We prove that general merge-and-shrink representations are strictly more powerful than linear ones by showing that there exist problem families that can be represented compactly with general merge-and-shrink representations but not with linear ones. We also give a precise bound that quantifies the necessary blowup incurred by conversions from general merge-and-shrink representations to linear r...

We conjecture a combinatorial formula for the monomial expansion of the image of any Schur function under the Bergeron–Garsia nabla operator. The formula involves nested labeled Dyck paths weighted by area and a suitable “diagonal inversion” statistic. Our model includes as special cases many previous conjectures connecting the nabla operator to quantum lattice paths. The combinatorics of the i...

Given an undirected graph G = (V,E) with a terminal set S ⊆ V , a weight function μ : ( S 2 ) → Z+ on terminal pairs, and an edge-cost a : E → Z+, the μweighted minimum-cost edge-disjoint S-paths problem (μ-CEDP) is to maximize ∑ P∈P μ(sP , tP )−a(P ) over all edge-disjoint sets P of S-paths, where sP , tP denote the ends of P and a(P ) is the sum of edge-cost a(e) over edges e in P . Our main ...

Shown below is pseudocode for Dijkstra’s algorithm. The input is a directed graph G = (V,E) with nonnegative edge weights wt(u, v) for every edge (u, v) ∈ E, and a distinguished node s, called the source (or start) node. The algorithm computes, for each node u ∈ V , the weight of a shortest path from s to u. (It can be easily modified to compute, for each node u, the predecessor of u in a short...

Two ordered Hamiltonian paths in the n-dimensional hypercube Qn are said to be independent if i-th vertices of the paths are distinct for every 1 ≤ i ≤ 2n. Similarly, two s-starting Hamiltonian cycles are independent if i-th vertices of the cycle are distinct for every 2 ≤ i ≤ 2n. A set S of Hamiltonian paths and sstarting Hamiltonian cycles are mutually independent if every two paths or cycles...

This paper includes different approaches for adaptive wavelet threshold (Bayes Shrink and Normal Shrink) and a robust wavelet domain method for noise filtering in ultrasound images. The search for efficient image denoising methods is still a valid challenge at the crossing of functional analysis and statistics. The proposed work in this paper extends the existing technique by improving the thre...