Improved Pattern-Avoidance Bounds for Greedy BSTs via Matrix Decomposition

Greedy BST (or simply Greedy) is an online self-adjusting binary search tree defined in the geometric view ([Lucas, 1988; Munro, 2000; Demaine, Harmon, Iacono, Kane, Patrascu, SODA 2009). Along with Splay trees (Sleator, Tarjan 1985), Greedy is considered the most promising candidate for being dynamically optimal, i.e., starting with any initial tree, their access costs on any sequence is conjectured to be within $O(1)$ factor of the offline optimal. However, in the past four decades, the question has remained elusive even for highly restricted input. In this paper, we prove new bounds on the cost of Greedy in the ''pattern avoidance'' regime. Our new results include: The (preorder) traversal conjecture for Greedy holds up to a factor of $O(2^{\alpha(n)})$, improving upon the bound of $2^{\alpha(n)^{O(1)}}$ in (Chalermsook et al., FOCS 2015). This is the best known bound obtained by any online BSTs. We settle the postorder traversal conjecture for Greedy. The deque conjecture for Greedy holds up to a factor of $O(\alpha(n))$, improving upon the bound $2^{O(\alpha(n))}$ in (Chalermsook, et al., WADS 2015). The split conjecture holds for Greedy up to a factor of $O(2^{\alpha(n)})$. Key to all these results is to partition (based on the input structures) the execution log of Greedy into several simpler-to-analyze subsets for which classical forbidden submatrix bounds can be leveraged. Finally, we show the applicability of this technique to handle a class of increasingly complex pattern-avoiding input sequences, called $k$-increasing sequences. As a bonus, we discover a new class of permutation matrices whose extremal bounds are polynomially bounded. This gives a partial progress on an open question by Jacob Fox (2013).Comment: Accepted to SODA 202

Modulation of Cytoskeleton, Protein Trafficking, and Signaling Pathways by Metabolites from Cucurbitaceae, Ericaceae, and Rosaceae Plant Families

One promising frontier within the field of Medical Botany is the study of the bioactivity of plant metabolites on human health. Although plant metabolites are metabolic byproducts that commonly regulate ecological interactions and biochemical processes in plant species, such metabolites also elicit profound effects on the cellular processes of human and other mammalian cells. In this regard, due to their potential as therapeutic agents for a variety of human diseases and induction of toxic cellular responses, further research advances are direly needed to fully understand the molecular mechanisms induced by these agents. Herein, we focus our investigation on metabolites from the Cucurbitaceae, Ericaceae, and Rosaceae plant families, for which several plant species are found within the state of Florida in Hillsborough County. Specifically, we compare the molecular mechanisms by which metabolites and/or plant extracts from these plant families modulate the cytoskeleton, protein trafficking, and cell signaling to mediate functional outcomes, as well as a discussion of current gaps in knowledge. Our efforts to lay the molecular groundwork in this broad manner hold promise in supporting future research efforts in pharmacology and drug discovery