Constructing Linear-Sized Spectral Sparsification in Almost-Linear Time

Fast almost-linear-sized nets for boxes in the plane

Let B be any set of n axis-aligned boxes in R, d ≥ 1. For any point p, we define the subset Bp of B as Bp = {B ∈ B : p ∈ B}. A box B in Bp is said to be stabbed by p. A subset N ⊆ B is a (1/c)-net for B if Np 6= ∅ for any p ∈ R such that |Bp| ≤ n/c. The number of distinct subsets Bp is O((2n)), so the set system described above has so-called finite VCdimension d. This ensures that there always ...

Building PQR trees in almost-linear time

In 1976, Booth and Leuker invented the PQ trees as a compact way of storing and manipulating all the permutations on n elements that keep consecutive the elements in certain given sets C1, C2, . . . , Cm. Such permutations are called valid. This problem finds applications in DNA physical mapping, interval graph recognition, logic circuit optimization and data retrieval, among others. PQ trees c...

Symbolic Program Analysis in Almost-Linear Time

A Linear-time Algorithm for Sparsification of Unweighted Graphs

Given an undirected graph G and an error parameter ε > 0, the graph sparsification problem requires sampling edges in G and giving the sampled edges appropriate weights to obtain a sparse graph Gε with the following property: the weight of every cut in Gε is within a factor of (1± ε) of the weight of the corresponding cut in G. If G is unweighted, an O(m logn)-time algorithm for constructing Gε...

Almost multiplicative linear functionals and approximate spectrum

We define a new type of spectrum, called δ-approximate spectrum, of an element a in a complex unital Banach algebra A and show that the δ-approximate spectrum σ_δ (a) of a is compact. The relation between the δ-approximate spectrum and the usual spectrum is investigated. Also an analogue of the classical Gleason-Kahane-Zelazko theorem is established: For each ε>0, there is δ>0 such that if ϕ is...