A novel encoding method into sequence-pair

The sequence-pair was proposed to represent a rectangle packing and a placement, and is used to place modules automatically in VLSI layout design. Several decoding methods of sequence-pair were proposed. However, encoding methods are not found except the original one called "gridding". The gridding requires almost O(n³) time for a packing of n rectangular modules and it is hard to implement. Therefore, we propose a novel method to encode a given rectangle packing into a sequence-pair in O(n log n) time. We also propose a linear time method to obtain a sequence-pair from a given rectangular dissection represented by a Q-sequence, a recently proposed representation method of rectangular dissection. The proposed methods can be used for the compaction keeping topology, for example, in the post-process of the force directed relaxation, a method used in module placement, and so on.