In this paper, we show a construction of locality-sensitive hash functions without false negatives, i.e., which ensure collision for every pair of points within a given radius (R) in (d) dimensional space equipped with (l_p) norm when (p \in [1,\infty]). Furthermore, we show how to use these hash functions to solve the (c)-approximate nearest neighbor search problem without false negatives. Namely, if there is a point at distance (R), we will certainly report it and points at distance greater than (cR) will not be reported for (c=Ω(\sqrt{d},d^{1-\frac{1}{p}})). The constructed algorithms work: - with preprocessing time (\mathcal{O}(n log(n))) and sublinear expected query time,

• with preprocessing time (\mathcal{O}(\mathrm{poly}(n))) and expected query time (\mathcal{O}(log(n))). Our paper reports progress on answering the open problem presented by Pagh [8] who considered the nearest neighbor search without false negatives for the Hamming distance.