arXiv subjects

László Kozma

Publications and source records attributed to László Kozma.

At least 37 records · Page 2Linked to original sources

arXiv2021

Analysis of Smooth Heaps and Slim Heaps

The smooth heap is a recently introduced self-adjusting heap [Kozma, Saranurak, 2018] similar to the pairing heap [Fredman, Sedgewick, Sleator, Tarjan, 1986]. The smooth heap was obtained as a heap-counterpart of Greedy BST, a binary search tree updating strategy conjectured to be \emph{instance-optimal} [Lucas, 1988], [Munro, 2000]. Several adaptive properties of smooth heaps follow from this connection; moreover, the smooth heap itself has been conjectured to be instance-optimal within a certain class of heaps. Nevertheless, no general analysis of smooth heaps has existed until now, the only previous analysis showing that, when used in \emph{sorting mode} ($n$ insertions followed by $n$ delete-min operations), smooth heaps sort $n$ numbers in $O(n\lg n)$ time. In this paper we describe a simpler variant of the smooth heap we call the \emph{slim heap}. We give a new, self-contained analysis of smooth heaps and slim heaps in unrestricted operation, obtaining amortized bounds that match the best bounds known for self-adjusting heaps. Previous experimental work has found the pairing heap to dominate other data structures in this class in various settings. Our tests show that smooth heaps and slim heaps are competitive with pairing heaps, outperforming them in some cases, while being comparably easy to implement.

cs.DS↗

arXiv2020

Geometric group testing

Group testing is concerned with identifying $t$ defective items in a set of $m$ items, where each test reports whether a specific subset of items contains at least one defective. In non-adaptive group testing, the subsets to be tested are fixed in advance. By testing multiple items at once, the required number of tests can be made much smaller than $m$. In fact, for $t \in \mathcal{O}(1)$, the optimal number of (non-adaptive) tests is known to be $Θ(\log{m})$. In this paper, we consider the problem of non-adaptive group testing in a geometric setting, where the items are points in $d$-dimensional Euclidean space and the tests are axis-parallel boxes (hyperrectangles). We present upper and lower bounds on the required number of tests under this geometric constraint. In contrast to the general, combinatorial case, the bounds in our geometric setting are polynomial in $m$. For instance, our results imply that identifying a defective pair in a set of $m$ points in the plane always requires $Ω(m^{3/5})$ tests, and there exist configurations of $m$ points for which $\mathcal{O}(m^{2/3})$ tests are sufficient, whereas to identify a single defective point in the plane, $Θ(m^{1/2})$ tests are always necessary and sometimes sufficient.

cs.CG↗

arXiv2020

Splay trees on trees

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 $(1 + \frac{1}{t})$-approximation of an optimal size-$n$ STT for a given search distribution can be computed in time $O(n^{2t + 1})$ for all integers $t \geq 1$. 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.

cs.DS↗

arXiv2020

Exact exponential algorithms for two poset problems

Partially ordered sets (posets) are fundamental combinatorial objects with important applications in computer science. Perhaps the most natural algorithmic task, given a size-$n$ poset, is to compute its number of linear extensions. In 1991 Brightwell and Winkler showed this problem to be $\#P$-hard. In spite of extensive research, the fastest known algorithm is still the straightforward $O(n 2^n)$-time dynamic programming (an adaptation of the Bellman-Held-Karp algorithm for the TSP). Very recently, Dittmer and Pak showed that the problem remains $\#P$-hard for two-dimensional posets, and no algorithm was known to break the $2^n$-barrier even in this special case. The question of whether the two-dimensional problem is easier than the general case was raised decades ago by Möhring, Felsner and Wernisch, and others. In this paper we show that the number of linear extensions of a two-dimensional poset can be computed in time $O(1.8172^n)$. The related jump number problem asks for a linear extension of a poset, minimizing the number of neighboring incomparable pairs. The problem has applications in scheduling, and has been widely studied. In 1981 Pulleyblank showed it to be NP-complete. We show that the jump number problem can be solved (in arbitrary posets) in time $O(1.824^n)$. This improves (slightly) the previous best bound of Kratsch and Kratsch.

cs.DS↗

arXiv2020

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

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 $n^{O(n)} + O(n^3 \log \sum_c k_c )$ and requires $n^{Θ(n)}$ space. An interesting feature of the Cosmadakis-Papadimitriou algorithm is its \emph{logarithmic} dependence on the total length $\sum_c k_c$ of the tour, allowing the algorithm to handle instances with very long tours. The \emph{superexponential} dependence on the number of cities in both the time and space complexity, however, renders the algorithm impractical for all but the narrowest range of this parameter. In this paper we improve upon the Cosmadakis-Papadimitriou algorithm, giving an MV-TSP algorithm that runs in time $2^{O(n)}$, i.e.\ \emph{single-exponential} in the number of cities, using \emph{polynomial} space. 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.

