Many word processors and keyword search engines have a spelling correction feature. If you type in a misspelled word x, the word processor or search engine can suggest a correction y. The correction y should be a word that is close to x. One way to measure the similarity in spelling between two text strings is by “edit distance.” The notion of edit distance is useful in other fields as well. Fo...

Problem 1. [20 points] The adjacency matrix A of a graph G with n vertices as defined in lecture is an n×n matrix in which Ai,j is 1 if there is an edge from i to j and 0 if there is not. In lecture we saw how the smallest k where Ai,j 6= 0 describes the length of the shortest path from i to j. Given a combinatorial interpretation of the following statements about the adjacency matrix in terms ...

(b) If a process X(t) is strongly stationary, all of its moments have to be independent of t. The third moment of X(t), E(X(t)) = (EA)(cos(ωt) + sin(ωt)), is not a constant with respect to t unless EA = 0. Clearly we can find many examples of distributions for A so that the third moment is not zero while E(A) = 0. For example take A to be a discrete random variable with P (A = −1) = 1/3, P (A =...

Problem 3. The element 1 + 2 ∈ k[S3] is equal to (1 + ρ+ ρ2)/3. Thus M is the same as the image of (1 + ρ+ ρ2)/3, which I’ll denote by φ. Notice that if β ∈ L, then ρ(φβ) = (ρφ)β = φβ. This means that M is contained in the fixed field of ρ, i.e. M ⊆ L. Let’s show that in fact M = L. We need to prove the other inclusion, that L ⊆M . If β ∈ L, then ρ(β) = β, so φ(β) = (β + ρ(β) + ρ2(β))/3 = (β + ...

For linear least squares problems minx kAx ? bk2, where A is sparse except for a few dense rows, a straightforward application of Cholesky or QR factorization will lead to a catastrophic ll in the factor R. We consider handling such problems by a matrix stretching technique, where the dense rows are split into several more sparse rows. We develop both a recursive binary splitting algorithm and ...

This paper proposes and evaluates a novel, edge-based approach, which we call the kset Traffic Engineering (TE) method, to perform traffic engineering in OSPF networks by partitioning traffic into uneven k traffic sets. The traffic partitioning and splitting takes place only at network edges, leaving the core simple. We theoretically prove that if k is large enough, the k-set TE method achieves...

Given a ring of size n and a set K of traffic demands, the ring loading problem with demand splitting (RLPW) is to determine a routing to minimize the maximum load on the edges. In the problem, a demand between two nodes can be split into two flows and then be routed along the ring in different directions. If the two flows obtained by splitting a demand are restricted to integers, this restrict...