47,569 research outputs found
Improved Approximation Algorithms for Balanced Partitioning Problems
We present approximation algorithms for balanced partitioning problems. These problems are notoriously hard and we present new bicriteria approximation algorithms, that approximate the optimal cost and relax the balance constraint.
In the first scenario, we consider Min-Max k-Partitioning, the problem of dividing a graph into k equal-sized parts while minimizing the maximum cost of edges cut by a single part. Our approximation algorithm relaxes the size of the parts by (1+epsilon) and approximates the optimal cost by O(log^{1.5}(n) * log(log(n))), for every 0 < epsilon < 1. This is the first nontrivial algorithm for this problem that relaxes the balance constraint by less than 2.
In the second scenario, we consider strategies to find a minimum-cost mapping of a graph of processes to a hierarchical network with identical processors at the leaves. This Hierarchical Graph Partitioning problem has been studied recently by Hajiaghayi et al. who presented an (O(log(n)),(1+epsilon)(h+1)) approximation algorithm for constant network heights h. We use spreading metrics to give an improved (O(log(n)),(1+epsilon)h) approximation algorithm that runs in polynomial time for arbitrary network heights
Incremental Measurement of Structural Entropy for Dynamic Graphs
Structural entropy is a metric that measures the amount of information
embedded in graph structure data under a strategy of hierarchical abstracting.
To measure the structural entropy of a dynamic graph, we need to decode the
optimal encoding tree corresponding to the optimal hierarchical community
partitioning of the graph. However, the current structural entropy methods do
not support efficient incremental updating of encoding trees. To address this
issue, we propose Incre-2dSE, a novel incremental measurement framework that
dynamically adjusts the community partitioning and efficiently computes the
updated structural entropy for each snapshot of dynamic graphs. Incre-2dSE
consists of an online module and an offline module. The online module includes
dynamic measurement algorithms based on two dynamic adjustment strategies for
two-dimensional encoding trees, i.e., the naive adjustment strategy and the
node-shifting adjustment strategy, which supports theoretical analysis of the
updated structural entropy and incrementally adjusts the community partitioning
towards a lower structural entropy. In contrast, the offline module globally
constructs the encoding tree for the updated graph using static community
detection methods and calculates the structural entropy by definition. We
conduct experiments on an artificial dynamic graph dataset generated by Hawkes
Process and 3 real-world datasets. Experimental results confirm that our
dynamic measurement algorithms effectively capture the dynamic evolution of the
communities, reduce time consumption, and provide great interpretability
Multiscale Transforms for Signals on Simplicial Complexes
Our previous multiscale graph basis dictionaries/graph signal transforms --
Generalized Haar-Walsh Transform (GHWT); Hierarchical Graph Laplacian Eigen
Transform (HGLET); Natural Graph Wavelet Packets (NGWPs); and their relatives
-- were developed for analyzing data recorded on nodes of a given graph. In
this article, we propose their generalization for analyzing data recorded on
edges, faces (i.e., triangles), or more generally -dimensional
simplices of a simplicial complex (e.g., a triangle mesh of a manifold). The
key idea is to use the Hodge Laplacians and their variants for hierarchical
partitioning of a set of -dimensional simplices in a given simplicial
complex, and then build localized basis functions on these partitioned subsets.
We demonstrate their usefulness for data representation on both illustrative
synthetic examples and real-world simplicial complexes generated from a
co-authorship/citation dataset and an ocean current/flow dataset.Comment: 19 Pages, Comments welcom
Hierarchical path-finding for Navigation Meshes (HNA*)
Path-finding can become an important bottleneck as both the size of the virtual environments and the number of agents navigating them increase. It is important to develop techniques that can be efficiently applied to any environment independently of its abstract representation. In this paper we present a hierarchical NavMesh representation to speed up path-finding. Hierarchical path-finding (HPA*) has been successfully applied to regular grids, but there is a need to extend the benefits of this method to polygonal navigation meshes. As opposed to regular grids, navigation meshes offer representations with higher accuracy regarding the underlying geometry, while containing a smaller number of cells. Therefore, we present a bottom-up method to create a hierarchical representation based on a multilevel k-way partitioning algorithm (MLkP), annotated with sub-paths that can be accessed online by our Hierarchical NavMesh Path-finding algorithm (HNA*). The algorithm benefits from searching in graphs with a much smaller number of cells, thus performing up to 7.7 times faster than traditional A¿ over the initial NavMesh. We present results of HNA* over a variety of scenarios and discuss the benefits of the algorithm together with areas for improvement.Peer ReviewedPostprint (author's final draft
Magic-State Functional Units: Mapping and Scheduling Multi-Level Distillation Circuits for Fault-Tolerant Quantum Architectures
Quantum computers have recently made great strides and are on a long-term
path towards useful fault-tolerant computation. A dominant overhead in
fault-tolerant quantum computation is the production of high-fidelity encoded
qubits, called magic states, which enable reliable error-corrected computation.
We present the first detailed designs of hardware functional units that
implement space-time optimized magic-state factories for surface code
error-corrected machines. Interactions among distant qubits require surface
code braids (physical pathways on chip) which must be routed. Magic-state
factories are circuits comprised of a complex set of braids that is more
difficult to route than quantum circuits considered in previous work [1]. This
paper explores the impact of scheduling techniques, such as gate reordering and
qubit renaming, and we propose two novel mapping techniques: braid repulsion
and dipole moment braid rotation. We combine these techniques with graph
partitioning and community detection algorithms, and further introduce a
stitching algorithm for mapping subgraphs onto a physical machine. Our results
show a factor of 5.64 reduction in space-time volume compared to the best-known
previous designs for magic-state factories.Comment: 13 pages, 10 figure
Large constraint length high speed viterbi decoder based on a modular hierarchial decomposition of the deBruijn graph
A method of formulating and packaging decision-making elements into a long constraint length Viterbi decoder which involves formulating the decision-making processors as individual Viterbi butterfly processors that are interconnected in a deBruijn graph configuration. A fully distributed architecture, which achieves high decoding speeds, is made feasible by novel wiring and partitioning of the state diagram. This partitioning defines universal modules, which can be used to build any size decoder, such that a large number of wires is contained inside each module, and a small number of wires is needed to connect modules. The total system is modular and hierarchical, and it implements a large proportion of the required wiring internally within modules and may include some external wiring to fully complete the deBruijn graph. pg,14
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