arXiv subjects

Grigorios Koumoutsos

Publications and source records attributed to Grigorios Koumoutsos.

12 recordsLinked to original sources

arXiv2022

The Online Min-Sum Set Cover Problem

We consider the online Min-Sum Set Cover (MSSC), a natural and intriguing generalization of the classical list update problem. In Online MSSC, the algorithm maintains a permutation on $n$ elements based on subsets $S_1, S_2, \ldots$ arriving online. The algorithm serves each set $S_t$ upon arrival, using its current permutation $π_{t}$, incurring an access cost equal to the position of the first element of $S_t$ in $π_{t}$. Then, the algorithm may update its permutation to $π_{t+1}$, incurring a moving cost equal to the Kendall tau distance of $π_{t}$ to $π_{t+1}$. The objective is to minimize the total access and moving cost for serving the entire sequence. We consider the $r$-uniform version, where each $S_t$ has cardinality $r$. List update is the special case where $r = 1$. We obtain tight bounds on the competitive ratio of deterministic online algorithms for MSSC against a static adversary, that serves the entire sequence by a single permutation. First, we show a lower bound of $(r+1)(1-\frac{r}{n+1})$ on the competitive ratio. Then, we consider several natural generalizations of successful list update algorithms and show that they fail to achieve any interesting competitive guarantee. On the positive side, we obtain a $O(r)$-competitive deterministic algorithm using ideas from online learning and the multiplicative weight updates (MWU) algorithm. Furthermore, we consider efficient algorithms. We propose a memoryless online algorithm, called Move-All-Equally, which is inspired by the Double Coverage algorithm for the $k$-server problem. We show that its competitive ratio is $Ω(r^2)$ and $2^{O(\sqrt{\log n \cdot \log r})}$, and conjecture that it is $f(r)$-competitive. We also compare Move-All-Equally against the dynamic optimal solution and obtain (almost) tight bounds by showing that it is $Ω(r \sqrt{n})$ and $O(r^{3/2} \sqrt{n})$-competitive.

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arXiv2022

External Memory Planar Point Location with Fast Updates

We study dynamic planar point location in the External Memory Model or Disk Access Model (DAM). Previous work in this model achieves polylog query and polylog amortized update time. We present a data structure with $O( \log_B^2 N)$ query time and $O(\frac{1}{ B^{1-ε}} \log_B N)$ amortized update time, where $N$ is the number of segments, $B$ the block size and $ε$ is a small positive constant, under the assumption that all faces have constant size. This is a $B^{1-ε}$ factor faster for updates than the fastest previous structure, and brings the cost of insertion and deletion down to subconstant amortized time for reasonable choices of $N$ and $B$. Our structure solves the problem of vertical ray-shooting queries among a dynamic set of interior-disjoint line segments; this is well-known to solve dynamic planar point location for a connected subdivision of the plane with faces of constant size.

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arXiv2021

Worst-Case Efficient Dynamic Geometric Independent Set

We consider the problem of maintaining an approximate maximum independent set of geometric objects under insertions and deletions. We present data structures that maintain a constant-factor approximate maximum independent set for broad classes of fat objects in $d$ dimensions, where $d$ is assumed to be a constant, in sublinear \textit{worst-case} update time. This gives the first results for dynamic independent set in a wide variety of geometric settings, such as disks, fat polygons, and their high-dimensional equivalents. Our result is obtained via a two-level approach. First, we develop a dynamic data structure which stores all objects and provides an approximate independent set when queried, with output-sensitive running time. We show that via standard methods such a structure can be used to obtain a dynamic algorithm with \textit{amortized} update time bounds. Then, to obtain worst-case update time algorithms, we develop a generic deamortization scheme that with each insertion/deletion keeps (i) the update time bounded and (ii) the number of changes in the independent set constant. We show that such a scheme is applicable to fat objects by showing an appropriate generalization of a separator theorem. Interestingly, we show that our deamortization scheme is also necessary in order to obtain worst-case update bounds: If for a class of objects our scheme is not applicable, then no constant-factor approximation with sublinear worst-case update time is possible. We show that such a lower bound applies even for seemingly simple classes of geometric objects including axis-aligned rectangles in the plane.

