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László Kozma 0002

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8 papers

1 author row

STOC Conference 2024 Conference Paper

Optimization with Pattern-Avoiding Input

Benjamin Aram Berendsohn
László Kozma 0002
Michal Opler

Permutation pattern-avoidance is a central concept of both enumerative and extremal combinatorics. In this paper we study the effect of permutation pattern-avoidance on the complexity of optimization problems. In the context of the dynamic optimality conjecture (Sleator, Tarjan, STOC 1983), Chalermsook, Goswami, Kozma, Mehlhorn, and Saranurak (FOCS 2015) conjectured that the amortized search cost of an optimal binary search tree (BST) is constant whenever the search sequence is pattern-avoiding. The best known bound to date is 2 α( n )(1+ o (1)) recently obtained by Chalermsook, Pettie, and Yingchareonthawornchai (SODA 2024); here n is the BST size and α(·) the inverse-Ackermann function. In this paper we resolve the conjecture, showing a tight (1) bound. This indicates a barrier to dynamic optimality: any candidate online BST (e.g., splay trees or greedy trees) must match this optimum, but current analysis techniques only give superconstant bounds. More broadly, we argue that the easiness of pattern-avoiding input is a general phenomenon, not limited to BSTs or even to data structures. To illustrate this, we show that when the input avoids an arbitrary, fixed, a priori unknown pattern, one can efficiently compute: (1) a k -server solution of n requests from a unit interval, with total cost n (1/log k ) , in contrast to the worst-case Θ( n / k ) bound, and (2) a traveling salesman tour of n points from a unit box, of length (log n ), in contrast to the worst-case Θ(√ n ) bound; similar results hold for the euclidean minimum spanning tree , Steiner tree , and nearest-neighbor graphs. We show both results to be tight. Our techniques build on the Marcus-Tardos proof of the Stanley-Wilf conjecture, and on the recently emerging concept of twin-width .

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MFCS Conference 2022 Conference Paper

Fixed-Point Cycles and Approximate EFX Allocations

Benjamin Aram Berendsohn
Simona Boyadzhiyska
László Kozma 0002

We study edge-labelings of the complete bidirected graph K^↔_n with functions that map the set [d] = {1, …, d} to itself. We call a directed cycle in K^↔_n a fixed-point cycle if composing the labels of its edges in order results in a map that has a fixed point, and we say that a labeling is fixed-point-free if no fixed-point cycle exists. For a given d, we ask for the largest value of n, denoted R_f(d), for which there exists a fixed-point-free labeling of K^↔_n. Determining R_f(d) for all d > 0 is a natural Ramsey-type question, generalizing some well-studied zero-sum problems in extremal combinatorics. The problem was recently introduced by Chaudhury, Garg, Mehlhorn, Mehta, and Misra [EC 2021], who proved that d ≤ R_f(d) ≤ d⁴+d and showed that the problem has close connections to EFX allocations, a central problem of fair allocation in social choice theory. In this paper we show the improved bound R_f(d) ≤ d^{2 + o(1)}, yielding an efficient (1-ε)-EFX allocation with n agents and O((n/ε)^{0. 67}) unallocated goods; this improves the bound of O((n/ε)^{0. 8}) of Chaudhury, Garg, Mehlhorn, Mehta, and Misra. {Additionally, we prove the stronger upper bound 2d-2, in the case where all edge-labels are permutations. A very special case of this problem, that of finding zero-sum cycles in digraphs whose edges are labeled with elements of ℤ_d, was recently considered by Alon and Krivelevich [JGT 2021] and by Mészáros and Steiner [EJC 2021]. Our result improves the bounds obtained by these authors and extends them to labelings with elements of an arbitrary (not necessarily commutative) group, while also simplifying the proof. }

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SODA Conference 2022 Conference Paper

