1,483 research outputs found
Cluster Before You Hallucinate: Approximating Node-Capacitated Network Design and Energy Efficient Routing
We consider circuit routing with an objective of minimizing energy, in a
network of routers that are speed scalable and that may be shutdown when idle.
We consider both multicast routing and unicast routing. It is known that this
energy minimization problem can be reduced to a capacitated flow network design
problem, where vertices have a common capacity but arbitrary costs, and the
goal is to choose a minimum cost collection of vertices whose induced subgraph
will support the specified flow requirements. For the multicast (single-sink)
capacitated design problem we give a polynomial-time algorithm that is
O(log^3n)-approximate with O(log^4 n) congestion. This translates back to a
O(log ^(4{\alpha}+3) n)-approximation for the multicast energy-minimization
routing problem, where {\alpha} is the polynomial exponent in the dynamic power
used by a router. For the unicast (multicommodity) capacitated design problem
we give a polynomial-time algorithm that is O(log^5 n)-approximate with
O(log^12 n) congestion, which translates back to a O(log^(12{\alpha}+5)
n)-approximation for the unicast energy-minimization routing problem.Comment: 22 pages (full version of STOC 2014 paper
A Primal-Dual Algorithm for Multicommodity Flows and Multicuts in Treewidth-2 Graphs
We study the problem of multicommodity flow and multicut in treewidth-2 graphs and prove bounds on the multiflow-multicut gap. In particular, we give a primal-dual algorithm for computing multicommodity flow and multicut in treewidth-2 graphs and prove the following approximate max-flow min-cut theorem: given a treewidth-2 graph, there exists a multicommodity flow of value f with congestion 4, and a multicut of capacity c such that c ? 20 f. This implies a multiflow-multicut gap of 80 and improves upon the previous best known bounds for such graphs. Our algorithm runs in polynomial time when all the edges have capacity one. Our algorithm is completely combinatorial and builds upon the primal-dual algorithm of Garg, Vazirani and Yannakakis for multicut in trees and the augmenting paths framework of Ford and Fulkerson
Routing algorithms for recursively-defined data centre networks
The server-centric data centre network architecture can accommodate a wide variety of network topologies. Newly proposed topologies in this arena often require several rounds of analysis and experimentation in order that they might achieve their full potential as data centre networks. We propose a family of novel routing algorithms on two well-known data centre networks of this type, (Generalized) DCell and FiConn, using techniques that can be applied more generally to the class of networks we call completely connected recursively-defined networks. In doing so, we develop a classification of all possible routes from server-node to server-node on these networks, called general routes of order t, and find that for certain topologies of interest, our routing algorithms efficiently produce paths that are up to 16% shorter than the best previously known algorithms, and are comparable to shortest paths. In addition to finding shorter paths, we show evidence that our algorithms also have good load-balancing properties
ILP formulations for p-cycle design without candidate cycle enumeration
The concept of p-cycle (preconfigured protection cycle) allows fast and efficient span protection in wavelength division multiplexing (WDM) mesh networks. To design p-cycles for a given network, conventional algorithms need to enumerate cycles in the network to form a candidate set, and then use an integer linear program (ILP) to find a set of p-cycles from the candidate set. Because the size of the candidate set increases exponentially with the network size, candidate cycle enumeration introduces a huge number of ILP variables and slows down the optimization process. In this paper, we focus on p-cycle design without candidate cycle enumeration. Three ILPs for solving the problem of spare capacity placement (SCP) are first formulated. They are based on recursion, flow conservation, and cycle exclusion, respectively. We show that the number of ILP variables/constraints in our cycle exclusion approach only increases linearly with the network size. Then, based on cycle exclusion, we formulate an ILP for solving the joint capacity placement (JCP) problem. Numerical results show that our ILPs are very efficient in generating p-cycle solutions. © 2009 IEEE.published_or_final_versio
Exact Distance Oracles for Planar Graphs
We present new and improved data structures that answer exact node-to-node
distance queries in planar graphs. Such data structures are also known as
distance oracles. For any directed planar graph on n nodes with non-negative
lengths we obtain the following:
* Given a desired space allocation , we show how to
construct in time a data structure of size that answers
distance queries in time per query.
