3,538 research outputs found
Space-Efficient Routing Tables for Almost All Networks and the Incompressibility Method
We use the incompressibility method based on Kolmogorov complexity to
determine the total number of bits of routing information for almost all
network topologies. In most models for routing, for almost all labeled graphs
bits are necessary and sufficient for shortest path routing. By
`almost all graphs' we mean the Kolmogorov random graphs which constitute a
fraction of of all graphs on nodes, where is an arbitrary
fixed constant. There is a model for which the average case lower bound rises
to and another model where the average case upper bound
drops to . This clearly exposes the sensitivity of such bounds
to the model under consideration. If paths have to be short, but need not be
shortest (if the stretch factor may be larger than 1), then much less space is
needed on average, even in the more demanding models. Full-information routing
requires bits on average. For worst-case static networks we
prove a lower bound for shortest path routing and all
stretch factors in some networks where free relabeling is not allowed.Comment: 19 pages, Latex, 1 table, 1 figure; SIAM J. Comput., To appea
Mixing times of lozenge tiling and card shuffling Markov chains
We show how to combine Fourier analysis with coupling arguments to bound the
mixing times of a variety of Markov chains. The mixing time is the number of
steps a Markov chain takes to approach its equilibrium distribution. One
application is to a class of Markov chains introduced by Luby, Randall, and
Sinclair to generate random tilings of regions by lozenges. For an L X L region
we bound the mixing time by O(L^4 log L), which improves on the previous bound
of O(L^7), and we show the new bound to be essentially tight. In another
application we resolve a few questions raised by Diaconis and Saloff-Coste, by
lower bounding the mixing time of various card-shuffling Markov chains. Our
lower bounds are within a constant factor of their upper bounds. When we use
our methods to modify a path-coupling analysis of Bubley and Dyer, we obtain an
O(n^3 log n) upper bound on the mixing time of the Karzanov-Khachiyan Markov
chain for linear extensions.Comment: 39 pages, 8 figure
Circuit Transformations for Quantum Architectures
Quantum computer architectures impose restrictions on qubit interactions. We propose efficient circuit transformations that modify a given quantum circuit to fit an architecture, allowing for any initial and final mapping of circuit qubits to architecture qubits. To achieve this, we first consider the qubit movement subproblem and use the ROUTING VIA MATCHINGS framework to prove tighter bounds on parallel routing. In practice, we only need to perform partial permutations, so we generalize ROUTING VIA MATCHINGS to that setting. We give new routing procedures for common architecture graphs and for the generalized hierarchical product of graphs, which produces subgraphs of the Cartesian product. Secondly, for serial routing, we consider the TOKEN SWAPPING framework and extend a 4-approximation algorithm for general graphs to support partial permutations. We apply these routing procedures to give several circuit transformations, using various heuristic qubit placement subroutines. We implement these transformations in software and compare their performance for large quantum circuits on grid and modular architectures, identifying strategies that work well in practice
Routing for analog chip designs at NXP Semiconductors
During the study week 2011 we worked on the question of how to automate certain aspects of the design of analog chips. Here we focused on the task of connecting different blocks with electrical wiring, which is particularly tedious to do by hand. For digital chips there is a wealth of research available for this, as in this situation the amount of blocks makes it hopeless to do the design by hand. Hence, we set our task to finding solutions that are based on the previous research, as well as being tailored to the specific setting given by NXP.
This resulted in an heuristic approach, which we presented at the end of the
week in the form of a protoype tool. In this report we give a detailed account of the ideas we used, and describe possibilities to extend the approach
Traffic Analysis in Random Delaunay Tessellations and Other Graphs
In this work we study the degree distribution, the maximum vertex and edge
flow in non-uniform random Delaunay triangulations when geodesic routing is
used. We also investigate the vertex and edge flow in Erd\"os-Renyi random
graphs, geometric random graphs, expanders and random -regular graphs.
Moreover we show that adding a random matching to the original graph can
considerably reduced the maximum vertex flow.Comment: Submitted to the Journal of Discrete Computational Geometr
Trickle-down processes and their boundaries
It is possible to represent each of a number of Markov chains as an evolving
sequence of connected subsets of a directed acyclic graph that grow in the
following way: initially, all vertices of the graph are unoccupied, particles
are fed in one-by-one at a distinguished source vertex, successive particles
proceed along directed edges according to an appropriate stochastic mechanism,
and each particle comes to rest once it encounters an unoccupied vertex.
Examples include the binary and digital search tree processes, the random
recursive tree process and generalizations of it arising from nested instances
of Pitman's two-parameter Chinese restaurant process, tree-growth models
associated with Mallows' phi model of random permutations and with
Schuetzenberger's non-commutative q-binomial theorem, and a construction due to
Luczak and Winkler that grows uniform random binary trees in a Markovian
manner. We introduce a framework that encompasses such Markov chains, and we
characterize their asymptotic behavior by analyzing in detail their Doob-Martin
compactifications, Poisson boundaries and tail sigma-fields.Comment: 62 pages, 8 figures, revised to address referee's comment
Hardness of Token Swapping on Trees
Given a graph where every vertex has exactly one labeled token, how can we most quickly execute a given permutation on the tokens? In (sequential) token swapping, the goal is to use the shortest possible sequence of swaps, each of which exchanges the tokens at the two endpoints of an edge of the graph. In parallel token swapping, the goal is to use the fewest rounds, each of which consists of one or more swaps on the edges of a matching. We prove that both of these problems remain NP-hard when the graph is restricted to be a tree.
These token swapping problems have been studied by disparate groups of researchers in discrete mathematics, theoretical computer science, robot motion planning, game theory, and engineering. Previous work establishes NP-completeness on general graphs (for both problems), constant-factor approximation algorithms, and some poly-time exact algorithms for simple graph classes such as cliques, stars, paths, and cycles. Sequential and parallel token swapping on trees were first studied over thirty years ago (as "sorting with a transposition tree") and over twenty-five years ago (as "routing permutations via matchings"), yet their complexities were previously unknown.
We also show limitations on approximation of sequential token swapping on trees: we identify a broad class of algorithms that encompass all three known polynomial-time algorithms that achieve the best known approximation factor (which is 2) and show that no such algorithm can achieve an approximation factor less than 2
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