We present new faster algorithms for the exact solution of the shortest vector problem in arbitrary lattices. Our main result shows that the shortest vector in any n-dimensional lattice can be found in time 2 3.199n (and space 2 1.325n ), or in . We also present a practical variant of our algorithm which provably uses an amount of space proportional to τn, the "kissing" constant in dimension n.No upper bound on the running time of our second algorithm is currently known, but experimentally the algorithm seems to perform fairly well in practice, with running time 2 0.52n , and space complexity 2 0.2n .

We give deterministicÕ(2 2n )-timeÕ(2 n )-space algorithms to solve all the most important computational problems on point lattices in NP, including the shortest vector problem (SVP), closest vector problem (CVP), and shortest independent vectors problem (SIVP). This improves the n O(n) running time of the best previously known algorithms for CVP [R. Kannan, and gives a deterministic and asymptotically faster alternative to the 2 O(n) -time (and space) randomized algorithm for SVP of Ajtai, Kumar, and Sivakumar [Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, 2001, pp. 266-275]. The core of our algorithm is a new method to solve the closest vector problem with preprocessing (CVPP) that uses the Voronoi cell of the lattice (described as intersection of half-spaces) as the result of the preprocessing function. A direct consequence of our results is a derandomization of the best current polynomial time approximation algorithms for SVP and CVP, achieving a 2 O(n log log n/ log n) approximation factor. Introduction.A d-dimensional lattice Λ is a discrete subgroup of the Euclidean space R d , and is customarily represented as the set of all integer linear combinations of n ≤ d basis vectors B = [b 1 , . . . , b n ] ∈ R d×n . There are many famous algorithmic problems on point lattices, the most important of which are as follows:• The shortest vector problem (SVP): given a basis B, find a shortest nonzero vector in the lattice generated by B. • The closest vector problem (CVP): given a basis B and a target vector t ∈ R d , find a lattice vector generated by B that is closest to t. • The shortest independent vectors problem (SIVP): given a basis B, find n linearly independent lattice vectors in the lattice generated by B that are as short as possible. 1 Besides being classic mathematical problems in the study of the geometry of numbers [16], these problems play an important role in many computer science and communication theory applications. SVP and CVP have been used to solve many landmark algorithmic problems in theoretical computer science, like integer programming [41,36], factoring polynomials over the rationals [40], checking the solvability by radicals [39],

We give deterministic 2 O(n) -time algorithms to solve all the most important computational problems on point lattices in NP, including the Shortest Vector Problem (SVP), Closest Vector Problem (CVP), and Shortest Independent Vectors Problem (SIVP). This improves the n O(n) running time of the best previously known algorithms for CVP (Kannan, Math. Operation Research 12(3): 1987) and SIVP (Micciancio, Proc. of SODA, 2008), and gives a deterministic alternative to the 2 O(n) -time (and space) randomized algorithm for SVP of (Ajtai, Kumar and Sivakumar, STOC 2001). The core of our algorithm is a new method to solve the closest vector problem with preprocessing (CVPP) that uses the Voronoi cell of the lattice (described as intersection of half-spaces) as the result of the preprocessing function. In the process, we also give algorithms for several other lattice problems, including computing the kissing number of a lattice, and computing the set of all Voronoi relevant vectors. All our algorithms are deterministic, and have 2 O(n) time and space complexity 1

No abstract

Liposarcomas are the most common soft tissue tumors with various histological subtypes. They usually appear in the retroperitoneal region of the abdomen, but their symptomatology remains unclear and their diagnosis, as well as treatment challenging. A case of a 55-year-old female patient with dyspnea and light diffuse abdominal pain caused by a giant retroperitoneal liposarcoma is presented. The patient had an unremarkable medical history, while the computed tomography scan revealed a large mass at the right portion of the abdomen, with its upper limits to the lower edge and the gate portion of the liver. The mass was in contact with the right kidney, the inferior vena cava and the right renal vein, causing mild dilation of the right kidney pelvis. Without any evidence of intra-abdominal metastases, the tumor was surgically resected. The histological analysis of the tumor revealed a well-differentiated liposarcoma. The patient had an uneventful recovery and was discharged on the 10th postoperative day. Until today (4 years later) she remains asymptomatic, without any signs of recurrence. The retroperitoneal liposarcoma is a clinical entity with unclear clinical symptoms and the physician should consider including it in the differential diagnosis of a majority of symptoms, such as dyspnea.