An important index widely used to analyze social and information networks is betweenness centrality. In this paper, first given a directed network $G$ and a vertex $r\in V(G)$, we present a novel adaptive algorithm for estimating betweenness score of $r$. Our algorithm first computes two subsets of the vertex set of $G$, called $\mathcal{RF}(r)$ and $\mathcal{RT}(r)$, that define the sample spaces of the start-points and the end-points of the samples. Then, it adaptively samples from $\mathcal{RF}(r)$ and $\mathcal{RT}(r)$ and stops as soon as some condition is satisfied. The stopping condition depends on the samples met so far, $|\mathcal{RF}(r)|$ and $|\mathcal{RT}(r)|$. We show that compared to the well-known existing methods, our algorithm gives a more efficient $(λ,δ)$-approximation. Then, we propose a novel algorithm for estimating $k$-path centrality of $r$. Our algorithm is based on computing two sets $\mathcal{RF}(r)$ and $\mathcal{D}(r)$. While $\mathcal{RF}(r)$ defines the sample space of the source vertices of the sampled paths, $\mathcal{D}(r)$ defines the sample space of the other vertices of the paths. We show that in order to give a $(λ,δ)$-approximation of the $k$-path score of $r$, our algorithm requires considerably less samples. Moreover, it processes each sample faster and with less memory. Finally, we empirically evaluate our proposed algorithms and show their superior performance. Also, we show that they can be used to efficiently compute centrality scores of a set of vertices.