1 1 Introduction An n-dimensional lattice L is a set of vectors fPni=1 aivi j ai 2 Zg where v1; v2; : : : ; vn 2 Rm is a setof independent vectors called the basis for the lattice. The same lattice could have many bases. Given a basis for an n-dimensional lattice, the ...
We study four problems from the geometry of numbers, the shortest vector problem (SVP), the closest vector problem (CVP), the successive minima problem (SMP), and the shortest independent vectors problem (SIVP). Extending and generalizing results of Ajtai, Kumar, and Sivakumar we present probabi...
, theclosest vector problem , thesuccessive minima problem , and theshortest independent vectors problem ( )toSap, establishing probabilistic single exponential time algorithms for them. The result generalize and extend previous results of Ajtai, Kumar and Sivakumar. The results on and are new for a...
Any finite set of linearly independent vectors b_1, b_2, \cdots , b_m\in \mathbb {R}^N generates a lattice: {\mathcal L}= \left\{ \sum _{i=1}^{m}z_i b_i|\;z_i\in \mathbb {Z}\right\} . Denote by B the matrix whose column vectors are the b_i’s. We say B ...
Returned vector (by itself) isn’t really useful, however, we can use Vector.ConditionalSelect(lt_vec, ij_vec, ikj_vec) vector operation to extract required values from ij_vec and ikj_vec vectors into a new vector – r_vec. This operation returns a new vector where values are equ...
only according to the number of edges connected to the node and independent of the edge costs. This seems like a reasonable choice for moving in a street network, if we consider that the decision of direction of the random walker in an intersection is not affected by the lengths of the ...
Two grid paths aresymmetricif they share the same start and end point and one can be derived from the other by swapping the order of the constituent vectors. As a clarifying example, consider the problem instance in Figure 1. Each highlighted path is symmetric to the others since they all ...
Next, the shortest vectors of the root lattice E6 may be taken as a derived spherical design of E7. As the parameters we compute (smallest degree of a non-trivial polynomial in the ideal, smallest possible maximum degree of polynomials in any generating set) are independent of the choice of...
In most vehicle routing and crew scheduling applications solved by column generation, the subproblem corresponds to a shortest path problem with resource constraints (SPPRC) or one of its variants. This chapter proposes a classification and a generic formulation for the SPPRCs, briefly discusses compl...
bucket elegantly solves the problem of finding the next free slot in a bucket. Specifically, the technique (a) does not require a variable holding the size of the vector, and (b) requires a constant number of instructions that is independent from the number of values that are in the ...