Splay trees on trees

Benjamin Aram Berendsohn
László Kozma 0002

Search trees on trees (STTs) are a far-reaching generalization of binary search trees (BSTs), allowing the efficient exploration of tree-structured domains. (BSTs are the special case in which the underlying domain is a path.) Trees on trees have been extensively studied under various guises in computer science and discrete mathematics. Recently Bose, Cardinal, Iacono, Koumoutsos, and Langerman (SODA 2020) considered adaptive STTs and observed that, apart from notable exceptions, the machinery developed for BSTs in the past decades does not readily transfer to STTs. In particular, they asked whether the optimal STT can be efficiently computed or approximated (by analogy to Knuth's algorithm for optimal BSTs), and whether natural self-adjusting BSTs such as Splay trees (Sleator, Tarjan, 1983) can be extended to this more general setting. We answer both questions affirmatively. First, we show that a -approximation of an optimal size- n STT for a given search distribution can be computed in time ( n 2 t + 1 ) for all integers t ≥ 1. Second, we identify a broad family of STTs with linear rotation-distance, allowing the generalization of Splay trees to the STT setting. We show that our generalized Splay satisfies a static optimality theorem, asymptotically matching the cost of the optimal STT in an online fashion, i. e. without knowledge of the search distribution. Our results suggest an extension of the dynamic optimality conjecture for Splay trees to the broader setting of trees on trees.

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SODA Conference 2019 Conference Paper

A time- and space-optimal algorithm for the many-visits TSP

André Berger
László Kozma 0002
Matthias Mnich
Roland Vincze

The many-visits traveling salesperson problem (MV-TSP) asks for an optimal tour of n cities that visits each city c a prescribed number k c of times. Travel costs may be asymmetric, and visiting a city twice in a row may incur a non-zero cost. The MV-TSP problem finds applications in scheduling, geometric approximation, and Hamiltonicity of certain graph families. The fastest known algorithm for MV-TSP is due to Cosmadakis and Papadimitriou (SICOMP, 1984). It runs in time n O ( n ) + O ( n 3 log Σ c k c ) and requires n O ( n ) space. The interesting feature of the Cosmadakis-Papadimitriou algorithm is its logarithmic dependence on the total length Σ c k c of the tour, allowing the algorithm to handle instances with very long tours, beyond what is tractable in the standard TSP setting. However, its superexponential dependence on the number of cities in both its time and space complexity renders the algorithm impractical for all but the narrowest range of this parameter. In this paper we significantly improve on the Cosmadakis-Papadimitriou algorithm, giving an MV-TSP algorithm that runs in time 2 O ( n ), i. e. single-exponential in the number of cities, with polynomial space. The space requirement of our algorithm is (essentially) the size of the output, and assuming the Exponential-time Hypothesis (ETH), the time requirement is optimal. Our algorithm is deterministic, and arguably both simpler and easier to analyse than the original approach of Cosmadakis and Papadimitriou. It involves an optimization over directed spanning trees and a recursive, centroid-based decomposition of trees.

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MFCS Conference 2018 Conference Paper

Pairing heaps: the forward variant

Dani Dorfman
Haim Kaplan
László Kozma 0002
Uri Zwick

The pairing heap is a classical heap data structure introduced in 1986 by Fredman, Sedgewick, Sleator, and Tarjan. It is remarkable both for its simplicity and for its excellent performance in practice. The "magic" of pairing heaps lies in the restructuring that happens after the deletion of the smallest item. The resulting collection of trees is consolidated in two rounds: a left-to-right pairing round, followed by a right-to-left accumulation round. Fredman et al. showed, via an elegant correspondence to splay trees, that in a pairing heap of size n all heap operations take O(log n) amortized time. They also proposed an arguably more natural variant, where both pairing and accumulation are performed in a combined left-to-right round (called the forward variant of pairing heaps). The analogy to splaying breaks down in this case, and the analysis of the forward variant was left open. In this paper we show that inserting an item and deleting the minimum in a forward-variant pairing heap both take amortized time O(log(n) * 4^(sqrt(log n))). This is the first improvement over the O(sqrt(n)) bound showed by Fredman et al. three decades ago. Our analysis relies on a new potential function that tracks parent-child rank-differences in the heap.

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STOC Conference 2018 Conference Paper

Smooth heaps and a dual view of self-adjusting data structures

László Kozma 0002
Thatchaphol Saranurak

We present a new connection between self-adjusting binary search trees (BSTs) and heaps, two fundamental, extensively studied, and practically relevant families of data structures (Allen,Munro, 1978; Sleator, Tarjan, 1983; Fredman, Sedgewick, Sleator, Tarjan, 1986; Wilber, 1989; Fredman, 1999; Iacono, Özkan, 2014). Roughly speaking, we map an arbitrary heap algorithm within a broad and natural model, to a corresponding BST algorithm with the same cost on a dual sequence of operations (i.e. the same sequence with the roles of time and key-space switched). This is the first general transformation between the two families of data structures.