As a consequence, we obtain an improvement over the fastest algorithm for
k-many distances in planar graphs whenever .
* We provide a linear-space exact distance oracle for planar graphs with
query time for any constant eps>0. This is the first such data
structure with provable sublinear query time.
* For edge lengths at least one, we provide an exact distance oracle of space
such that for any pair of nodes at distance D the query time is
. Comparable query performance had been observed
experimentally but has never been explained theoretically.
Our data structures are based on the following new tool: given a
non-self-crossing cycle C with nodes, we can preprocess G in
time to produce a data structure of size that can
answer the following queries in time: for a query node u, output
the distance from u to all the nodes of C. This data structure builds on and
extends a related data structure of Klein (SODA'05), which reports distances to
the boundary of a face, rather than a cycle.
The best distance oracles for planar graphs until the current work are due to
Cabello (SODA'06), Djidjev (WG'96), and Fakcharoenphol and Rao (FOCS'01). For
and space , we essentially improve the query
time from to .Comment: To appear in the proceedings of the 23rd ACM-SIAM Symposium on
Discrete Algorithms, SODA 201
Maximum Edge-Disjoint Paths in -sums of Graphs
We consider the approximability of the maximum edge-disjoint paths problem
(MEDP) in undirected graphs, and in particular, the integrality gap of the
natural multicommodity flow based relaxation for it. The integrality gap is
known to be even for planar graphs due to a simple
topological obstruction and a major focus, following earlier work, has been
understanding the gap if some constant congestion is allowed.
In this context, it is natural to ask for which classes of graphs does a
constant-factor constant-congestion property hold. It is easy to deduce that
for given constant bounds on the approximation and congestion, the class of
"nice" graphs is nor-closed. Is the converse true? Does every proper
minor-closed family of graphs exhibit a constant factor, constant congestion
bound relative to the LP relaxation? We conjecture that the answer is yes.
One stumbling block has been that such bounds were not known for bounded
treewidth graphs (or even treewidth 3). In this paper we give a polytime
algorithm which takes a fractional routing solution in a graph of bounded
treewidth and is able to integrally route a constant fraction of the LP
solution's value. Note that we do not incur any edge congestion. Previously
this was not known even for series parallel graphs which have treewidth 2. The
algorithm is based on a more general argument that applies to -sums of
graphs in some graph family, as long as the graph family has a constant factor,
constant congestion bound. We then use this to show that such bounds hold for
the class of -sums of bounded genus graphs
Path-Based partitioning methods for 3D Networks-on-Chip with minimal adaptive routing
© 2014 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.Combining the benefits of 3D ICs and Networks-on-Chip (NoCs) schemes provides a significant performance gain in Chip Multiprocessors (CMPs) architectures. As multicast communication is commonly used in cache coherence protocols for CMPs and in various parallel applications, the performance of these systems can be significantly improved if multicast operations are supported at the hardware level. In this paper, we present several partitioning methods for the path-based multicast approach in 3D mesh-based NoCs, each with different levels of efficiency. In addition, we develop novel analytical models for unicast and multicast traffic to explore the efficiency of each approach. In order to distribute the unicast and multicast traffic more efficiently over the network, we propose the Minimal and Adaptive Routing (MAR) algorithm for the presented partitioning methods. The analytical and experimental results show that an advantageous method named Recursive Partitioning (RP) outperforms the other approaches. RP recursively partitions the network until all partitions contain a comparable number of switches and thus the multicast traffic is equally distributed among several subsets and the network latency is considerably decreased. The simulation results reveal that the RP method can achieve performance improvement across all workloads while performance can be further improved by utilizing the MAR algorithm. Nineteen percent average and 42 percent maximum latency reduction are obtained on SPLASH-2 and PARSEC benchmarks running on a 64-core CMP.Ebrahimi, M.; Daneshtalab, M.; Liljeberg, P.; Plosila, J.; Flich Cardo, J.; Tenhunen, H. (2014). Path-Based partitioning methods for 3D Networks-on-Chip with minimal adaptive routing. IEEE Transactions on Computers. 63(3):718-733. doi:10.1109/TC.2012.255S71873363
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