Selected sequence-pair: an efficient decodable packing representation in linear time using sequence-pair

In this paper, we propose "selected sequence-pair" (SSP), a sequence-pair (seq-pair) with the limited number of subsequences called adjacent crosses. Its features are: (1) The smallest packing based on a given SSP can be obtained in O(n) time, where n is the number of rectangles. (2) An arbitrary packing can be represented by SSP. (3) The total representation number of SSP of size n is not more than that of rectangular dissection of the same size with n-└√(4n-1)┘ empty rooms (the necessary number of empty rooms to represent an arbitrary packing). To realize these features of SSP, we propose an algorithm to enumerate all adjacent crosses on a seq-pair in linear time of n + k (k is the number of adjacent crosses). Also we apply a conventional method to convert a seq-pair without adjacent crosses to an equivalent Q-sequence, representation of rectangular dissection, in O(n + k) time. A move operation to obtain an adjacent solution efficiently is proposed to perturb SSP for Simulated Annealing (SA). From experimental results, we confirmed the proposed method was carried out in linear time and was more efficient than the conventional method when SSP size got bigger.