AN EFFICIENT ALGORITHM FOR CHINESE POSTMAN WALK ON BI-DIRECTED DE BRUIJN GRAPHS

Sequence assembly from short reads is an important problem in biology. It is known that solving the sequence assembly problem exactly on a bi-directed de Bruijn graph or a string graph is intractable. However, finding a shortest double stranded DNA string (SDDNA) containing all the k-long words in the reads seems to be a good heuristic to get close to the original genome. This problem is equivalent to finding a cyclic Chinese Postman (CP) walk on the underlying unweightedbi- directed de Bruijn graph built from the reads. The Chinese Postman walk Problem (CPP) is solved by reducing it to a general bi-directed flow on this graph which runs in O(|E| (2)log(2)(|V |)) time. In this paper we show that the cyclic CPP on bi-directed graphs can be solved without reducing it to bi-directed flow. We present a Xi(p(|V |+| E|) log(|V |)+(dmaxp)(3)) time algorithm to solve the cyclic CPP on a weighted bi-directed de Bruijn graph, where p = max{|{v| d(in)(v) - d(out)( v) > 0}|, |{v| d(in)(v) - d(out)(v) < 0}|} and d(max) = max{| d(in)(v) - d(out)(v)}. Our algorithm performs asymptotically better than the bi-directed flow algorithm when the number of imbalanced nodes p is much less than the nodes in the bi-directed graph. From our experimental results on various datasets, we have noticed that the value of p/|V | lies between 0.08% and 0.13% with 95% probability. Many practical bi-directed de Bruijn graphs do not have cyclic CP walks. In such cases it is not clear how the bi-directed flow can be useful in identifying contigs. Our algorithm can handle such situations and identify maximal bi-directed sub-graphs that have CP walks. A Xi(p(|V | + | E|)) time heuristic algorithm based on these ideas has been implemented for the SDDNA problem. This algorithm was tested on short reads from a plant genome and achieves an approximation ratio of at most 1.0134. We also present a Xi((|V | + | E|) log(V)) time algorithm for the single source shortest path problem on bi-directed de Bruijn graphs, which may be of independent interest.