A family P = {π1,..., πq} of permutations of [n] = {1,..., n} is completely k-scrambling [Spencer, 1972; Füredi, 1996] if for any distinct k points x1,..., xk ∈ [n], permutations πi’s in P produce all k! possible orders on πi(x1),..., πi(xk). Let N ∗ (n, k) be the minimum size of such a family

Erdos and Szekeres showed that any permutation of length n +1contains a monotone subsequence of length k + 1. A simple example shows that there need be no more than (n mod k) such subsequences; we conjecture that this is the minimum number of such subsequences. We prove

In this paper, a hub refers to a non-terminal vertex of degree at least three. We study the minimum number of hubs needed in a network to guarantee certain flow demand constraints imposed between multiple pairs of sources and sinks. We prove that under the constraints, regardless of the size

Erdős asked in 1962 about the value of f(n, k, l), the minimum number of k-cliques in a graph with order n and independence number less than l. The case (k, l) = (3, 3) was solved by Lorden. Here we solve the problem (for all large n) for (3, l) with 4 ≤ l ≤ 7 and (k, 3) with 4 ≤ k ≤ 7

at least one landmark to be seen at any point along the way. Therefore, the problem becomes: find a path P from s to t such that the driver can see at least one landmark at any point along P and the number of landmarks the driver can stick to is minimized. There are two cases: (a) The same landmark

We describe a distributed position-based network protocol optimized for minimum energy consumption in mobile wireless networks that support peer-to-peer communications. Given any number of randomly deployed nodes over an area, we illustrate that a simple local optimization scheme executed at each

with the problem of the determination of the minimum number of required tracks when super-resolution subspace methods are applied. The results are validated on real data acquired in L-band by the E-SAR system of the German

Merrifield-Simmons index and minimum number of independent sets in short trees by

The minimum number of distinct eigenvalues, taken over all real symmetric matrices compatible with a given graph G, is denoted by q(G). Using other parameters related to G, bounds for q(G) are proven and then applied to deduce further properties of q(G). It is shown that there is a great number

Often, the training procedure for statistical machine translation models is based on maximum likelihood or related criteria. A general problem of this approach is that there is only a loose relation to the final translation quality on unseen text. In this paper, we analyze various training criteria which directly optimize translation quality.