3 research outputs found
Lower Bounds for External Memory Integer Sorting via Network Coding
Sorting extremely large datasets is a frequently occuring task in practice.
These datasets are usually much larger than the computer's main memory; thus
external memory sorting algorithms, first introduced by Aggarwal and Vitter
(1988), are often used. The complexity of comparison based external memory
sorting has been understood for decades by now, however the situation remains
elusive if we assume the keys to be sorted are integers. In internal memory,
one can sort a set of integer keys of bits each in
time using the classic Radix Sort algorithm, however in external memory, there
are no faster integer sorting algorithms known than the simple comparison based
ones. In this paper, we present a tight conditional lower bound on the
complexity of external memory sorting of integers. Our lower bound is based on
a famous conjecture in network coding by Li and Li, who conjectured that
network coding cannot help anything beyond the standard multicommodity flow
rate in undirected graphs. The only previous work connecting the Li and Li
conjecture to lower bounds for algorithms is due to Adler et al. Adler et al.
indeed obtain relatively simple lower bounds for oblivious algorithms (the
memory access pattern is fixed and independent of the input data).
Unfortunately obliviousness is a strong limitations, especially for integer
sorting: we show that the Li and Li conjecture implies an
lower bound for internal memory oblivious sorting when the keys are bits. This is in sharp contrast to the classic (non-oblivious) Radix Sort
algorithm. Indeed going beyond obliviousness is highly non-trivial; we need to
introduce several new methods and involved techniques, which are of their own
interest, to obtain our tight lower bound for external memory integer sorting
On the Partition Bound for Undirected Unicast Network Information Capacity
One of the important unsolved problems in information theory is the
conjecture that network coding has no rate benefit over routing in undirected
unicast networks. Three known bounds on the symmetric rate in undirected
unicast information networks are the sparsest cut, the LP bound and the
partition bound. In this paper, we present three results on the partition
bound. We show that the decision version problem of computing the partition
bound is NP-complete. We give complete proofs of optimal routing schemes for
two classes of networks that attain the partition bound. Recently, the
conjecture was proved for a new class of networks and it was shown that all the
network instances for which the conjecture is proved previously are elements of
this class. We show the existence of a network for which the partition bound is
tight, achievable by routing and is not an element of this new class of
networks.Comment: ISIT'20 (c) 2020 IEEE. Personal use of this material is permitted.
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Network Coding Gaps for Completion Times of Multiple Unicasts
We study network coding gaps for the problem of makespan minimization of
multiple unicasts. In this problem distinct packets at different nodes in a
network need to be delivered to a destination specific to each packet, as fast
as possible. The network coding gap specifies how much coding packets together
in a network can help compared to the more natural approach of routing.
While makespan minimization using routing has been intensely studied for the
multiple unicasts problem, no bounds on network coding gaps for this problem
are known. We develop new techniques which allow us to upper bound the network
coding gap for the makespan of unicasts, proving this gap is at most
polylogarithmic in . Complementing this result, we show there exist
instances of unicasts for which this coding gap is polylogarithmic in .
Our results also hold for average completion time, and more generally any
norm of completion times