Selection on $X_1 + X_1 + \cdots X_m$ via Cartesian product tree

Selection on the Cartesian product is a classic problem in computer science. Recently, an optimal algorithm for selection on $X+Y$, based on soft heaps, was introduced. By combining this approach with layer-ordered heaps (LOHs), an algorithm using a balanced binary tree of $X+Y$ selections was proposed to perform $k$-selection on $X_1+X_2+\cdots+X_m$ in $o(n\cdot m + k\cdot m)$, where $X_i$ have length $n$. Here, that $o(n\cdot m + k\cdot m)$ algorithm is combined with a novel, optimal LOH-based algorithm for selection on $X+Y$ (without a soft heap). Performance of algorithms for selection on $X_1+X_2+\cdots+X_m$ are compared empirically, demonstrating the benefit of the algorithm proposed here.