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 or vector $F$ of $n$ symbols and then one symbol arrives at a time in a stream. After each symbol has ar
rived we must output some function of $F$ and the $n$-length suffix of the arriving stream. Cell probe bounds of $Omega(deltalg{n}/w)$ have previously been shown for both convolution and Hamming distance in this setting, where $delta$ is the size of a symbol in bits and $winOmega(lg{n})$ is the cell size in bits. However, when $delta$ is a constant, as it is in many natural situations, these previous results no longer give us non-trivial bounds. We introduce a new lop-sided information transfer proof technique which enables us to prove meaningful lower bounds even for constant size input alphabets. We use our new framework to prove an amortised cell probe lower bound of $Omega(lg^2 n/(wcdot lg lg n))$ time per arriving bit for an online version of a well studied problem known as pattern matching with address errors. This is the first non-trivial cell probe lower bound for any online problem on bit streams that still holds when the cell sizes are large. We also show the same bound for online convolution conditioned on a new combinatorial conjecture related to Toeplitz matrices.
We give tight cell-probe bounds for the time to compute convolution, multiplication and Hamming distance in a stream. The cell probe model is a particularly strong computational model and subsumes, for example, the popular word RAM model. We first
consider online convolution where the task is to output the inner product between a fixed $n$-dimensional vector and a vector of the $n$ most recent values from a stream. One symbol of the stream arrives at a time and the each output must be computed before the next symbols arrives. Next we show bounds for online multiplication where the stream consists of pairs of digits, one from each of two $n$ digit numbers that are to be multiplied. One pair arrives at a time and the task is to output a single new digit from the product before the next pair of digits arrives. Finally we look at the online Hamming distance problem where the Hamming distance is outputted instead of the inner product. For each of these three problems, we give a lower bound of $Omega(frac{delta}{w}log n)$ time on average per output, where $delta$ is the number of bits needed to represent an input symbol and $w$ is the cell or word size. We argue that these bound are in fact tight within the cell probe model.
We give cell-probe bounds for the computation of edit distance, Hamming distance, convolution and longest common subsequence in a stream. In this model, a fixed string of $n$ symbols is given and one $delta$-bit symbol arrives at a time in a stream.
After each symbol arrives, the distance between the fixed string and a suffix of most recent symbols of the stream is reported. The cell-probe model is perhaps the strongest model of computation for showing data structure lower bounds, subsuming in particular the popular word-RAM model. * We first give an $Omega((delta log n)/(w+loglog n))$ lower bound for the time to give each output for both online Hamming distance and convolution, where $w$ is the word size. This bound relies on a new encoding scheme and for the first time holds even when $w$ is as small as a single bit. * We then consider the online edit distance and longest common subsequence problems in the bit-probe model ($w=1$) with a constant sized input alphabet. We give a lower bound of $Omega(sqrt{log n}/(loglog n)^{3/2})$ which applies for both problems. This second set of results relies both on our new encoding scheme as well as a carefully constructed hard distribution. * Finally, for the online edit distance problem we show that there is an $O((log n)^2/w)$ upper bound in the cell-probe model. This bound gives a contrast to our new lower bound and also establishes an exponential gap between the known cell-probe and RAM model complexities.
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 Omega((d
/w)*log n) time on average per output, where d 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.
We investigate the problem of deterministic pattern matching in multiple streams. In this model, one symbol arrives at a time and is associated with one of s streaming texts. The task at each time step is to report if there is a new match between a f
ixed pattern of length m and a newly updated stream. As is usual in the streaming context, the goal is to use as little space as possible while still reporting matches quickly. We give almost matching upper and lower space bounds for three distinct pattern matching problems. For exact matching we show that the problem can be solved in constant time per arriving symbol and O(m+s) words of space. For the k-mismatch and k-difference problems we give O(k) time solutions that require O(m+ks) words of space. In all three cases we also give space lower bounds which show our methods are optimal up to a single logarithmic factor. Finally we set out a number of open problems related to this new model for pattern matching.
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. The proof is considerably more complex than that for exact matching.
