# Selection on X_1 + X_1 + ⋯ 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+⋯+X_m in o(n· m + k· m), where X_i have length n. Here, that o(n· m + k· 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+⋯+X_m are compared empirically, demonstrating the benefit of the algorithm proposed here.

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