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arXiv2020

Dynamic Geometric Independent Set

We present fully dynamic approximation algorithms for the Maximum Independent Set problem on several types of geometric objects: intervals on the real line, arbitrary axis-aligned squares in the plane and axis-aligned $d$-dimensional hypercubes. It is known that a maximum independent set of a collection of $n$ intervals can be found in $O(n\log n)$ time, while it is already \textsf{NP}-hard for a set of unit squares. Moreover, the problem is inapproximable on many important graph families, but admits a \textsf{PTAS} for a set of arbitrary pseudo-disks. Therefore, a fundamental question in computational geometry is whether it is possible to maintain an approximate maximum independent set in a set of dynamic geometric objects, in truly sublinear time per insertion or deletion. In this work, we answer this question in the affirmative for intervals, squares and hypercubes. First, we show that for intervals a $(1+\varepsilon)$-approximate maximum independent set can be maintained with logarithmic worst-case update time. This is achieved by maintaining a locally optimal solution using a constant number of constant-size exchanges per update. We then show how our interval structure can be used to design a data structure for maintaining an expected constant factor approximate maximum independent set of axis-aligned squares in the plane, with polylogarithmic amortized update time. Our approach generalizes to $d$-dimensional hypercubes, providing a $O(4^d)$-approximation with polylogarithmic update time. Those are the first approximation algorithms for any set of dynamic arbitrary size geometric objects; previous results required bounded size ratios to obtain polylogarithmic update time. Furthermore, it is known that our results for squares (and hypercubes) cannot be improved to a $(1+\varepsilon)$-approximation with the same update time.

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arXiv2020

Memoryless Algorithms for the Generalized $k$-server Problem on Uniform Metrics

We consider the generalized $k$-server problem on uniform metrics. We study the power of memoryless algorithms and show tight bounds of $Θ(k!)$ on their competitive ratio. In particular we show that the \textit{Harmonic Algorithm} achieves this competitive ratio and provide matching lower bounds. This improves the $\approx 2^{2^k}$ doubly-exponential bound of Chiplunkar and Vishwanathan for the more general setting of uniform metrics with different weights.

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arXiv2020

Sublinear Explicit Incremental Planar Voronoi Diagrams

A data structure is presented that explicitly maintains the graph of a Voronoi diagram of $N$ point sites in the plane or the dual graph of a convex hull of points in three dimensions while allowing insertions of new sites/points. Our structure supports insertions in $\tilde O (N^{3/4})$ expected amortized time, where $\tilde O$ suppresses polylogarithmic terms. This is the first result to achieve sublinear time insertions; previously it was shown by Allen et al. that $Θ(\sqrt{N})$ amortized combinatorial changes per insertion could occur in the Voronoi diagram but a sublinear-time algorithm was only presented for the special case of points in convex position.

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arXiv2019

Competitive Online Search Trees on Trees

We consider the design of adaptive data structures for searching elements of a tree-structured space. We use a natural generalization of the rotation-based online binary search tree model in which the underlying search space is the set of vertices of a tree. This model is based on a simple structure for decomposing graphs, previously known under several names including elimination trees, vertex rankings, and tubings. The model is equivalent to the classical binary search tree model exactly when the underlying tree is a path. We describe an online $O(\log \log n)$-competitive search tree data structure in this model, matching the best known competitive ratio of binary search trees. Our method is inspired by Tango trees, an online binary search tree algorithm, but critically needs several new notions including one which we call Steiner-closed search trees, which may be of independent interest. Moreover our technique is based on a novel use of two levels of decomposition, first from search space to a set of Steiner-closed trees, and secondly from these trees into paths.

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arXiv2019

Belga B-trees

We revisit self-adjusting external memory tree data structures, which combine the optimal (and practical) worst-case I/O performances of B-trees, while adapting to the online distribution of queries. Our approach is analogous to undergoing efforts in the BST model, where Tango Trees (Demaine et al. 2007) were shown to be $O(\log\log N)$-competitive with the runtime of the best offline binary search tree on every sequence of searches. Here we formalize the B-Tree model as a natural generalization of the BST model. We prove lower bounds for the B-Tree model, and introduce a B-Tree model data structure, the Belga B-tree, that executes any sequence of searches within a $O(\log \log N)$ factor of the best offline B-tree model algorithm, provided $B=\log^{O(1)}N$. We also show how to transform any static BST into a static B-tree which is faster by a $Θ(\log B)$ factor; the transformation is randomized and we show that randomization is necessary to obtain any significant speedup.

