• Corpus ID: 204904789

Selection on $X_1+X_2+\cdots + X_m$ with layer-ordered heaps

@article{Kreitzberg2019SelectionO,
  title={Selection on \$X\_1+X\_2+\cdots + X\_m\$ with layer-ordered heaps},
  author={Patrick Kreitzberg and Kyle A. Lucke and Oliver Serang},
  journal={arXiv: Data Structures and Algorithms},
  year={2019}
}

• Patrick Kreitzberg, Kyle A. Lucke, O. Serang
• Published 26 October 2019
• Computer Science
• arXiv: Data Structures and Algorithms

Selection on $X_1+X_2+\cdots + X_m$ is an important problem with many applications in areas such as max-convolution, max-product Bayesian inference, calculating most probable isotopes, and computing non-parametric test statistics, among others. Faster-than-naive approaches exist for $m=2$: Johnson \& Mizoguchi (1978) find the smallest $k$ values in $A+B$ with runtime $O(n \log(n))$. Frederickson \& Johnson (1982) created a method for finding the $k$ smallest values in $A+B$ with runtime $O(n + \min(k,n)\log(\frac{k}{\min(k,n)}))$. In 1993, Frederickson published an optimal algorithm for selection on $A+B$, which runs in $O(n+k)$. In 2018, Kaplan \emph{et al.} described another optimal algorithm in terms Chazelle's of soft heaps. No fast methods exist for $m>2$. Johnson \& Mizoguchi (1978) introduced a method to compute the minimal $k$ terms when $m>2$, but that method runs in $O(m\cdot n^{\frac{m}{2}} \log(n))$ and is inefficient when $m \gg 1$.
In this paper, we introduce the first efficient methods for problems where $m>2$. We introduce the ``layer-ordered heap,'' a simple special class of heap with which we produce a new, fast selection algorithm on the Cartesian product. Using this new algorithm to perform $k$-selection on the Cartesian product of $m$ arrays of length $n$ has runtime $\in o(m\cdot n + k\cdot m)$. We also provide implementations of the algorithms proposed and their performance in practice.