Co-linear Chaining with Overlaps and Gap Costs

Abstract

Motivation: Co-linear chaining has proven to be a powerful technique for finding approximately optimal alignments and approximating edit distance. It is used as an intermediate step in numerous mapping tools that follow seed-and-extend strategy. Despite this popularity, subquadratic time algorithms for the case where chains support anchor overlaps and gap costs are not currently known. Moreover, a theoretical connection between co-linear chaining cost and edit distance remains unknown. Results: We present algorithms to solve the co-linear chaining problem with anchor overlaps and gap costs in Õ(n) time, where n denotes the count of anchors. We establish the first theoretical connection between co-linear chaining cost and edit distance. Specifically, we prove that for a fixed set of anchors under a carefully designed chaining cost function, the optimal 'anchored' edit distance equals the optimal co-linear chaining cost. Finally, we demonstrate experimentally that optimal co-linear chaining cost under the proposed cost function can be computed significantly faster than edit distance, and achieves high correlation with edit distance for closely as well as distantly related sequences.

Sequence aligners can guarantee accuracy in almostO(mlogn) time: a rigorous average-case analysis of the seed-chain-extend heuristic

“…We only use k-mer seeds in this study, although other types of seeds are possible (Keich et al, 2004;Kie lbasa et al, 2011). An optimal increasing subsequence of possibly overlapping anchors based on some score is then collected into a chain, where increasing is defined with the standard precedence relationship (Jain et al, 2022) between k-mer anchors (See Figure 5a and Chaining below). The chain is extended into a full alignment by aligning between anchor gaps in the chain.…”

“…Extension and chaining runtimes Given sorted anchors, let T Chain be the time spent finding an optimal chain. T Chain depends on the objective function (Mäkinen and Sahlin, 2020;Jain et al, 2022;Abouelhoda and Ohlebusch, 2005;Otto et al, 2011). Since our gap costs are linear, T Chain = O(N log N ) where N is the number of anchors (Abouelhoda and Ohlebusch, 2005).…”

Chaining of Maximal Exact Matches in Graphs

“…Namely, we obtain an O(m + n + k 2 |V | + |E| + kN log N ) time algorithm for computing a longest common subsequence (LCS) between a query string Q and a path of G, where m = |Q|, n is the total length of node labels, k is the width (minimum number of paths covering the nodes) of G, and N is the number of maximal exact matches (MEMs) between Q and the node labels (node MEMs). For the case with two strings as input, a recent formulation of co-linear chaining [19] captures unit cost edit distance. There has been an attempt to extend the results to graphs considering gap costs [6], but it appears difficult to make such formulation fully symmetric (due to there being exponential many paths between two anchors).…”

Sequence to graph alignment using gap-sensitive co-linear chaining

“…Co-linear chaining is a mathematically rigorous approach to do clustering of anchors. It is well studied for the case of sequence-to-sequence alignment [1,11,12,16,30,34,43], and is widely used in present-day long read to reference sequence aligners [18,23,38,40].…”

“…However, the problem formulations in these works did not include gap cost. Without penalizing gaps, co-linear chaining is less effective [16]. A challenge in enforcing gap cost is that measuring gap between two loci in a DAG is not a constant-time operation like in a sequence.…”