  The Ulam's metric is the minimal number of moves consisting in removal of one
element from a permutation and its subsequent reinsertion in different place,
to go between two given permutations. Thet elements that are not moved create
longest common subsequence of permutations. Aldous and Diaconis, in their
paper, pointed that Ulam's metric had been introduced in the context of
questions concerning sorting and tossing cards. In this paper we define and
study Ulam's metric in highier dimensions: for dimension one the considered
object is a pair of permutations, for dimension k it is a pair of k-tuples of
permutations. Over encodings by k-tuples of permutations we define two dually
related hierarchies. Our very first motivation come from Murata at al. paper,
in which pairs of permutations were used as representation of topological
relation between rectangles packed into minimal area with application to VLSI
physical design. Our results concern hardness, approximability, and
parametrized complexity inside the hierarchies.
