Discrete Quantum Walks on the Symmetric Group

The theory of random walks on finite graphs is well developed with numerous applications. In quantum walks, the propagation is governed by quantum mechanical rules; generalizing random walks to the quantum setting. They have been successfully applied in the development of quantum algorithms. In particular, to solve problems that can be mapped to searching or property testing on some specific graph. In this paper we investigate the discrete time coined quantum walk (DTCQW) model using tools from non-commutative Fourier analysis. Specifically, we are interested in characterizing the DTCQW on Cayley graphs generated by the symmetric group (\sym) with appropriate generating sets. The lack of commutativity makes it challenging to find an analytical description of the limiting behavior with respect to the spectrum of the walk-operator. We determine certain characteristics of these walks using a path integral approach over the characters of \sym

Sorting under Forbidden Comparisons

In this paper we study the problem of sorting under forbidden comparisons where some pairs of elements may not be compared (forbidden pairs). Along with the set of elements V the input to our problem is a graph G(V, E), whose edges represents the pairs that we can compare in constant time. Given a graph with n vertices and m =(n2) - q edges we propose the first non-trivial deterministic algorithm which makes O((q + n) log n) comparisons with a total complexity of O(n2 + qω/2), where ω is the exponent in the complexity of matrix multiplication. We also propose a simple randomized algorithm for the problem which makes Õ(n2/√q + n+n√q) probes with high probability. When the input graph is random we show that Õ(min (n3/2, pn2)) probes suffice, where p is the edge probability

Distributed Matrix Tiling using a Hypergraph Labeling Formulation

Partitioning large matrices is an important problem in distributed linear algebra computing, used in ML among others. Briefly, our goal is to perform a sequence of matrix algebra operations in a distributed manner on these large matrices. However, not all partitioning schemes work well with different matrix algebra operations and their implementations (algorithms). This is a type of data tiling problem. In this paper we consider a data tiling problem using hypergraphs. We prove some hardness results and give a theoretical characterization of its complexity on random instances. Additionally, we develop a greedy algorithm and experimentally show its efficacy

Locality-Aware Qubit Routing for the Grid Architecture

Due to the short decohorence time of qubits available in the NISQ-era, it is essential to pack (minimize the size and or the depth of) a logical quantum circuit as efficiently as possible given a sparsely coupled physical architecture. In this work we introduce a locality-aware qubit routing algorithm based on a graph theoretic framework. Our algorithm is designed for the grid and certain \u27grid-like\u27 architectures. We experimentally show the competitiveness of algorithm by comparing it against the approximate token swapping algorithm, which is used as a primitive in many state-of-the-art quantum trans pilers. Our algorithm produces circuits of comparable depth (better on random permutations) while being an order of magnitude faster than a typical implementation of the approximate token swapping algorithm

An Adjacency Labeling Scheme based on a Decomposition of Trees into Caterpillars

In this paper we look at the problem of adjacency labeling of graphs. Given a family of undirected graphs the problem is to determine an encoding-decoding scheme for each member of the family such that we can decode the adjacency information of any pair of vertices only from their encoded labels. Further, we want the length of each label to be short (logarithmic in n, the number of vertices) and the encoding-decoding scheme to be computationally efficient. We propose a simple tree-decomposition based encoding scheme and use it give an adjacency labeling of size O(klog klog n) -bits. Here k is the clique-width of the graph family. We also extend the result to a certain family of k-probe graphs

An Adjacency Labeling Scheme based on a Decomposition of Trees into Caterpillars

In this paper we look at the problem of adjacency labeling of graphs. Given a family of undirected graphs the problem is to determine an encoding-decoding scheme for each member of the family such that we can decode the adjacency information of any pair of vertices only from their encoded labels. Further, we want the length of each label to be short (logarithmic in n, the number of vertices) and the encoding-decoding scheme to be computationally efficient. We propose a simple tree-decomposition based encoding scheme and use it give an adjacency labeling of size O(klog klog n) -bits. Here k is the clique-width of the graph family. We also extend the result to a certain family of k-probe graphs

New Results on Routing via Matchings on Graphs

In this paper we present some new complexity results on the routing time of a graph under the routing via matching model. This is a parallel routing model which was introduced by Alon et al. [1]. The model can be viewed as a communication scheme on a distributed network. The nodes in the network can communicate via matchings (a step), where a node exchanges data (pebbles) with its matched partner. Let G be a connected graph with vertices labeled from {1, …, n} and the destination vertices of the pebbles are given by a permutation π. The problem is to find a minimum step routing scheme for the input permutation π. This is denoted as the routing time rt(G, π) of G given π. In this paper we characterize the complexity of some known problems under the routing via matching model and discuss their relationship to graph connectivity and clique number. We also introduce some new problems in this domain, which may be of independent interest

A Sorting Network on Trees

Sorting networks are a class of parallel oblivious sorting algorithms. Not only do they have interesting theoretical properties but they can be fabricated. A sorting network is a sequence of parallel compare-exchange operations using comparators which are grouped into stages. This underlying graph defines the topology of the network. The majority of results on sorting networks concern the unrestricted case where the underlying graph is the complete graph. Prior results are also known for paths, hypercubes, and meshes. In this paper we introduce a sorting network whose underlying topology is a tree and formalize the concept of sorting networks on a restricted graph topology by introducing a new parameter for graphs called its sorting number. The main result of the paper is a description of an O(min(nΔ2,n2)) depth sorting network on a tree with maximum degree Δ

Computing Maximal Layers of Points in Eᶠ⁽ⁿ⁾

In this paper we present a randomized algorithm for computing the collection of maximal layers for a point set in Ek (k = f(n)). The input to our algorithm is a point set P = {p1,..., pn} with pi ∈ Ek. The proposed algorithm achieves a runtime of O (formula presented) when P is a random order and a runtime of O(formula presented) for an arbitrary P. Both bounds hold in expectation. Additionally, the run time is bounded by O(knk) in the worst case. This is the first non-trivial algorithm whose run-time remains polynomial whenever f(n) is bounded by some polynomial in n while remaining sub-quadratic in n for constant k (in expectation). The algorithm is implemented using a new data-structure for storing and answering dominance queries over the set of incomparable points