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 he 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 un-weighted hi-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(vertical bar E vertical bar(2) log(2)(vertical bar V vertical bar)) time. In this paper we show that the cyclic CPP on bi-directed graphs can be solved without reducing it to hi-directed flow. We present a Theta(p(vertical bar V vertical bar+vertical bar E vertical bar) log(vertical bar V vertical bar)+(d(max)p)(3)) time algorithm to solve the cyclic CPP on a weighted bi-directed de Bruijn graph, where p = max{vertical bar{v vertical bar d(in)(v) d(out)(v) > 0}vertical bar, vertical bar{v vertical bar d(in) (v) - d(out)(v) < 0}vertical bar} and d(max) = max{vertical bar 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/vertical bar V vertical bar lies between 0.08% and 0.13% with 95% probability. Many practical hi-directed de Bruijn graphs do not have cyclic CP walks. In such cases it is not clear how the hi-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. We also present a Theta((vertical bar V vertical bar + vertical bar E vertical bar) log(V)) time algorithm for the single source shortest path problem on bi-directed de Bruijn graphs, which may be of independent interest.