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arXiv2017

Weighted k-Server Bounds via Combinatorial Dichotomies

The weighted $k$-server problem is a natural generalization of the $k$-server problem where each server has a different weight. We consider the problem on uniform metrics, which corresponds to a natural generalization of paging. Our main result is a doubly exponential lower bound on the competitive ratio of any deterministic online algorithm, that essentially matches the known upper bounds for the problem and closes a large and long-standing gap. The lower bound is based on relating the weighted $k$-server problem to a certain combinatorial problem and proving a Ramsey-theoretic lower bound for it. This combinatorial connection also reveals several structural properties of low cost feasible solutions to serve a sequence of requests. We use this to show that the generalized Work Function Algorithm achieves an almost optimum competitive ratio, and to obtain new refined upper bounds on the competitive ratio for the case of $d$ different weight classes.

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arXiv2017

Nested Convex Bodies are Chaseable

In the Convex Body Chasing problem, we are given an initial point $v_0$ in $R^d$ and an online sequence of $n$ convex bodies $F_1, ..., F_n$. When we receive $F_i$, we are required to move inside $F_i$. Our goal is to minimize the total distance travelled. This fundamental online problem was first studied by Friedman and Linial (DCG 1993). They proved an $Ω(\sqrt{d})$ lower bound on the competitive ratio, and conjectured that a competitive ratio depending only on d is possible. However, despite much interest in the problem, the conjecture remains wide open. We consider the setting in which the convex bodies are nested: $F_1 \supset ... \supset F_n$. The nested setting is closely related to extending the online LP framework of Buchbinder and Naor (ESA 2005) to arbitrary linear constraints. Moreover, this setting retains much of the difficulty of the general setting and captures an essential obstacle in resolving Friedman and Linial's conjecture. In this work, we give the first $f(d)$-competitive algorithm for chasing nested convex bodies in $R^d$.

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arXiv2017

Competitive Algorithms for Generalized k-Server in Uniform Metrics

The generalized k-server problem is a far-reaching extension of the k-server problem with several applications. Here, each server $s_i$ lies in its own metric space $M_i$. A request is a k-tuple $r = (r_1,r_2,\dotsc,r_k)$ and to serve it, we need to move some server $s_i$ to the point $r_i \in M_i$, and the goal is to minimize the total distance traveled by the servers. Despite much work, no f(k)-competitive algorithm is known for the problem for k > 2 servers, even for special cases such as uniform metrics and lines. Here, we consider the problem in uniform metrics and give the first f(k)-competitive algorithms for general k. In particular, we obtain deterministic and randomized algorithms with competitive ratio $O(k 2^k)$ and $O(k^3 \log k)$ respectively. Our deterministic bound is based on a novel application of the polynomial method to online algorithms, and essentially matches the long-known lower bound of $2^k-1$. We also give a $2^{2^{O(k)}}$-competitive deterministic algorithm for weighted uniform metrics, which also essentially matches the recent doubly exponential lower bound for the problem.

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arXiv2017

The $(h,k)$-Server Problem on Bounded Depth Trees

We study the $k$-server problem in the resource augmentation setting i.e., when the performance of the online algorithm with $k$ servers is compared to the offline optimal solution with $h \leq k$ servers. The problem is very poorly understood beyond uniform metrics. For this special case, the classic $k$-server algorithms are roughly $(1+1/ε)$-competitive when $k=(1+ε) h$, for any $ε>0$. Surprisingly however, no $o(h)$-competitive algorithm is known even for HSTs of depth 2 and even when $k/h$ is arbitrarily large. We obtain several new results for the problem. First we show that the known $k$-server algorithms do not work even on very simple metrics. In particular, the Double Coverage algorithm has competitive ratio $Ω(h)$ irrespective of the value of $k$, even for depth-2 HSTs. Similarly the Work Function Algorithm, that is believed to be optimal for all metric spaces when $k=h$, has competitive ratio $Ω(h)$ on depth-3 HSTs even if $k=2h$. Our main result is a new algorithm that is $O(1)$-competitive for constant depth trees, whenever $k =(1+ε)h$ for any $ε> 0$. Finally, we give a general lower bound that any deterministic online algorithm has competitive ratio at least 2.4 even for depth-2 HSTs and when $k/h$ is arbitrarily large. This gives a surprising qualitative separation between uniform metrics and depth-2 HSTs for the $(h,k)$-server problem, and gives the strongest known lower bound for the problem on general metrics.

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