Given a box of some specified size and a number of pieces of some specified shape, the anti-slide problem considers how to pack the pieces such that none of the pieces in the box can slide in any direction. The object is to find such a sparsest packing. In this paper, we consider the problem for the case of a two-dimensional square box using T-tetromino pieces. We show that, for a square box of side length n, the number of pieces in a sparsest packing is exactly $lfloor 2n/3 floor$ when n≢0 (mod 3), and is between 2n/3-1 and n-1 when n≡0 (mod 3).

The online interval coloring problem has been extensively studied for many years. Kierstead and Trotter (Congressus Numerantium 33, 1981) proved that their algorithm is an optimal online algorithm for this problem. The number of colors used by the algorithm is at most 3ω(G)-2, where ω(G) is the size of the maximum clique in a given graph G. Also, they presented an instance for which the number of colors used by any online algorithm is at least 3ω(G)-2. This instance includes intervals with various lengths, which cannot be applied to the case when the lengths of the given intervals are restricted to one, i.e., the online unit interval coloring problem. In this case, the current best upper and lower bounds on the number of colors used by an online algorithm are 2ω(G)-1 and 3ω(G)/2 respectively by Epstein and Levy (ICALP2005). In this letter, we conduct a complete performance analysis of the Kierstead-Trotter algorithm for online unit interval coloring, and prove it is NOT optimal. Specifically, we provide an upper bound of 3ω(G)-3 on the number of colors used by their algorithm. Moreover, the bound is the best possible.

Given a set of n disjoint intervals on a line and an integer k, we want to find k points in the intervals so that the minimum pairwise distance of the k points is maximized. Intuitively, given a set of n disjoint time intervals on a timeline, each of which is a time span we are allowed to check something, and an integer k, which is the number of times we will check something, we plan k checking times so that the checks occur at equal time intervals as much as possible, that is, we want to maximize the minimum time interval between the k checking times. We call the problem the k-dispersion problem on intervals. If we need to choose exactly one point in each interval, so k=n, and the disjoint intervals are given in the sorted order on the line, then two O(n) time algorithms to solve the problem are known. In this paper we give the first O(n) time algorithm to solve the problem for any constant k. Our algorithm works even if the disjoint intervals are given in any (not sorted) order. If the disjoint intervals are given in the sorted order on the line, then, by slightly modifying the algorithm, one can solve the problem in O(log n) time. This is the first sublinear time algorithm to solve the problem. Also we show some results on the k-dispersion problem on disks, including an FPTAS.

The dispersion problem is a variant of the facility location problem. Given a set P of n points and an integer k, we intend to find a subset S of P with |S|=k such that the cost minp∈S{cost(p)} is maximized, where cost(p) is the sum of the distances from p to the nearest c points in S. We call the problem the dispersion problem with partial c sum cost, or the PcS-dispersion problem. In this paper we present two algorithms to solve the P2S-dispersion problem(c=2) if all points of P are on a line. The running times of the algorithms are O(kn2 log n) and O(n log n), respectively. We also present an algorithm to solve the PcS-dispersion problem if all points of P are on a line. The running time of the algorithm is O(knc+1).

The online unit clustering problem is one of the most basic clustering problems proposed by Chan and Zarrabi-Zadeh (WAOA2007 and Theory of Computing Systems 45(3), 2009). Several variants of this problem have been extensively studied. In this letter, we propose a new variant of the online unit clustering problem, called the online unit clustering problem with capacity constraints. For this problem, we use competitive analysis to evaluate the performance of an online algorithm. Then, we develop an online algorithm whose competitive ratio is at most 3.178, and show that a lower bound on the competitive ratio of any online algorithm is 2.