On page $8$ in these slides (http://www.math.unicaen.fr/~nitaj/LatticeMalaysia2014-2.pdf) it is told that if we want to solve $$x_1a_1+\dots+x_na_n=N$$ where $|x_i|<\frac{2^{n/4}N^\frac1{n+1}}{\sqrt n}$ holds then we have a polynomial time algorithm for this.

What is the complexity of the algorithm? Is there a reference?

Can we replace $|x_i|<\frac{2^{n/4}N^\frac1{n+1}}{\sqrt n}$ by $0\leq x_i<\frac{2^{n/4}N^\frac1{n+1}}{\sqrt n}$?

On page $8$ in these slides (http://www.math.unicaen.fr/~nitaj/LatticeMalaysia2014-2.pdf) it is told that if we want to solve $$x_1a_1+\dots+x_na_n=N$$ where $|x_i|<\frac{2^{n/4}N^\frac1{n+1}}{\sqrt n}$ holds then we have a polynomial time algorithm for this.

(1) What is the complexity of the algorithm? Is there a reference?

(2) Can we replace $|x_i|<\frac{2^{n/4}N^\frac1{n+1}}{\sqrt n}$ by $0\leq x_i<\frac{2^{n/4}N^\frac1{n+1}}{\sqrt n}$?