1,412 research outputs found
An ETH-Tight Exact Algorithm for Euclidean TSP
We study exact algorithms for {\sc Euclidean TSP} in . In the
early 1990s algorithms with running time were presented for
the planar case, and some years later an algorithm with
running time was presented for any . Despite significant interest in
subexponential exact algorithms over the past decade, there has been no
progress on {\sc Euclidean TSP}, except for a lower bound stating that the
problem admits no algorithm unless ETH fails. Up to
constant factors in the exponent, we settle the complexity of {\sc Euclidean
TSP} by giving a algorithm and by showing that a
algorithm does not exist unless ETH fails.Comment: To appear in FOCS 201
Parity balance of the -th dimension edges in Hamiltonian cycles of the hypercube
Let be an integer, and let . An -th dimension
edge in the -dimensional hypercube is an edge such that
differ just at their -th entries. The parity of an -th
dimension edge \edg{v_1}{v_2} is the number of 1's modulus 2 of any of its
vertex ignoring the -th entry. We prove that the number of -th dimension
edges appearing in a given Hamiltonian cycle of with parity zero
coincides with the number of edges with parity one. As an application of this
result it is introduced and explored the conjecture of the inscribed squares in
Hamiltonian cycles of the hypercube: Any Hamiltonian cycle in contains
two opposite edges in a 4-cycle. We prove this conjecture for , and
for any Hamiltonian cycle containing more than edges in the same
dimension. This bound is finally improved considering the equi-independence
number of , which is a concept introduced in this paper for bipartite
graphs
Architectural Considerations for a Self-Configuring Routing Scheme for Spontaneous Networks
Decoupling the permanent identifier of a node from the node's
topology-dependent address is a promising approach toward completely scalable
self-organizing networks. A group of proposals that have adopted such an
approach use the same structure to: address nodes, perform routing, and
implement location service. In this way, the consistency of the routing
protocol relies on the coherent sharing of the addressing space among all nodes
in the network. Such proposals use a logical tree-like structure where routes
in this space correspond to routes in the physical level. The advantage of
tree-like spaces is that it allows for simple address assignment and
management. Nevertheless, it has low route selection flexibility, which results
in low routing performance and poor resilience to failures. In this paper, we
propose to increase the number of paths using incomplete hypercubes. The design
of more complex structures, like multi-dimensional Cartesian spaces, improves
the resilience and routing performance due to the flexibility in route
selection. We present a framework for using hypercubes to implement indirect
routing. This framework allows to give a solution adapted to the dynamics of
the network, providing a proactive and reactive routing protocols, our major
contributions. We show that, contrary to traditional approaches, our proposal
supports more dynamic networks and is more robust to node failures
Embedding multidimensional grids into optimal hypercubes
Let and be graphs, with , and a one to one map of their vertices. Let , where is the distance
between vertices and of . Now let = , over all such maps .
The parameter is a generalization of the classic and well studied
"bandwidth" of , defined as , where is the path on
points and . Let
be the -dimensional grid graph with integer values through in
the 'th coordinate. In this paper, we study in the case when and is the hypercube
of dimension , the hypercube of
smallest dimension having at least as many points as . Our main result is
that
provided for each . For such , the bound
improves on the previous best upper bound . Our methods include
an application of Knuth's result on two-way rounding and of the existence of
spanning regular cyclic caterpillars in the hypercube.Comment: 47 pages, 8 figure
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