# Tagged Questions

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63 views

### approximate coordinates in a one dimensional lattice

suppose I have a finite set of real numbers ${r_1, \ldots r_n \in \mathbf{R} }$ and a single real number $x \in \mathbf{R}$. Is there a fast algorithm for finding integer numbers ${i_1, \ldots i_n \in ...

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51 views

### Finding special vectors generated by a matrix

Let $G\in \Bbb Z^{n\times n}$ be a unimodular matrix.
Are there any efficient algorithms to find the maximum norm of a vector $v$ that satisfies $\langle\Delta(v),v\rangle=0$ over all vectors $v\in ...

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**1**answer

212 views

### Main problems on lattice-basis reduction algorithms (such as LLL)?

What are the main open problems on lattice-basis reduction algorithms (such as LLL)? I am looking for problems satisfying the following two conditions:
(a) their solution would likely be of some ...

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**2**answers

237 views

### Lattice reduction on an orthonormal lattice?

Suppose you are given an inner product on a vector space and given a set of linearly independent vectors, and that you have been promised that the lattice they span has an orthonormal basis. Can you ...

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**1**answer

504 views

### Lovász $\delta$ condition for LLL Algorithm

http://en.wikipedia.org/wiki/Lenstra%E2%80%93Lenstra%E2%80%93Lov%C3%A1sz_lattice_basis_reduction_algorithm
What is the importance of the $\delta$ parameter for LLL bases called LovĂˇsz condition?
...

**8**

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**1**answer

637 views

### The Number of Short Vectors in a Lattice

Given a lattice $L = \bigoplus_{i=1}^{m} \mathbb{Z}v_i$ (the $v_i$ are linearly independent vectors in $\mathbb{R}^n$) and a number $c > 0$, can one quickly compute or find a good estimate on the ...

**4**

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468 views

### diameter of Voronoi cell of the lattice ? What about R^n ? What about small n =2,3,4 ?What about random lattice ?

Consider a lattice in R^n.
Consider Voronoi cell of it.
What is known about diameter ? About the shape ? What are good references ?
As far as I understand they are not easy to compute.
May be in ...

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175 views

### how to find vertex of parallelotope closest to given point P in R^n ? (Or minimize quadratic form over {+-1}) Is it NP ?

Consider a parallelotope in R^n and some point "P" in R^n.
What algorithms (except of brute force) can be suggested to find the closest vertex of paralleloptope to "P" ?
Is it NP ?
Parallelotope ...

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150 views

### How to find closest point to restricted lattice on the plane ? ( m*h1 + n*h2, for 0<m,n<N)

Consider finite piece of lattice i.e. points of the form m*h1 + n*h2, for 0<m,n<N h1, h2 -some vectors. Consider some point "P" on the plane. How to find ...

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872 views

### Closest vector problem (=nearest lattice point) is trivial for “reduced lattice” ?

Consider some lattice in R^n. Take some point "P" in R^n (which does not belong to this lattice in general). The problem is to find "nearest" lattice point. The problem is known NP-hard in general it ...

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**1**answer

248 views

### How “often” does LLL-reduction produce “optimal” result ? Is there condition (or informal understanding) on lattice that it LLL is optimal ?

Consider some lattice in R^4 (C^4) or C^8.
Famous "lattice reduction" procedures (like LLL latice reduction)
produces some "reduced basis". However in general there results are not "the best ...

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**2**answers

389 views

### Lattice reduction in R^3 (R^4) or what is fundamental domain for SL(3,Z) , (SL(4,Z)) ?

Consider a lattice in R^3.
Is the some "canonical" way or ways to choose basis in it ?
I mean in R^2 we can choose a basis |h_1| < |h_2| and |(h_2, h_1)| < 1/2 |h_1|.
Considering lattices with ...

**4**

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**0**answers

208 views

### Domination in Nice Lattices

Let an integer vector be nice when it has only two nonzero components, which sum to zero. So (0, 0, 3, 0, -3) and (-1, 0, 1, 0, 0) are examples of nice vectors in $n=5$ dimensions.
Call a lattice ...

**2**

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252 views

### Integer relation detection for Subset Sum or NPP?

Is there a way to encode an instance of Subset Sum or the Number Partition Problem so that a (small) solution to an integer relation yields an answer? If not definitely, then in some probabilistic ...