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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, wasintroduced. By combining this approach with layer-ordered heaps (LOHs), analgorithm using a balanced binary tree of $X+Y$ selections was proposed toperform $k$-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 iscombined 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 thealgorithm proposed here.

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