A static dictionary is a data structure for storing subsets of a finite universe U , so that membership queries can be answered efficiently. We study this problem in a unit cost RAM model with word size Ω(log |U |), and show that for n-element subsets, constant worst case query time can be obtained using B +O(log log |U |) + o(n) bits of storage, where B = dlog2 (|U | n ) e is the minimum numbe...

For integers 0 6 t 6 k 6 v − t, let X be a v-set, and let Wtk(v) be a ( v t ) × ( v k ) inclusion matrix where rows and columns are indexed by t-subsets and k-subsets of X, respectively, and for row T and column K, Wtk(v)(T,K) = 1 if T ⊆ K and zero otherwise. Since Wtk(v) is a full rank matrix, by reordering the columns of Wtk(v) we can write Wtk(v) = (S|N), where N denotes a set of independent...

We study the large scale geometry of the mapping class group, MCG. Our main result is that for any asymptotic cone of MCG, the maximal dimension of locally compact subsets coincides with the maximal rank of free abelian subgroups of MCG. An application is a proof of Brock-Farb’s Rank Conjecture which asserts that MCG has quasi-flats of dimension N if and only if it has a rank N free abelian sub...

A variant of the classical knapsack problem is considered in which each item associated with an integer weight and a qualitative level. We define dominance relation over feasible subsets given set show that this defines preorder. propose dynamic programming algorithm to compute entire non-dominated rank cardinality vectors we state two greedy algorithms, efficiently single efficient solution.

The minimum rank of a graph G is the ranks all symmetric adjacency matrices G. We present new combinatorial bound for an arbitrary based on enumerating certain subsets vertices satisfying matroid theoretic properties. also some computational and theoretical challenges associated with computing rank. This includes conjecture that this actually holds equality graphs. History: “Challenge” paper wa...

Data fusion involves the integration of multiple related datasets. The statistical file-matching problem is a canonical data in multivariate analysis, where objective to characterise joint distribution set variables when only strict subsets marginal distributions have been observed. Estimation covariance matrix full challenging given missing-data pattern. Factor analysis models use lower-dimens...

We consider the computational complexity of some problems dealing with matrix rank. Let E ; S be subsets of a commutative ring R. we want to determine maxrank S (M) = max (a 1 ;a 2 ;:::;at)2S t rank M (a 1 ; a 2 ; : : : a t) and minrank S (M) = min (a 1 ;a 2 ;:::;at)2S t rank M (a 1 ; a 2 ; : : : a t): There are also variants of these problems that specify more about the structure of M , or ins...

The Hadamard Extension $\mathbb H({\mathrm {m}})$ of an notation="LaTeX">$n \times k$ matrix m is the collection all products subsets its rows. This construction essential for source identification (parameter estimation) a mixture not...

We find the automorphism group of moduli space parabolic bundles on a smooth curve (with fixed determinant and system weights). This is generated by: automorphisms marked curve, tensoring with line bundle, taking dual, Hecke transforms (using filtrations given by structure). A Torelli theorem for arbitrary rank generic weights also obtained. These results are extended to classification biration...