# 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} }

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…

## 2 Citations

Optimal construction of a layer-ordered heap

- Computer ScienceArXiv
- 2020

A few algorithms for constructing LOHs are introduced, their complexity is analyzed, and it is demonstrated that one algorithm is optimal for building a LOH of any rank $\alpha$.

Fast exact computation of the k most abundant isotope peaks with layer-ordered heaps

- Computer ScienceAnalytical chemistry
- 2020

A novel algorithm for calculating the most abundant $k$ isotopologue peaks of a compound is presented, which uses Serang's optimal method of selection on Cartesian products.

## References

SHOWING 1-10 OF 16 REFERENCES

Selection from heaps, row-sorted matrices and X+Y using soft heaps

- Computer ScienceSOSA
- 2019

A new optimal "output-sensitive" algorithm is obtained that performs only $O(m+\sum_{i=1}^m \log(k_i+1)$ comparisons, where k_i$ is the number of items of the $i$-th list that belong to the overall set of~$k$ smallest items.

Optimal construction of a layer-ordered heap

- Computer ScienceArXiv
- 2020

A few algorithms for constructing LOHs are introduced, their complexity is analyzed, and it is demonstrated that one algorithm is optimal for building a LOH of any rank $\alpha$.

A Bounded p-norm Approximation of Max-Convolution for Sub-Quadratic Bayesian Inference on Additive Factors

- Computer ScienceJ. Mach. Learn. Res.
- 2016

The method with fastest known worst-case runtime is extended, which can be applied to nonnegative vectors by numerically approximating the Chebyshev norm, and is used to derive two numerically stable methods based on the idea of computing p-norms via fast convolution.

An Optimal Algorithm for Selection in a Min-Heap

- Computer ScienceInf. Comput.
- 1993

The result establishes a further example of a partial order for which the kth smallest element can be determined in time proportional to the information theory lower bound.

A Fast Numerical Method for Max-Convolution and the Application to Efficient Max-Product Inference in Bayesian Networks

- Computer ScienceJ. Comput. Biol.
- 2015

The numerical max-convolution method can be applied to specialized classes of hidden Markov models to reduce the runtime of computing the Viterbi path from nk(2) to nk log(k), and has potential application to the all-pairs shortest paths problem.

Fast exact computation of the k most abundant isotope peaks with layer-ordered heaps

- Computer ScienceAnalytical chemistry
- 2020

A novel algorithm for calculating the most abundant $k$ isotopologue peaks of a compound is presented, which uses Serang's optimal method of selection on Cartesian products.

The soft heap: an approximate priority queue with optimal error rate

- Computer ScienceJACM
- 2000

A simple variant of a priority queue, called a soft heap, is introduced, which is optimal for any value of ε in a comparison-based model and can be used to compute exact or approximate medians and percentiles optimally.

How Good is the Information Theory Bound in Sorting?

- Mathematics, Computer ScienceTheor. Comput. Sci.
- 1976

Necklaces, Convolutions, and X + Y

- MathematicsESA
- 2006

Subquadratic algorithms that, given two necklace each with n beads at arbitrary positions, compute the optimal rotation of the necklaces to best align the beads to shed some light on the classic sorting X + Y problem.