Related papers: Quantum Speedup for Sampling Random Spanning Trees
We consider the problem of augmenting an $n$-vertex tree with one shortcut in order to minimize the diameter of the resulting graph. The tree is embedded in an unknown space and we have access to an oracle that, when queried on a pair of…
Span program is a linear-algebraic model of computation originally proposed for studying the complexity theory. Recently, it has become a useful tool for designing quantum algorithms. In this paper, we present a time-efficient…
We develop a novel method for measuring the similarity between complete weighted graphs, which are probed by means of discrete-time quantum walks. Directly probing complete graphs using discrete-time quantum walks is intractable due to the…
In this paper we further investigate the well-studied problem of finding a perfect matching in a regular bipartite graph. The first non-trivial algorithm, with running time $O(mn)$, dates back to K\"{o}nig's work in 1916 (here $m=nd$ is the…
An algorithm observes the trajectories of random walks over an unknown graph $G$, starting from the same vertex $x$, as well as the degrees along the trajectories. For all finite connected graphs, one can estimate the number of edges $m$ up…
Probabilistic distributions over spanning trees in directed graphs are a fundamental model of dependency structure in natural language processing, syntactic dependency trees. In NLP, dependency trees often have an additional root…
We create a variety of new quantum algorithms that use Grover's algorithm and similar techniques to give polynomial speedups over their classical counterparts. We begin by introducing a set of tools that carefully minimize the impact of…
We study the potential utility of classical techniques of spectral sparsification of graphs as a preprocessing step for digital quantum algorithms, in particular, for Hamiltonian simulation. Our results indicate that spectral sparsification…
We develop a new algorithmic technique that allows to transfer some constant time approximation algorithms for general graphs into random order streaming algorithms. We illustrate our technique by proving that in random order streams with…
Dense subgraphs of sparse graphs (communities), which appear in most real-world complex networks, play an important role in many contexts. Computing them however is generally expensive. We propose here a measure of similarities between…
We significantly improve known time bounds for solving the minimum cut problem on undirected graphs. We use a ``semi-duality'' between minimum cuts and maximum spanning tree packings combined with our previously developed random sampling…
Many applications require randomly sampling bipartite graphs with fixed degrees, or randomly sampling incidence matrices with fixed row and column sums. Although several sampling algorithms exist, the ``curveball'' algorithm is the most…
Quantum walks are standard tools for searching graphs for marked vertices, and they often yield quadratic speedups over a classical random walk's hitting time. In some exceptional cases, however, the system only evolves by sign flips,…
Up to now, relatively few exponential quantum speed-ups have been achieved. Out of them, the welded tree problem (Childs, Cleve, Deotto, Farhi, Gutmann, and Spielman'2003) is one of the unusual examples, as the exponential speed-up is…
Leverage score sampling is crucial to the design of randomized algorithms for large-scale matrix problems, while the computation of leverage scores is a bottleneck of many applications. In this paper, we propose a quantum algorithm to…
This paper initiates the study of quantum algorithms for matroid property problems. It is shown that quadratic quantum speedup is possible for the calculation problem of finding the girth or the number of circuits (bases, flats,…
In this paper we describe a randomized algorithm which returns a maximal spanning forest of an unknown {\em weighted} undirected graph making $O(n)$ $\mathsf{CUT}$ queries in expectation. For weighted graphs, this is optimal due to a result…
Many problems that can be solved in quadratic time have bit-parallel speed-ups with factor $w$, where $w$ is the computer word size. A classic example is computing the edit distance of two strings of length $n$, which can be solved in…
We introduce two quantum algorithms for solving structured prediction problems. We first show that a stochastic gradient descent that uses the quantum minimum finding algorithm and takes its probabilistic failure into account solves the…
In recent years, the problem of computing the frequencies of the induced $k$-vertex subgraphs of a graph, or \emph{$k$-graphlets}, has become central. One approach for this problem is to sample $k$-graphlets randomly. Classic algorithms for…