ISSN 0032-9460, Problems of Information Transmission, 2017, Vol. 53, No. 1, pp. 55–72.
Pleiades Publishing, Inc., 2017.
Original Russian Text
K.Yu. Gorbunov, V.A. Lyubetsky, 2017, published in Problemy Peredachi Informatsii, 2017, Vol. 53, No. 1, pp. 60–78.
Linear Algorithm for Minimal
Rearrangement of Structures
K. Yu. Gorbunov
and V. A. Lyubetsky
Kharkevich Institute for Information Transmission Problems,
Russian Academy of Sciences, Moscow, Russia
Received December 29, 2014; in ﬁnal form, April 25, 2016
Abstract—We propose a linear time and linear space algorithm which constructs a minimal
sequence of operations rearranging one structure (directed graph of cycles and paths) into
another. Structures in such a sequence may have a varying number of edges; a list of operations
is ﬁxed and includes deletion and insertion of a fragment of a structure. We give a complete
proof that the algorithm is correct, i.e., ﬁnds the corresponding minimum.
In Section 2 we present formulations and deﬁnitions related to the problem considered in the
paper and problems close to it. There are many publications devoted to such problems, among
which we note the fundamental work . Below we only mention some papers most close to the
present paper; see also . These problems are evoked by biological and medical subjects, so terms
from these areas are commonly used in their settings. However, we consider the mathematical
component of these problems only; the obtained solutions can also be applied, for example, in
In  there were proposed operations for rearranging chromosome structure (see Section 2); in
what follows they are referred to as standard operations and are a part of our more general set
of operations. In  there is also given an algorithm to compute the number of operations in the
minimal (in the number of operations) sequence transforming one structure into another if the
structures consist of paths (“linear chromosomes”) only; the algorithm is almost linear in time, and
no estimate for the runtime is presented. These operations being applied to paths only correspond
to reversals, translocations, fusions, and ﬁssions previously considered in . In the case where
all intermediate structures also consist of linear chromosomes, an algorithm for computing the
minimum number of operation is also given in . The general case of genomes having equal gene
content and equal cost of operations was solved in  using ideas from [3, 4].
In the case of genomes with unequal gene content one needs additional operations: deletion and
insertion of strings of unique genes, which were proposed in [6, 7].
Papers [6–8] uses the notion of an adjacency graph: its vertices are adjacencies of extremities
(heads and tails) of genes that belong to both structures and ends of the original paths. Two
adjacencies from diﬀerent structures are joined by an edge if they represent the head or tail of
the same gene. Furthermore, an end of the original path is assumed to be adjacent to an “empty
extremity” (telomere); between adjacent extremities of common genes of two structures there can
The research was carried out at the Institute for Information Transmission Problems of the Russian
Academy of Sciences at the expense of the Russian Science Foundation, project no. 14-50-00150.