Related papers: Randomized selection with quintary partitions
We consider low-space algorithms for the classic Element Distinctness problem: given an array of $n$ input integers with $O(\log n)$ bit-length, decide whether or not all elements are pairwise distinct. Beame, Clifford, and Machmouchi [FOCS…
Feature selection can facilitate the learning of mixtures of discrete random variables as they arise, e.g. in crowdsourcing tasks. Intuitively, not all workers are equally reliable but, if the less reliable ones could be eliminated, then…
Given a sequence $A$ of $n$ numbers and an integer (target) parameter $1\leq i\leq n$, the (exact) selection problem asks to find the $i$-th smallest element in $A$. An element is said to be $(i,j)$-mediocre if it is neither among the top…
We consider the problem of sorting $n$ items, given the outcomes of $m$ pre-existing comparisons. We present a simple and natural deterministic algorithm that runs in $O(m + \log T)$ time and does $O(\log T)$ comparisons, where $T$ is the…
Selection and sorting the Cartesian sum, $X+Y$, are classic and important problems. Here, a new algorithm is presented, which generates the top $k$ values of the form $X_i+Y_j$. The algorithm relies only on median-of-medians and is simple…
We consider the problem of learning an unknown partition of an $n$ element universe using rank queries. Such queries take as input a subset of the universe and return the number of parts of the partition it intersects. We give a simple…
We consider the fundamental problem of internally sorting a sequence of $n$ elements. In its best theoretical setting QuickMergesort, a combination Quicksort with Mergesort with a Median-of-$\sqrt{n}$ pivot selection, requires at most $n…
We show how to find all $k$ marked elements in a list of size $N$ using the optimal number $O(\sqrt{N k})$ of quantum queries and only a polylogarithmic overhead in the gate complexity, in the setting where one has a small quantum memory.…
Sequential minimum optimization is a machine-learning global search training algorithm. It is applicable when the functional dependence of the cost function on a tunable parameter given the other parameters can be cheaply determined. This…
We consider the problem of ranking $N$ objects starting from a set of noisy pairwise comparisons provided by a crowd of equal workers. We assume that objects are endowed with intrinsic qualities and that the probability with which an object…
We consider the problem of partial order production: arrange the elements of an unknown totally ordered set T into a target partially ordered set S, by comparing a minimum number of pairs in T. Special cases include sorting by comparisons,…
In an undirected graph, a $k$-cut is a set of edges whose removal breaks the graph into at least $k$ connected components. The minimum weight $k$-cut can be computed in $O(n^{O(k)})$ time, but when $k$ is treated as part of the input,…
Quantum $k$-minimum finding is a fundamental subroutine with numerous applications in combinatorial problems and machine learning. Previous approaches typically assume oracle access to exact function values, making it challenging to…
The Quickselect algorithm (also called FIND) is a fundamental algorithm for selecting ranks or quantiles within a set of data. Gr\"ubel and R\"osler showed that the number of key comparisons required by Quickselect considered as a process…
The k-means algorithm is a well-known method for partitioning n points that lie in the d-dimensional space into k clusters. Its main features are simplicity and speed in practice. Theoretically, however, the best known upper bound on its…
We present numerical results for the probability of bad cases for Quicksort, i.e. cases of input data for which the sorting cost considerably exceeds that of the average. Dynamic programming was used to compute solutions of the recurrence…
We are given n base elements and a finite collection of subsets of them. The size of any subset varies between p to k (p < k). In addition, we assume that the input contains all possible subsets of size p. Our objective is to find a…
Min-entropy sampling gives a bound on the min-entropy of a randomly chosen subset of a string, given a bound on the min-entropy of the whole string. K\"onig and Renner showed a min-entropy sampling theorem that holds relative to quantum…
One of the fundamental problem in the theory of sorting is to find the pessimistic number of comparisons sufficient to sort a given number of elements. Currently 16 is the lowest number of elements for which we do not know the exact value.…
Given a random variable $O \in \mathbb{R}$ and a set of experts $E$, we describe a method for finding a subset of experts $S \subseteq E$ whose aggregated opinion best predicts the outcome of $O$. Therefore, the problem can be regarded as a…