Given a sequence of integers, we want to find a longest increasing subsequence of the sequence. It is known that this problem can be solved in \(O\left (n \log n\right )\) time and space. Our goal in this paper is to reduce the space consumption while keeping the time complexity small. For \(\sqrt {n} \le s \le n\) , we present algorithms that use \(O\left (s \log n\right )\) bits and \(O\left (\frac {1}{s} \cdot n^{2} \cdot \log n\right )\) time for computing the length of a longest increasing subsequence, and \(O\left (\frac {1}{s} \cdot n^{2} \cdot \log ^{2} n\right )\) time for finding an actual subsequence. We also show that the time complexity of our algorithms is optimal up to polylogarithmic factors in the framework of sequential access algorithms with the prescribed amount of space.

An explicit study of min-wise independent permutation families, together with their variants --- k-restricted, approximate, etc. --- was initiated by Broder, et al[4]. In this paper, we give a lower bound for the size of k-restricted min-wise independent permutation family. A family F of permutations on [0,n-1]=(0,1,...,n-1) is said to be k-restricted min-wise independent if for any subset X ⊆ [0,n-1] with |X| ≤ k and any x ∈ X, Pr[min(π(X))=π(x)] = 1/|X|, when π is randomly chosen from F according to a probability distribution D on the family F. For the minimum size of a family of k-restricted min-wise independent permutations, upper bounds of O(nk) for any fixed k have been shown for uniform and biased probability distributions on F. We show that if a family F of permutations on [0,n-1] is k-restricted min-wise independent, then |F| ≥ m(n-1,k-1), where m(n,d) = ∑i=0d/2(ni) if d is even; m(n,d)= ∑i=0(d-1)/2(ni) + (n-1(d-1)/2) otherwise. The lower bound for the size of F still holds when we allow an arbitrary probability distribution on F. Our proof technique is based on linear algebra methods, and can be regarded as a generalization of the result by Alon, Babai, and Itai[1], i.e., if random variables X1,X2,...,Xn: Ω → (0,1) are k-wise independent and Pr[Xi=1] = pi is neither 0 nor 1, then |Ω| ≥ m(n,k). By applying our proof technique, we also derive lower bounds for the sample size of the related notions, e.g., k-wise symmetrically independent distributions, k-rankwise independent permutation families, etc.

We analyze upper and lower bounds on size of Boolean conjunctions necessary and sufficient to approximate a given DNF formula by accuracy slightly better than 1/2 (here we define the size of a Boolean conjunction as the number of distinct variables on which it depends). Such an analysis determines the performance of a naive search algorithm that exhausts Boolean conjunctions in the order of their sizes. In fact, our analysis does not depend on kinds of symmetric functions to be exhausted: instead of conjunctions, counting either disjunctions, parity functions, majority functions, or even general symmetric functions, derives the same learning results from similar analyses.

Let M(m,n) be the minimum number of comparators needed in a comparator network that merges m elements x 1 ≤ x 2 ≤ … ≤ x m and n elements y 1 ≤ y 2 ≤ … ≤ y m , where n ≥ m. Batcher's odd-even merge yields the following upper bound: M(m,n) ≤ ½(m + n)log 2 m + O(n); in particular, M(n,n) ≤ n log 2 n + o(n) We prove the following lower bound that matches the upper bound above asymptotically as n ≥ m →∞; M(m,n) ≥ ½(m+n)log 2 m - O(m) in particular, M(n,n) ≥ n log 2 - O(n). Our proof technique extends to give similarily tight lower bounds for the size of monotone Boolean circuits for merging, and for the size of switching networks capable of realizing the set of permutations that arise from merging.

Let M(m,n) be the minimum number of comparators needed in a comparator network that merges m elements x/sub 1/or=m. Batcher's odd-even merge yields the following upper bound: M(m,n)or=m to infinity :M(m,n)>or=/sup 1///sub 2/(m+n)log/sub 2/(m+1)-O(m); in particular, M(n,n)>or=nlog/sub 2/n-O(n). The authors' proof technique extends to give similarly tight lower bounds for the size of monotone Boolean circuits for merging, and for the size of switching networks capable of realizing the set of permutations that arise from merging.>

It has been shown by A. Yao (1990) that every language in ACC is recognized by a sequence of depth-2 probabilistic circuits with a symmetric gate at the root and n/sup polylog/(n) AND gates of fan-in polylog (n) at the leaves. The authors simplify Yao's proof and strengthen his results: every language in ACC is recognized by a sequence of depth-2 deterministic circuits with a symmetric gate at the root and n/sup polylog/(n) AND gates of fan-in polylog(n) at the leaves. They also analyze and improve modulus-amplifying polynomials constructed by S. Toda (1989) and Yao: this yields smaller circuits in Yao's and the present results on ACC.>