New Analytical Methods for Online Binary Search Trees

This thesis aims to analyze the costs of various binary search tree (BST) algorithms. BST is one of the most fundamental data structures. The long-standing dynamic optimality conjecture states that there exists an online self-adjusting BST algorithm such that, starting with any initial tree, its access cost on any sequence X is O(OPT(X)+n) where OPT(X) is the cost of serving X on an offline optimal algorithm and n is the number of keys. Despite attempts from many groups of researchers, we believe that it is still far from being proven, primarily due to the lack of systematic techniques on the analytical side of algorithms. To address this, we explore the power of two analyzing methods in the context of binary search trees: potential functions and extremal combinatorics. In the first part, we focus on the potential function method, which is widely used but mysterious in how it works. We introduce new charging schemes based on inversion counting, a popular potential function for list updates that has not been used in a BST context. An inversion potential function is arguably the most natural potential function, as it captures the difference between the states of two different algorithms. We illustrate our techniques in the context of both list updates and BSTs: (1) systematically deriving many known list update results and (2) unifying, strengthening, and deriving new bounds for BSTs. In the second part, we explore and extend the use of extremal combinatorics in an amortized analysis of BSTs. The technique exploits the specific structures of the input of BSTs. This method encodes the execution log of BSTs in the form of a matrix and applies powerful theorems from forbidden matrix literature to upper-bound the cost. We propose an extra preprocessing step that decomposes the matrix into several simpler submatrices. We show the applications of the BST Greedy algorithm, one of the most promising candidates for being dynamically optimal, including deriving improved bounds for long-standing conjectures such as preorder traversal, postorder traversal, and path conjectures

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

Co-Bipartite Neighborhood Edge Elimination Orderings

In SODA 2001, Raghavan and Spinrad introduced robust algorithms as a way to solve hard combinatorial graph problems in polynomial time even when the input graph falls slightly outside a graph class for which a polynomial-time algorithm exists. As a leading example, the Maximum Clique problem on unit disk graphs (intersection graphs of unit disks in the plane) was shown to have a robust, polynomial-time algorithm by proving that such graphs admit a co-bipartite neighborhood edge elimination ordering (CNEEO). This begs the question whether other graph classes also admit a CNEEO. In this paper, we answer this question positively, and identify many graph classes that admit a CNEEO, including several graph classes for which no polynomial-time recognition algorithm exists (unless P=NP). As a consequence, we obtain robust, polynomial-time algorithms for Maximum Clique on all identified graph classes. We also prove some negative results, and identify graph classes that do not admit a CNEEO. This implies an almost-perfect dichotomy for subclasses of perfect graphs

New binary search tree bounds via geometric inversions

| openaire: EC/H2020/759557/EU//ALGOComThe long-standing dynamic optimality conjecture postulates the existence of a dynamic binary search tree (BST) that is Op1q-competitive to all other dynamic BSTs. Despite attempts from many groups of researchers, we believe the conjecture is still far-fetched. One of the main reasons is the lack of the “right” potential functions for the problem: existing results that prove various consequences of dynamic optimality rely on very different potential function techniques, while proving dynamic optimality requires a single potential function that can be used to derive all these consequences. In this paper, we propose a new potential function, that we call extended (geometric) inversion. Inversion is arguably the most natural potential function principle that has been used in competitive analysis but has never been used in the context of BSTs. We use our potential function to derive new results, as well as streamlining/strengthening existing results. First, we show that a broad class of BST algorithms (including Greedy and Splay) are Op1qcompetitive to Move-to-Root algorithm and therefore have simulation embedding property – a new BST property that was recently introduced and studied by Levy and Tarjan (SODA 2019). This result, besides substantially expanding the list of BST algorithms having this property, gives the first potential function proof of the simulation embedding property for BSTs (thus unifying apparently different kinds of results). Moreover, our analysis is the first where the costs of two dynamic binary search trees are compared against each other directly and systematically. Secondly, we use our new potential function to unify and strengthen known BST bounds, e.g., showing that Greedy satisfies the weighted dynamic finger property within a multiplicative factor of p5 ` op1qq.Peer reviewe

Co-Bipartite Neighborhood Edge Elimination Orderings

In SODA 2001, Raghavan and Spinrad introduced robust algorithms as a way to solve hard combinatorial graph problems in polynomial time even when the input graph falls slightly outside a graph class for which a polynomial-time algorithm exists. As a leading example, the Maximum Clique problem on unit disk graphs (intersection graphs of unit disks in the plane) was shown to have a robust, polynomial-time algorithm by proving that such graphs admit a co-bipartite neighborhood edge elimination ordering (CNEEO). This begs the question whether other graph classes also admit a CNEEO. In this paper, we answer this question positively, and identify many graph classes that admit a CNEEO, including several graph classes for which no polynomial-time recognition algorithm exists (unless P=NP). As a consequence, we obtain robust, polynomial-time algorithms for Maximum Clique on all identified graph classes. We also prove some negative results, and identify graph classes that do not admit a CNEEO. This implies an almost-perfect dichotomy for subclasses of perfect graphs