arXiv subjects

Wanchote Jiamjitrak

Publications and source records attributed to Wanchote Jiamjitrak.

3 recordsLinked to original sources

arXiv2025

Width Parameters for Minimum Flow Decomposition

Minimum flow decomposition (MFD) is the strongly NP-hard problem of finding a smallest set of integer weighted $s$-$t$ paths in an $s$-$t$ DAG $G$ whose weighted sum is equal to a given flow $f$ on $G$. Despite its many practical applications, we lack an understanding of graph structures that make MFD easy or hard. Recent progress is due to Cáceres et al. [ACM TALG 2024], who showed that the DAG width, the minimum number of paths to cover all edges, plays an essential role in the approximation of the problem. Our first set of results regard the computational complexity of MFD parameterised by the width. This question was previously open, because MFD on width-1 DAGs (paths) is trivially solvable, and the existing NP-hardness proofs use DAGs of unbounded width. We show that MFD on width-2 DAGs is already NP-hard and that MFD on width-3 DAGs is strongly NP-hard. Our main contribution complements these hardness bounds, as we show that weak NP-hardness is the best we can hope for on width-2 DAGs. In fact, we prove the more general statement that MFD with unary coded input can be solved in quasi-polynomial time on DAGs of constant parallel-width, which includes width-2 DAGs. The parallel-width of a DAG $G$ (par-width$(G)$) was defined by Deligkas and Meir [MFCS 2017] as the size of the largest minimal $s$-$t$ cut-set. We obtain these results by, a) interpreting flow decompositions as a sequence of certain digraph minor operations defined by Deligkas and Meir [MFCS 2017], and b) defining a new notion of width of a flow network, flow-width of $(G,f)$, defined as the minimum number of paths covering all edges of $G$, where every edge $e$ can be covered by at most $f(e)$ paths. Using (a) and (b), we show as an intermediate result, an improved upper bound $(\lfloor\log \Vert f\Vert\rfloor+1) \cdot \text{par-width}(G)$ for MFD, where $\Vert f\Vert$ is the largest flow weight of all edges.

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arXiv2023

The Group Access Bounds for Binary Search Trees

The access lemma (Sleator and Tarjan, JACM 1985) is a property of binary search trees that implies interesting consequences such as static optimality, static finger, and working set property. However, there are known corollaries of the dynamic optimality that cannot be derived via the access lemma, such as the dynamic finger, and any $o(\log n)$-competitive ratio to the optimal BST where $n$ is the number of keys. In this paper, we introduce the group access bound that can be defined with respect to a reference group access tree. Group access bounds generalize the access lemma and imply properties that are far stronger than those implied by the access lemma. For each of the following results, there is a group access tree whose group access bound Is $O(\sqrt{\log n})$-competitive to the optimal BST. Achieves the $k$-finger bound with an additive term of $O(m \log k \log \log n)$ (randomized) when the reference tree is an almost complete binary tree. Satisfies the unified bound with an additive term of $O(m \log \log n)$. Matches the unified bound with a time window $k$ with an additive term of $O(m \log k \log \log n)$ (randomized). Furthermore, we prove simulation theorem: For every group access tree, there is an online BST algorithm that is $O(1)$-competitive with its group access bound. In particular, any new group access bound will automatically imply a new BST algorithm achieving the same bound. Thereby, we obtain an improved $k$-finger bound (reference tree is an almost complete binary tree), an improved unified bound with a time window $k$, and matching the best-known bound for Unified bound in the BST model. Since any dynamically optimal BST must achieve the group access bounds, we believe our results provide a new direction towards proving $o(\log n)$-competitiveness of Splay tree and Greedy.

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arXiv2022

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 $O(2^{α(n)})$, improving upon the bound of $2^{α(n)^{O(1)}}$ 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 $O(α(n))$, improving upon the bound $2^{O(α(n))}$ in (Chalermsook, et al., WADS 2015). The split conjecture holds for Greedy up to a factor of $O(2^{α(n)})$. 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).

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