Availability of extensive genetics data across multiple individuals and populations is driving the growing importance of graph based reference representations. Aligning sequences to graphs is a fundamental operation on several types of sequence graphs (variation graphs, assembly graphs, pan-genomes, etc.) and their biological applications. Though research on sequence to graph alignments is nascent, it can draw from related work on pattern matching in hypertext. In this paper, we study sequence to graph alignment problems under Hamming and edit distance models, and linear and affine gap penalty functions, for multiple variants of the problem that allow changes in query alone, graph alone, or in both. We prove that when changes are permitted in graphs either standalone or in conjunction with changes in the query, the sequence to graph alignment problem is N P-complete under both Hamming and edit distance models for alphabets of size ≥ 2. For the case where only changes to the sequence are permitted, we present an O(|V | + m|E|) time algorithm, where m denotes the query size, and V and E denote the vertex and edge sets of the graph, respectively. Our result is generalizable to both linear and affine gap penalty functions, and improves upon the run-time complexity of existing algorithms.
We study dynamic algorithms for maintaining spectral vertex sparsifiers of graphs with respect to a set of terminals T of our choice. Such objects preserve pairwise resistances, solutions to systems of linear equations, and energy of electrical flows between the terminals in T . We give a data structure that supports insertions and deletions of edges, and terminal additions, all in sublinear time. We then show the applicability of our result to the following problems.(1) A data structure for dynamically maintaining solutions to Laplacian systems Lx = b, where L is the graph Laplacian matrix and b is a demand vector. For a bounded degree, unweighted graph, we support modifications to both L and b while providing access to ǫapproximations to the energy of routing an electrical flow with demand b, as well as query access to entries of a vectorx such that x − L † b L ≤ ǫ L † b L inÕ(n 11/12 ǫ −5 ) expected amortized update and query time.(2) A data structure for maintaining fully dynamic All-Pairs Effective Resistance. For an intermixed sequence of edge insertions, deletions, and resistance queries, our data structure returns (1 ± ǫ)-approximation to all the resistance queries against an oblivious adversary with high probability. Its expected amortized update and query times areÕ(min(m 3/4 , n 5/6 ǫ −2 )ǫ −4 ) on an unweighted graph, andÕ(n 5/6 ǫ −6 ) on weighted graphs.The key ingredients in these results are (1) the intepretation of Schur complement as a sum of random walks, and (2) a suitable choice of terminals based on the behavior of these random walks to make sure that the majority of walks are local, even when the graph itself is highly connected and (3) maintenance of these local walks and numerical solutions using data structures.These results together represent the first data structures for maintaining key primitives from the Laplacian paradigm for graph algorithms in sublinear time without assumptions on the underlying graph topologies. The importance of routines such as effective resistance, electrical flows, and Laplacian solvers in the static setting make us optimistic that some of our components can provide new building blocks for dynamic graph algorithms.
In this paper, we prove convergence of a sticky particle method for the modified Camassa-Holm equation (mCH) with cubic nonlinearity in one dimension. As a byproduct, we prove global existence of weak solutions u with regularity: u and ux are space-time BV functions. The total variation of m(•, t) = u(•, t) − uxx(•, t) is bounded by the total variation of the initial data m 0. We also obtain W 1,1 (R)-stability of weak solutions when solutions are in L ∞ (0, ∞; W 2,1 (R)). (Notice that peakon weak solutions are not in W 2,1 (R).) Finally, we provide some examples of nonuniqueness of peakon weak solutions to the mCH equation.
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