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Crosslinking in parallel
A crosslink is a double link established between the two entries of an edge in an adjacency list representation of a graph. Crosslinks play important roles in several parallel algorithms as they provide constant time access between the two entries of an edge; the existence of crosslinks is usually assumed. We consider the problem of establishing crosslinks in a crosslink-less adjacency list for graphs that belong to a class of graphs called the linearly contractible graphs, and show that cross-links can be established optimally in O(log n log*n) time using a CREW PRAM and optimally in O(log n) time using a CRCW PRAM for such graphs
Capacitated Vehicle Routing with Non-Uniform Speeds
The capacitated vehicle routing problem (CVRP) involves distributing
(identical) items from a depot to a set of demand locations, using a single
capacitated vehicle. We study a generalization of this problem to the setting
of multiple vehicles having non-uniform speeds (that we call Heterogenous
CVRP), and present a constant-factor approximation algorithm.
The technical heart of our result lies in achieving a constant approximation
to the following TSP variant (called Heterogenous TSP). Given a metric denoting
distances between vertices, a depot r containing k vehicles with possibly
different speeds, the goal is to find a tour for each vehicle (starting and
ending at r), so that every vertex is covered in some tour and the maximum
completion time is minimized. This problem is precisely Heterogenous CVRP when
vehicles are uncapacitated.
The presence of non-uniform speeds introduces difficulties for employing
standard tour-splitting techniques. In order to get a better understanding of
this technique in our context, we appeal to ideas from the 2-approximation for
scheduling in parallel machine of Lenstra et al.. This motivates the
introduction of a new approximate MST construction called Level-Prim, which is
related to Light Approximate Shortest-path Trees. The last component of our
algorithm involves partitioning the Level-Prim tree and matching the resulting
parts to vehicles. This decomposition is more subtle than usual since now we
need to enforce correlation between the size of the parts and their distances
to the depot
Throughput Optimal On-Line Algorithms for Advanced Resource Reservation in Ultra High-Speed Networks
Advanced channel reservation is emerging as an important feature of ultra
high-speed networks requiring the transfer of large files. Applications include
scientific data transfers and database backup. In this paper, we present two
new, on-line algorithms for advanced reservation, called BatchAll and BatchLim,
that are guaranteed to achieve optimal throughput performance, based on
multi-commodity flow arguments. Both algorithms are shown to have
polynomial-time complexity and provable bounds on the maximum delay for
1+epsilon bandwidth augmented networks. The BatchLim algorithm returns the
completion time of a connection immediately as a request is placed, but at the
expense of a slightly looser competitive ratio than that of BatchAll. We also
present a simple approach that limits the number of parallel paths used by the
algorithms while provably bounding the maximum reduction factor in the
transmission throughput. We show that, although the number of different paths
can be exponentially large, the actual number of paths needed to approximate
the flow is quite small and proportional to the number of edges in the network.
Simulations for a number of topologies show that, in practice, 3 to 5 parallel
paths are sufficient to achieve close to optimal performance. The performance
of the competitive algorithms are also compared to a greedy benchmark, both
through analysis and simulation.Comment: 9 pages, 8 figure
Fast Mixing of Parallel Glauber Dynamics and Low-Delay CSMA Scheduling
Glauber dynamics is a powerful tool to generate randomized, approximate
solutions to combinatorially difficult problems. It has been used to analyze
and design distributed CSMA (Carrier Sense Multiple Access) scheduling
algorithms for multi-hop wireless networks. In this paper we derive bounds on
the mixing time of a generalization of Glauber dynamics where multiple links
are allowed to update their states in parallel and the fugacity of each link
can be different. The results can be used to prove that the average queue
length (and hence, the delay) under the parallel Glauber dynamics based CSMA
grows polynomially in the number of links for wireless networks with
bounded-degree interference graphs when the arrival rate lies in a fraction of
the capacity region. We also show that in specific network topologies, the
low-delay capacity region can be further improved.Comment: 12 page
Scheduling MapReduce Jobs under Multi-Round Precedences
We consider non-preemptive scheduling of MapReduce jobs with multiple tasks
in the practical scenario where each job requires several map-reduce rounds. We
seek to minimize the average weighted completion time and consider scheduling
on identical and unrelated parallel processors. For identical processors, we
present LP-based O(1)-approximation algorithms. For unrelated processors, the
approximation ratio naturally depends on the maximum number of rounds of any
job. Since the number of rounds per job in typical MapReduce algorithms is a
small constant, our scheduling algorithms achieve a small approximation ratio
in practice. For the single-round case, we substantially improve on previously
best known approximation guarantees for both identical and unrelated
processors. Moreover, we conduct an experimental analysis and compare the
performance of our algorithms against a fast heuristic and a lower bound on the
optimal solution, thus demonstrating their promising practical performance
Parallel Batch-Dynamic Graph Connectivity
In this paper, we study batch parallel algorithms for the dynamic
connectivity problem, a fundamental problem that has received considerable
attention in the sequential setting. The most well known sequential algorithm
for dynamic connectivity is the elegant level-set algorithm of Holm, de
Lichtenberg and Thorup (HDT), which achieves amortized time per
edge insertion or deletion, and time per query. We
design a parallel batch-dynamic connectivity algorithm that is work-efficient
with respect to the HDT algorithm for small batch sizes, and is asymptotically
faster when the average batch size is sufficiently large. Given a sequence of
batched updates, where is the average batch size of all deletions, our
algorithm achieves expected amortized work per
edge insertion and deletion and depth w.h.p. Our algorithm
answers a batch of connectivity queries in expected
work and depth w.h.p. To the best of our knowledge, our algorithm
is the first parallel batch-dynamic algorithm for connectivity.Comment: This is the full version of the paper appearing in the ACM Symposium
on Parallelism in Algorithms and Architectures (SPAA), 201
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