2,406 research outputs found
Polynomial time algorithms for multicast network code construction
The famous max-flow min-cut theorem states that a source node s can send information through a network (V, E) to a sink node t at a rate determined by the min-cut separating s and t. Recently, it has been shown that this rate can also be achieved for multicasting to several sinks provided that the intermediate nodes are allowed to re-encode the information they receive. We demonstrate examples of networks where the achievable rates obtained by coding at intermediate nodes are arbitrarily larger than if coding is not allowed. We give deterministic polynomial time algorithms and even faster randomized algorithms for designing linear codes for directed acyclic graphs with edges of unit capacity. We extend these algorithms to integer capacities and to codes that are tolerant to edge failures
Reconstructing Generalized Staircase Polygons with Uniform Step Length
Visibility graph reconstruction, which asks us to construct a polygon that
has a given visibility graph, is a fundamental problem with unknown complexity
(although visibility graph recognition is known to be in PSPACE). We show that
two classes of uniform step length polygons can be reconstructed efficiently by
finding and removing rectangles formed between consecutive convex boundary
vertices called tabs. In particular, we give an -time reconstruction
algorithm for orthogonally convex polygons, where and are the number of
vertices and edges in the visibility graph, respectively. We further show that
reconstructing a monotone chain of staircases (a histogram) is fixed-parameter
tractable, when parameterized on the number of tabs, and polynomially solvable
in time under reasonable alignment restrictions.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Reconstructing a phylogenetic level-1 network from quartets
We describe a method that will reconstruct an unrooted binary phylogenetic
level-1 network on n taxa from the set of all quartets containing a certain
fixed taxon, in O(n^3) time. We also present a more general method which can
handle more diverse quartet data, but which takes O(n^6) time. Both methods
proceed by solving a certain system of linear equations over GF(2).
For a general dense quartet set (containing at least one quartet on every
four taxa) our O(n^6) algorithm constructs a phylogenetic level-1 network
consistent with the quartet set if such a network exists and returns an (O(n^2)
sized) certificate of inconsistency otherwise. This answers a question raised
by Gambette, Berry and Paul regarding the complexity of reconstructing a
level-1 network from a dense quartet set
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