### Browsing by Author "Jiamjitrak, Wanchote Po"

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Item New Analytical Methods for Online Binary Search Trees(Aalto University, 2023) Jiamjitrak, Wanchote Po; Chalermsook, Parinya, Asst. Prof., Aalto University, Department of Computer Science, Finland; Tietotekniikan laitos; Department of Computer Science; Combinatorics of Efficient Computations; Perustieteiden korkeakoulu; School of Science; Chalermsook, Parinya, Asst. Prof., Aalto University, Department of Computer Science, FinlandThis 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.Item New binary search tree bounds via geometric inversions(Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2020-08-01) Chalermsook, Parinya; Jiamjitrak, Wanchote Po; Department of Computer Science; Grandoni, Fabrizio; Herman, Grzegorz; Sanders, Peter; Professorship Chalermsook ParinyaThe 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.Item On Finding Balanced Bicliques via Matchings(Springer, 2020) Chalermsook, Parinya; Jiamjitrak, Wanchote Po; Orgo, Ly; Department of Computer Science; Adler, Isolde; Müller, Haiko; Professorship Chalermsook Parinya; Department of Computer ScienceIn the Maximum Balanced Biclique Problem (MBB), we are given an n-vertex graph G= (V, E), and the goal is to find a balanced complete bipartite subgraph with q vertices on each side while maximizing q. The MBB problem is among the first known NP-hard problems, and has recently been shown to be NP-hard to approximate within a factor n1 - o ( 1 ), assuming the Small Set Expansion hypothesis [Manurangsi, ICALP 2017]. An O(n/ log n) approximation follows from a simple brute-force enumeration argument. In this paper, we provide the first approximation guarantees beyond brute-force: (1) an O(n/ log 2n) efficient approximation algorithm, and (2) a parameterized approximation that returns, for any r∈ N, an r-approximation algorithm in time exp(O(nrlogr)). To obtain these results, we translate the subgraph removal arguments of [Feige, SIDMA 2004] from the context of finding a clique into one of finding a balanced biclique. The key to our proof is the use of matching edges to guide the search for a balanced biclique.