In 1992, Dwork and Naor proposed that e-mail messages be accompanied by easy-to-check proofs of computational effort in order to discourage junk e-mail, now known as spam. They proposed specific CPU-bound functions for this purpose. Burrows suggested that, since memory access speeds vary across machines much less than do CPU speeds, memory-bound functions may behave more equitably than CPU-bound functions; this approach was first explored by Abadi, Burrows, Manasse, and Wobber [8].

Abstract: We give an exponential lower bound for the size of any linear-time Boolean branching program computing an explicitly given function. More precisely, we prove that for all positive integers k and for all sufficiently small ε> 0, if n is sufficiently large then there is no Boolean (or 2-way) branching program of size less than 2 εn which, for all inputs X ⊆ {0,1,...,n − 1}, computes in time kn the parity of the number of elements of the set of all pairs 〈x,y 〉 with the property x ∈ X, y ∈ X, x < y, x + y ∈ X. For the proof of this fact we show that if A = (ai, j) n i=0, j=0 is a random n by n matrix over the field with 2 elements with the condition that “A is constant on each minor diagonal,” then with high probability the rank of each δn by δn submatrix of A is at least cδ|logδ | −2n, where c> 0 is an absolute constant and n is sufficiently large with respect to δ.

We present several applications of quantum amplitude amplification to finding claws and collisions in ordered or unordered functions. Our algorithms generalize those of Brassard, Høyer, and Tapp, and imply an O(N 3/4 log N) quantum upper bound for the element distinctness problem in the comparison complexity model. This contrasts with Θ(N log N) classical complexity. We also prove a lower bound of Ω ( √ N) comparisons for this problem and derive bounds for a number of related problems. 1

In spite of extensive and continuing research, for various geometric search problems (such as nearest neighbor search), the best algorithms known have performance that degrades exponentially in the dimension. This phenomenon is sometimes called the curse of dimensionality. Recent results [38, 37, 40] show that in some sense it is possible to avoid the curse of dimensionality for the approximate nearest neighbor search problem. But must the exact nearest neighbor search problem suffer this curse? We provide some evidence in support of the curse. Specifically we investigate the exact nearest neighbor search problem and the related problem of exact partial match within the asymmetric communication model first used by Miltersen [43] to study data structure problems. We derive non-trivial asymptotic lower bounds for the exact problem that stand in contrast to known algorithms for approximate nearest neighbor search. 1

We obtain the first non-trivial time-space tradeoff lower bound for functions f : {0, 1}^n &rarr; {0, 1} on general branching programs by exhibiting a Boolean function f that requires exponential size to be computed by any branching program of length (1 + &epsilon;)n, for some constant &epsilon; > 0. We also give the first separation result between the syntactic and semantic read-k models [BRS93] for k > 1 by showing that polynomial-size semantic read-twice branching programs can compute functions that require exponential size on any syntactic read-k branching program. We also show...

The subject of this article is differential compression, the algorithmic task of finding common strings between versions of data and using them to encode one version compactly by describing it as a set of changes from its companion. A main goal of this work is to present new differencing algorithms that (i) operate at a fine granularity (the atomic unit of change), (ii) make no assumptions about the format or alignment of input data, and (iii) in practice use linear time, use constant space, and give good compression. We present new algorithms, which do not always compress optimally but use considerably less time or space than existing algorithms. One new algorithm runs in O(n) time and O(1) space in the worst case (where each unit of space contains n# bits), as compared to

We prove the first time-space lower bound tradeoffs for randomized computation of decision problems. The bounds hold even in the case that the computation is allowed to have arbitrary probability of error on a small fraction of inputs. Our techniques are an extension of those used by Ajtai [Ajt99a, Ajt99b] in his time-space tradeoffs for deterministic RAM algorithms computing element distinctness and for Boolean branching programs computing a natural quadratic form. Ajtai's bounds were of the following form...

We prove the first time-space lower bound tradeoffs for randomized computation of decision problems.

We extend recent techniques for time-space tradeoff lower bounds using multiparty communication complexity ideas. Using these arguments, for inputs from large domains we prove larger tradeoff lower bounds than previously known for general branching programs, yielding time lower bounds of the form T = n) when space S = n , up from T = n log n) for the best previous results. We also prove the first unrestricted separation of the power of general and oblivious branching programs by proving that 1GAP , which is trivial on general branching programs, has a time-space tradeoff of the form T = (n=S)) on oblivious Finally, using time-space tradeoffs for branching programs, we improve the lower bounds on query time of data structures for nearest neighbor problems in d dimensions from d= log n), proved in the cell-probe model [8, 5], to d) or log d= log log d) or even d log d) (depending on the metric space involved) in slightly less general but more reasonable data structure models.

We derive worst case upper bounds on the minimum distance of parallel concatenated convolutional codes, serial concatenated convolutional codes, repeat-and-accumulate codes, and various generalizations. We conclude from the bounds that in all these cases the relative minimum distance goes to zero as the block length tends to innity.