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SODA Conference 2017 Conference Paper

Maximum Scatter TSP in Doubling Metrics

László Kozma 0002
Tobias Mömke

We study the problem of finding a tour of n points in which every edge is long. More precisely, we wish to find a tour that visits every point exactly once, maximizing the length of the shortest edge in the tour. The problem is known as Maximum Scatter TSP, and was introduced by Arkin et al. (SODA 1997), motivated by applications in manufacturing and medical imaging. Arkin et al. gave a 0. 5- approximation for the metric version of the problem and showed that this is the best possible ratio achievable in polynomial time (assuming P ≠ NP ). Arkin et al. raised the question of whether a better approximation ratio can be obtained in the Euclidean plane. We answer this question in the affirmative in a more general setting, by giving a (1-∊)-approximation algorithm for d -dimensional doubling metrics, with running time Õ( n 3 + 2 O ( K log K ) ), where As a corollary we obtain (i) an efficient polynomial- time approximation scheme ( EPTAS ) for all constant dimensions d, (ii) a polynomial-time approximation scheme ( PTAS ) for dimension d = log log n/c, for a sufficiently large constant c, and (iii) a PTAS for constant d and ∊ = Ω(1/log log n ). Furthermore, we show the dependence on d in our approximation scheme to be essentially optimal, unless Satisfiability can be solved in subexponential time.

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FOCS Conference 2015 Conference Paper

Pattern-Avoiding Access in Binary Search Trees

Parinya Chalermsook
Mayank Goswami 0001
László Kozma 0002
Kurt Mehlhorn
Thatchaphol Saranurak

The dynamic optimality conjecture is perhaps the most fundamental open question about binary search trees (BST). It postulates the existence of an asymptotically optimal online BST, i. e. One that is constant factor competitive with any BST on any input access sequence. The two main candidates for dynamic optimality in the literature are splay trees [Sleator and Tarjan, 1985], and Greedy [Lucas, 1988, Munro, 2000, Demaine et al. 2009]. Despite BSTs being among the simplest data structures in computer science, and despite extensive effort over the past three decades, the conjecture remains elusive. Dynamic optimality is trivial for almost all sequences: the optimum access cost of most length-n sequences is Theta(n log n), achievable by any balanced BST. Thus, the obvious missing step towards the conjecture is an understanding of the "easy" access sequences, and indeed the most fruitful research direction so far has been the study of specific sequences, whose "easiness" is captured by a parameter of interest. For instance, splay provably achieves the bound of O(nd) when d roughly measures the distances between consecutive accesses (dynamic finger), the average entropy (static optimality), or the delays between multiple accesses of an element(working set). The difficulty of proving dynamic optimality is witnessed by other highly restricted special cases that remain unresolved, one prominent example is the traversal conjecture [Sleator and Tarjan, 1985], which states that preorder sequences (whose optimum is linear) are linear-time accessed by splay trees, no online BST is known to satisfy this conjecture. In this paper, we prove two different relaxations of the traversal conjecture for Greedy: (i) Greedy is almost linear for preorder traversal, (ii) if a linear-time preprocessing is allowed, Greedy is in fact linear. These statements are corollaries of our more general results that express the complexity of access sequences in terms of a pattern avoidance parameter k. Pattern avoidance is a well-established concept in combinatorics, and the classes of input sequences thus defined are rich, e. g. The k = 3 case includes preorder sequences. For any sequence X with parameter k, our most general result shows that Greedy achieves the cost n*2(Α(n))O(k) where Α is the inverse Ackermann function. Furthermore, a broad subclass of parameter-k sequences has a natural combinatorial interpretation as k-decomposable sequences. For this class of inputs, we obtain an n*2O(k) bound for Greedy when preprocessing is allowed. For k = 3, these results imply (i) and (ii). To our knowledge, these are the first upper bounds for Greedy that are not known to hold for any other online BST. To obtain these results we identify an input-revealing property of Greedy. Informally, this means that the execution log partially reveals the structure of the access sequence. This property facilitates the use of rich technical tools from forbidden sub matrix theory. Further studying the intrinsic complexity of k-decomposable sequences, we make several observations. First, in order to obtain an offline optimal BST, it is enough to bound Greedy on non-decomposable access sequences. Furthermore, we show that the optimal cost for k-decomposable sequences is Theta(n log k), which is well below the proven performance of all known BST algorithms. Hence, sequences in this class can be seen as a "candidate counterexample" to dynamic optimality.

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