1,966 research outputs found
Bounds and exact values in network encoding complexity with two sinks
For an acyclic directed network with multiple pairs of sources and sinks and a set of Menger's paths connecting each pair of source and sink, it is well known that the number of mergings among these Menger's paths is closely related to network encoding complexity. In this paper, we focus on networks with two distinct sinks and we derive bounds on and exact values of two functions relevant to encoding complexity for such networks. © 2011 IEEE.published_or_final_versio
A Graph Theoretical Approach to Network Encoding Complexity
Consider an acyclic directed network with sources and
distinct sinks . For , let denote the
min-cut between and . Then, by Menger's theorem, there exists a
group of edge-disjoint paths from to , which will be referred
to as a group of Menger's paths from to in this paper. Although
within the same group they are edge-disjoint, the Menger's paths from different
groups may have to merge with each other. It is known that by choosing Menger's
paths appropriately, the number of mergings among different groups of Menger's
paths is always bounded by a constant, which is independent of the size and the
topology of . The tightest such constant for the all the above-mentioned
networks is denoted by when all 's are
distinct, and by when all 's are in
fact identical. It turns out that and are closely
related to the network encoding complexity for a variety of networks, such as
multicast networks, two-way networks and networks with multiple sessions of
unicast. Using this connection, we compute in this paper some exact values and
bounds in network encoding complexity using a graph theoretical approach.Comment: 44 pages, 22 figure
A graph theoretical approach to network encoding complexity
For an acyclic directed network with multiple pairs of sources and sinks and a group of edge-disjoint paths connecting each pair of source and sink, it is known that the number of mergings among different groups of edge-disjoint paths is closely related to network encoding complexity. Using this connection, we derive exact values of and bounds on two functions relevant to encoding complexity for such networks. © 2012 IEICE Institute of Electronics Informati.published_or_final_versio
Quantum Network Coding
Since quantum information is continuous, its handling is sometimes
surprisingly harder than the classical counterpart. A typical example is
cloning; making a copy of digital information is straightforward but it is not
possible exactly for quantum information. The question in this paper is whether
or not quantum network coding is possible. Its classical counterpart is another
good example to show that digital information flow can be done much more
efficiently than conventional (say, liquid) flow.
Our answer to the question is similar to the case of cloning, namely, it is
shown that quantum network coding is possible if approximation is allowed, by
using a simple network model called Butterfly. In this network, there are two
flow paths, s_1 to t_1 and s_2 to t_2, which shares a single bottleneck channel
of capacity one. In the classical case, we can send two bits simultaneously,
one for each path, in spite of the bottleneck. Our results for quantum network
coding include: (i) We can send any quantum state |psi_1> from s_1 to t_1 and
|psi_2> from s_2 to t_2 simultaneously with a fidelity strictly greater than
1/2. (ii) If one of |psi_1> and |psi_2> is classical, then the fidelity can be
improved to 2/3. (iii) Similar improvement is also possible if |psi_1> and
|psi_2> are restricted to only a finite number of (previously known) states.
(iv) Several impossibility results including the general upper bound of the
fidelity are also given.Comment: 27pages, 11figures. The 12page version will appear in 24th
International Symposium on Theoretical Aspects of Computer Science (STACS
2007
Quantum network communication -- the butterfly and beyond
We study the k-pair communication problem for quantum information in networks
of quantum channels. We consider the asymptotic rates of high fidelity quantum
communication between specific sender-receiver pairs. Four scenarios of
classical communication assistance (none, forward, backward, and two-way) are
considered. (i) We obtain outer and inner bounds of the achievable rate regions
in the most general directed networks. (ii) For two particular networks
(including the butterfly network) routing is proved optimal, and the free
assisting classical communication can at best be used to modify the directions
of quantum channels in the network. Consequently, the achievable rate regions
are given by counting edge avoiding paths, and precise achievable rate regions
in all four assisting scenarios can be obtained. (iii) Optimality of routing
can also be proved in classes of networks. The first class consists of directed
unassisted networks in which (1) the receivers are information sinks, (2) the
maximum distance from senders to receivers is small, and (3) a certain type of
4-cycles are absent, but without further constraints (such as on the number of
communicating and intermediate parties). The second class consists of arbitrary
backward-assisted networks with 2 sender-receiver pairs. (iv) Beyond the k-pair
communication problem, observations are made on quantum multicasting and a
static version of network communication related to the entanglement of
assistance.Comment: 15 pages, 17 figures. Final versio
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