized randomized $\Omega(\lg n)$ lower bound per operation for several data structural problems on $n$ elements, including partial sums, dynamic connectivity among disjoint paths, and several other dynamic graph problems (by simple reductions).

### Constructions of binary constant-weight cyclic codes and cyclically permutable codes

• Computer Science
IEEE Trans. Inf. Theory
• 1992
It is shown that cyclically permutable codes provide a natural solution to the problem of constructing protocol-sequence sets for the M-active-out-of-T-users collision channel without feedback.

### The cell probe complexity of dynamic data structures

• Computer Science
STOC '89
• 1989
New lower and upper bounds on the time per operation are proved to implement solutions to some familiar dynamic data structure problems including list representation, subset ranking, partial sums, and the set union problem.