cs.DS↗

arXiv2019

Finding and counting permutations via CSPs

Permutation patterns and pattern avoidance have been intensively studied in combinatorics and computer science, going back at least to the seminal work of Knuth on stack-sorting (1968). Perhaps the most natural algorithmic question in this area is deciding whether a given permutation of length $n$ contains a given pattern of length $k$. In this work we give two new algorithms for this well-studied problem, one whose running time is $n^{k/4 + o(k)}$, and a polynomial-space algorithm whose running time is the better of $O(1.6181^n)$ and $O(n^{k/2 + 1})$. These results improve the earlier best bounds of $n^{0.47k + o(k)}$ and $O(1.79^n)$ due to Ahal and Rabinovich (2000) resp. Bruner and Lackner (2012) and are the fastest algorithms for the problem when $k \in Ω(\log{n})$. We show that both our new algorithms and the previous exponential-time algorithms in the literature can be viewed through the unifying lens of constraint-satisfaction. Our algorithms can also count, within the same running time, the number of occurrences of a pattern. We show that this result is close to optimal: solving the counting problem in time $f(k) \cdot n^{o(k/\log{k})}$ would contradict the exponential-time hypothesis (ETH). For some special classes of patterns we obtain improved running times. We further prove that $3$-increasing and $3$-decreasing permutations can, in some sense, embed arbitrary permutations of almost linear length, which indicates that an algorithm with sub-exponential running time is unlikely, even for patterns from these restricted classes.

cs.DS↗

arXiv2019

Faster and simpler algorithms for finding large patterns in permutations

Permutation patterns and pattern avoidance have been intensively studied in combinatorics and computer science, going back at least to the seminal work of Knuth on stack-sorting (1968). Perhaps the most natural algorithmic question in this area is deciding whether a given permutation of length $n$ contains a given pattern of length $k$. In this work we give two new algorithms for this well-studied problem, one whose running time is $n^{0.44k+o(k)}$, and one whose running time is the better of $O(1.6181^n)$ and $n^{k/2+o(k)}$. These results improve the earlier best bounds of Ahal and Rabinovich (2000), and Bruner and Lackner (2012), and are the fastest algorithms for the problem when $k = Ω(\log n)$. When $k = o(\log n)$, the parameterized algorithm of Guillemot and Marx (2013) dominates. Our second algorithm uses polynomial space and is significantly simpler than all previous approaches with comparable running times, including an $n^{k/2+o(k)}$ algorithm proposed by Guillemot and Marx. Our approach can be summarized as follows: "for every matching of the even-valued entries of the pattern, try to match all odd-valued entries left-to-right". For the special case of patterns that are Jordan-permutations, we show an improved, subexponential running time.

cs.DS↗

arXiv2019

Hamiltonicity below Dirac's condition

Dirac's theorem (1952) is a classical result of graph theory, stating that an $n$-vertex graph ($n \geq 3$) is Hamiltonian if every vertex has degree at least $n/2$. Both the value $n/2$ and the requirement for every vertex to have high degree are necessary for the theorem to hold. In this work we give efficient algorithms for determining Hamiltonicity when either of the two conditions are relaxed. More precisely, we show that the Hamiltonian cycle problem can be solved in time $c^k \cdot n^{O(1)}$, for some fixed constant $c$, if at least $n-k$ vertices have degree at least $n/2$, or if all vertices have degree at least $n/2-k$. The running time is, in both cases, asymptotically optimal, under the exponential-time hypothesis (ETH). The results extend the range of tractability of the Hamiltonian cycle problem, showing that it is fixed-parameter tractable when parameterized below a natural bound. In addition, for the first parameterization we show that a kernel with $O(k)$ vertices can be found in polynomial time.

cs.DS↗

arXiv2018

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

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. Roughly speaking, we map an arbitrary heap algorithm within a 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. There is a rich theory of dynamic optimality for BSTs (i.e. the theory of competitiveness between BST algorithms). The lack of an analogous theory for heaps has been noted in the literature. Through our connection, we transfer all instance-specific lower bounds known for BSTs to a general model of heaps, initiating a theory of dynamic optimality for heaps. On the algorithmic side, we obtain a new, simple and efficient heap algorithm, which we call the smooth heap. We show the smooth heap to be the heap-counterpart of Greedy, the BST algorithm with the strongest proven and conjectured properties from the literature, widely believed to be instance-optimal. Assuming the optimality of Greedy, the smooth heap is also optimal within our model of heap algorithms. As corollaries of results known for Greedy, we obtain instance-specific upper bounds for the smooth heap, with applications in adaptive sorting. Intriguingly, the smooth heap, although derived from a non-practical BST algorithm, is simple and easy to implement (e.g. it stores no auxiliary data besides the keys and tree pointers). It can be seen as a variation on the popular pairing heap data structure, extending it with a "power-of-two-choices" type of heuristic.

