As described in the previous section, a linear ordering of the contigs, which is optimal with respect to the adjacency edge weights, can be computed using a suitable optimization algorithm.
We require first that each adjacency edge be chosen: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$x_{e} = 1\:, \qquad \forall~e \in E_{a}\:.
For the remaining edges let e = { v i, v j } ∈ A with v i ∈ { l i, r i } and v j ∈ { l j, r j } be an adjacency between contigs c i and c j. Then the weight of this adjacency edge is defined as where the (symmetric) function w r (v i, v j ) defines a likelihood score for the contigs c i and c j being adjacent, with respect to their connectors v i and v j.
A compound graph (Fig. 2) C = (V, E, F) consists of nodes V, adjacency edges E and inclusion edges F (Dogrusoz et al., 2009).
If a breakpoint corresponds to any aberrant adjacency, edges are also added to E L. If a gene duplication has occurred in the target genome, it causes an increased copy number and aberrant adjacencies flanking the gene.
Matching edges have weights defined by the normalized similarity σ, all adjacency edges have weight 1, and all self edges have weight 0. Notice that any edge in G corresponds to two matching edges in H. Now we describe the ILP.
We split these edges into two subsets: the intra contig edges I ={{ l1, r1},..., { l n, r n }} which connect for each contig its left and its right connector; and the set of adjacency edges A = E\ I that connect the contigs among each other.
(Q G)) is the graph obtained by inserting a new vertex into each edge of G, then joining with edges those pairs of new vertices on adjacent edges of G. (T(G)) has as its vertices, the edges and vertices of G. Adjacency in (T(G)) is defined as adjacency or incidence for the corresponding elements of G.
(Q(G)) is obtained from G by inserting a new vertex into each edge of G, then joining with edges those pairs of new vertices on adjacent edges of G. (T(G)) has as its vertices, the edges and vertices of G. Adjacency in (T(G)) is defined as adjacency or incidence for the corresponding elements of G.
(3) (Q G)) is the graph obtained by inserting a new vertex into each edge of G, then joining with edges those pairs of new vertices on adjacent edges of G. (4) (T(G)) has as its vertices, the edges and vertices of G. Adjacency in (T(G)) is defined as adjacency or incidence for the corresponding elements of G.
To abstract BP algorithm into RDDs, our key observation is that the message update process Equation 1 can be more efficiently represented by RDDs on an induced line graph from the original graph which represents the adjacencies between edges of original graph.
