Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Arranging numbers on circles to reach maximum total variations. | Arranging numbers on circles to reach maximum total variations Ying-Jie Liao Min-Zheng Shieh Shi-Chun Tsai Department of Computer Science National Chiao Tung University Hsinchu 30050 Taiwan yjliao mzhsieh sctsai @ Submitted Jan 15 2007 Accepted Jun 10 2007 Published Jun 28 2007 Mathematics Subject Classification 05A05 05B30 Abstract The dartboard problem is to arrange n numbers on a circle to obtain maximum risk which is the sum of the q-th power of the absolute differences of adjacent numbers for q 1. Curtis showed that the dartboard problem admits a greedy algorithm. We generalize the dartboard problem by considering more circles and the goal is to arrange kn number on k circles to obtain the maximum risk. In this paper we characterize an optimal arrangement for k 2 and show that the generalized dartboard problem also admits a greedy algorithm. 1 Introduction Darts is a very popular game. Players throw darts and score points corresponding to the sector the darts just landed on. The traditional dartboard is circular and partitioned into several sectors as shown in figure 1. When playing darts players often aim at the high score sectors. But for ordinary players it is hard to land the dart on the desired sectors. The risk of aiming at an area can be measured by the difference between the scores of adjacent sectors. As the larger the difference is the higher the risk is and the game becomes more challenging. The total risk of a dartboard is the sum over the risks of all sectors. The so called dartboard problem as discussed in Curtis paper 4 is to find a cyclic permutation T n1 an of a multiset a15 ang on a circle which maximizes the risk function L2n 1 l - ai-1 lq where n0 nn and q 1. The dartboard problem has been studied for a while. Eiselt and Laporte 5 used a branch-and-bound algorithm 1 to find optimal permutations for the dartboard problem on 1 2 . 20g for q 1 and q 2 and they observed that the traditional dartboard score arrangement is not .