cs.DS↗

arXiv2018

Multi-finger binary search trees

We study multi-finger binary search trees (BSTs), a far-reaching extension of the classical BST model, with connections to the well-studied $k$-server problem. Finger search is a popular technique for speeding up BST operations when a query sequence has locality of reference. BSTs with multiple fingers can exploit more general regularities in the input. In this paper we consider the cost of serving a sequence of queries in an optimal (offline) BST with $k$ fingers, a powerful benchmark against which other algorithms can be measured. We show that the $k$-finger optimum can be matched by a standard dynamic BST (having a single root-finger) with an $O(\log{k})$ factor overhead. This result is tight for all $k$, improving the $O(k)$ factor implicit in earlier work. Furthermore, we describe new online BSTs that match this bound up to a $(\log{k})^{O(1)}$ factor. Previously only the "one-finger" special case was known to hold for an online BST (Iacono, Langerman, 2016; Cole et al., 2000). Splay trees, assuming their conjectured optimality (Sleator and Tarjan, 1983), would have to match our bounds for all $k$. Our online algorithms are randomized and combine techniques developed for the $k$-server problem with a multiplicative-weights scheme for learning tree metrics. To our knowledge, this is the first time when tools developed for the $k$-server problem are used in BSTs. As an application of our $k$-finger results, we show that BSTs can efficiently serve queries that are close to some recently accessed item. This is a (restricted) form of the unified property (Iacono, 2001) that was previously not known to hold for any BST algorithm, online or offline.

cs.DS↗

arXiv2018

Improved bounds for multipass pairing heaps and path-balanced binary search trees

We revisit multipass pairing heaps and path-balanced binary search trees (BSTs), two classical algorithms for data structure maintenance. The pairing heap is a simple and efficient "self-adjusting" heap, introduced in 1986 by Fredman, Sedgewick, Sleator, and Tarjan. In the multipass variant (one of the original pairing heap variants described by Fredman et al.) the minimum item is extracted via repeated pairing rounds in which neighboring siblings are linked. Path-balanced BSTs, proposed by Sleator (Subramanian, 1996), are a natural alternative to Splay trees (Sleator and Tarjan, 1983). In a path-balanced BST, whenever an item is accessed, the search path leading to that item is re-arranged into a balanced tree. Despite their simplicity, both algorithms turned out to be difficult to analyse. Fredman et al. showed that operations in multipass pairing heaps take amortized $O(\log{n} \cdot \log\log{n} / \log\log\log{n})$ time. For searching in path-balanced BSTs, Balasubramanian and Raman showed in 1995 the same amortized time bound of $O(\log{n} \cdot \log\log{n} / \log\log\log{n})$, using a different argument. In this paper we show an explicit connection between the two algorithms and improve the two bounds to $O\left(\log{n} \cdot 2^{\log^{\ast}{n}} \cdot \log^{\ast}{n}\right)$, respectively $O\left(\log{n} \cdot 2^{\log^{\ast}{n}} \cdot (\log^{\ast}{n})^2 \right)$, where $\log^{\ast}(\cdot)$ denotes the very slowly growing iterated logarithm function. These are the first improvements in more than three, resp. two decades, approaching in both cases the information-theoretic lower bound of $Ω(\log{n})$.

cs.DS↗

arXiv2018

Pairing heaps: the forward variant

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 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 $O(\log{n} \cdot 4^{\sqrt{\log{n}}} )$. 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.

cs.DS↗

arXiv2018

On non-positive curvature properties of the Hilbert metric

In this paper, we consider different types of non-positive curvature properties of the Hilbert metric of a convex domain in R^n. First, we survey the relationships among the concepts and prove that in the case of Hilbert metric some of them are equivalent. Furthermore, we show some condition which implies the rigidity feature: if the Hilbert metric is Berwald, i.e., its Finslerian Chern connection reduces to a linear one, then the domain is an ellipsoid and the metric is Riemannian.

math.DG↗

arXiv2018

Selection from heaps, row-sorted matrices and $X+Y$ using soft heaps

We use soft heaps to obtain simpler optimal algorithms for selecting the $k$-th smallest item, and the set of~$k$ smallest items, from a heap-ordered tree, from a collection of sorted lists, and from $X+Y$, where $X$ and $Y$ are two unsorted sets. Our results match, and in some ways extend and improve, classical results of Frederickson (1993) and Frederickson and Johnson (1982). In particular, for selecting the $k$-th smallest item, or the set of~$k$ smallest items, from a collection of~$m$ sorted lists we obtain a new optimal "output-sensitive" algorithm that performs only $O(m+\sum_{i=1}^m \log(k_i+1))$ comparisons, where $k_i$ is the number of items of the $i$-th list that belong to the overall set of~$k$ smallest items.

