Gruber P. Convex and Discrete Geometry

420 Geometry of Numbers
Corollary 28.4. There is a polynomial time algorithm which, for any given basis
{a1,...,ad} of a rational lattice L in E d and any point x ∈ E d with rational coordinates,
finds a vector l ∈ L such that:
�x − l� ≤2 d 2 −1 min � �x − m� :m ∈ L � .
Proof. First, the LLL-theorem shows that there is a polynomial time algorithm which
finds an LLL-reduced basis {b1,...,bd}. Let{ ˆb1,..., ˆbd} be its Gram–Schmidt
orthogonalization. We will use the following relations:
(2)
Second,
b1=ˆb1
b2=µ21 ˆb1 + ˆb2
.................................
bd=µd1 ˆb1 +···+µdd−1 ˆbd−1 + ˆbd
ˆb1=b1
ˆb2=ν21b1 + b2
................................
ˆbd=νd1b1 +···+νdd−1bd−1 + bd.
(3) There is a polynomial time algorithm which finds for given rational x ∈ E d
a point l ∈ L such that:
x − l =
d�
i=1
Start with a representation of x of the form
x = �
λi ˆbi, where |λi| ≤ 1
for i = 1,...,d.
2
i
λ0i ˆbi.
Subtract ⌈λ0d⌋bd to get a representation of the form
x −⌈λ0d⌋bd = �
i
λ1i ˆbi where |λ1d| ≤ 1
2 .
Next subtract ⌈λ1d−1⌋bd−1, etc. Taking into account (2) we arrive, after d steps, at (3).
Third, it will be shown that:
(4) �x − l� ≤2 d 2 −1 �x − m� for any m ∈ L, where l is as in (3).
Let m ∈ L and write
x − m = �
i
µi ˆbi.
Let k be the largest index such that λk �= µk. Then
l − m =
k�
(λi − µi) ˆbi = (λk − µk)bk + lin. comb. of b1,...,bk−1 ∈ L
i=1
by (2). Thus λk −µk is a non-zero integer. In particular |λk −µk| ≥1. Since |λk| ≤ 1 2 ,
it follows that |µk| ≥ 1 2 . So,