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.Comment: accepted to FOCS'1

The (h,k)(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 $(1+1/\epsilon)$-competitive when $k=(1+\epsilon) h$, for any $\epsilon >0$. 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 $\Omega(h)$ 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 $\Omega(h)$ 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 $k =(1+\epsilon )h$ for any $\epsilon > 0$. 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.Comment: Appeared in SODA 201

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(1/B^(1-epsilon) log_B N) amortized update time, where N is the number of segments, B the block size and epsilon is a small positive constant, under the assumption that all faces have constant size. This is a B^(1-epsilon) 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

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 a data structure that maintains 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 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. For axis-aligned squares and hypercubes, our result improves upon all (recently announced) previous works. We obtain, in particular, a dynamic (4+?)-approximation for squares, with O(log? n) worst-case update time. 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 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

Memoryless Algorithms for the Generalized kk-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 $\Theta(k!)$ 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

Competitive algorithms for generalized k-server in uniform metrics

No abstract availabl

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

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 si lies in its own metric space Mi. A request is a k-tuple r = (r1, r2, … , rk), which is served by moving some server si to the point ri ∈ Mi, 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 k · 2k and O(k3log 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 2k − 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

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