cs.DS↗

arXiv2016

Maximum Scatter TSP in Doubling Metrics

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 $P \neq NP$). 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 $(1-ε)$-approximation algorithm for $d$-dimensional doubling metrics, with running time $\tilde{O}\big(n^3 + 2^{O(K \log K)}\big)$, where $K \leq \left( \frac{13}ε \right)^d$. 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 $ε= Ω(1/\log\log{n})$. Furthermore, we show the dependence on $d$ in our approximation scheme to be essentially optimal, unless Satisfiability can be solved in subexponential time.

cs.DS↗

arXiv2016

Hitting Set for hypergraphs of low VC-dimension

We study the complexity of the Hitting Set problem in set systems (hypergraphs) that avoid certain sub-structures. In particular, we characterize the classical and parameterized complexity of the problem when the Vapnik-Chervonenkis dimension (VC-dimension) of the input is small. VC-dimension is a natural measure of complexity of set systems. Several tractable instances of Hitting Set with a geometric or graph-theoretical flavor are known to have low VC-dimension. In set systems of bounded VC-dimension, Hitting Set is known to admit efficient and almost optimal approximation algorithms (Brönnimann and Goodrich, 1995; Even, Rawitz, and Shahar, 2005; Agarwal and Pan, 2014). In contrast to these approximation-results, a low VC-dimension does not necessarily imply tractability in the parameterized sense. In fact, we show that Hitting Set is W[1]-hard already on inputs with VC-dimension 2, even if the VC-dimension of the dual set system is also 2. Thus, Hitting Set is very unlikely to be fixed-parameter tractable even in this arguably simple case. This answers an open question raised by King in 2010. For set systems whose (primal or dual) VC-dimension is 1, we show that Hitting Set is solvable in polynomial time. To bridge the gap in complexity between the classes of inputs with VC-dimension 1 and 2, we use a measure that is more fine-grained than VC-dimension. In terms of this measure, we identify a sharp threshold where the complexity of Hitting Set transitions from polynomial-time-solvable to NP-hard. The tractable class that lies just under the threshold is a generalization of Edge Cover, and thus extends the domain of polynomial-time tractability of Hitting Set.

cs.DS↗

arXiv2016

Binary search trees and rectangulations

We revisit the classical problem of searching in a binary search tree (BST) using rotations, and present novel connections of this problem to a number of geometric and combinatorial structures. In particular, we show that the execution trace of a BST that serves a sequence of queries is in close correspondence with the flip-sequence between two rectangulations. (Rectangulations are well-studied combinatorial objects also known as mosaic floorplans.) We also reinterpret Small Manhattan Network, a problem with known connections to the BST problem, in terms of flips in rectangulations. We apply further transformations to the obtained geometric model, to arrive at a particularly simple view of the BST problem that resembles sequences of edge-relaxations in a shortest path algorithm. Our connections yield new results and observations for all structures concerned. In this draft we present some preliminary findings. BSTs with rotations are among the most fundamental and most thoroughly studied objects in computer science, nonetheless they pose long-standing open questions, such as the dynamic optimality conjecture of Sleator and Tarjan (STOC 1983). Our hope is that the correspondences presented in this paper provide a new perspective on this old problem and bring new tools to the study of dynamic optimality.

cs.DS↗

arXiv2016

The landscape of bounds for binary search trees

Binary search trees (BSTs) with rotations can adapt to various kinds of structure in search sequences, achieving amortized access times substantially better than the Theta(log n) worst-case guarantee. Classical examples of structural properties include static optimality, sequential access, working set, key-independent optimality, and dynamic finger, all of which are now known to be achieved by the two famous online BST algorithms (Splay and Greedy). (...) In this paper, we introduce novel properties that explain the efficiency of sequences not captured by any of the previously known properties, and which provide new barriers to the dynamic optimality conjecture. We also establish connections between various properties, old and new. For instance, we show the following. (i) A tight bound of O(n log d) on the cost of Greedy for d-decomposable sequences. The result builds on the recent lazy finger result of Iacono and Langerman (SODA 2016). On the other hand, we show that lazy finger alone cannot explain the efficiency of pattern avoiding sequences even in some of the simplest cases. (ii) A hierarchy of bounds using multiple lazy fingers, addressing a recent question of Iacono and Langerman. (iii) The optimality of the Move-to-root heuristic in the key-independent setting introduced by Iacono (Algorithmica 2005). (iv) A new tool that allows combining any finite number of sound structural properties. As an application, we show an upper bound on the cost of a class of sequences that all known properties fail to capture. (v) The equivalence between two families of BST properties. The observation on which this connection is based was known before - we make it explicit, and apply it to classical BST properties. (...)

cs.DS↗