Summary: ``The (parameterized) {\it Minimum Link-Length Rectilinear Spanning Path} problem in the $d$-dimensional Euclidean space $\Bbb R^d$ ($d$-RSP), for a given set $S$ of $n$ points in $\Bbb R^d$ and a positive integer $k$, is to find a piecewise-linear path $P$ with at most $k$ line-segments that covers (i.e., contains) all points in $S$, where all line-segments in $P$ are axis-parallel. We first prove that the problem 2-RSP is NP-complete, improving the previously known result that the problem 10-RSP is NP-complete. We then consider a constrained $d$-RSP problem in which each line-segment $s$ in the spanning path must cover all the points in the given set $S$ that share the same line with $s$. We present a new parameterized algorithm with running time $O^*((2d)^k)$ for the constrained $d$-RSP problem, which significantly improves the previous best result and is the first parameterized algorithm of running time $O^*(2^{O(k)})$ for the constrained $d$-RSP problem for a fixed $d$. We show that these results can be extended to the Minimum Link-Length Rectilinear Traveling Salesman problem.''