Better Analysis of GREEDY Binary Search Tree on Decomposable Sequences

In their seminal paper [Sleator and Tarjan, J.ACM, 1985], the authors conjectured that the splay tree is dynamically optimal binary search tree (BST). In spite of decades of intensive research, the problem remains open. Perhaps a more basic question, which has also attracted much attention, is if there exists any dynamically optimal BST algorithm. One such candidate is GREEDY which is a simple and intuitive BST algorithm [Lucas, Rutgers Tech. Report, 1988; Munro, ESA, 2000; Demaine, Harmon, Iacono, Kane and Patrascu, SODA, 2009]. [Demaine et al., SODA, 2009] showed a novel connection between a geometric problem. Since dynamic optimality conjecture in its most general form remains elusive despite much effort, researchers have studied this problem on special sequences. Recently, [Chalermsook, Goswami, Kozma, Mehlhorn and Saranurak, FOCS, 2015] studied a type of sequences known as $k$-{\em decomposable sequences} in this context, where $k$ parametrizes easiness of the sequence. Using tools from forbidden submatrix theory, they showed that GREEDY takes $n2^{O(k^2)}$ time on this sequence and explicitly raised the question of improving this bound. In this paper, we show that GREEDY takes $O(n \log{k})$ time on $k$-decomposable sequences. In contrast to the previous approach, ours is based on first principles. One of the main ingredients of our result is a new construction of a lower bound certificate on the performance of any algorithm. This certificate is constructed using the execution of GREEDY, and is more nuanced and possibly more flexible than the previous independent set certificate of Demaine et al. This result, which is applicable to all sequences, may be of independent interest and may lead to further progress in analyzing GREEDY on $k$-decomposable as well as general sequences.