215 research outputs found
On-line partitioning of width w posets into w^O(log log w) chains
An on-line chain partitioning algorithm receives the elements of a poset one
at a time, and when an element is received, irrevocably assigns it to one of
the chains. In this paper, we present an on-line algorithm that partitions
posets of width into chains. This improves over
previously best known algorithms using chains by Bosek and
Krawczyk and by Bosek, Kierstead, Krawczyk, Matecki, and Smith. Our algorithm
runs in time, where is the width and is the size of
a presented poset.Comment: 16 pages, 10 figure
Space Efficient Breadth-First and Level Traversals of Consistent Global States of Parallel Programs
Enumerating consistent global states of a computation is a fundamental
problem in parallel computing with applications to debug- ging, testing and
runtime verification of parallel programs. Breadth-first search (BFS)
enumeration is especially useful for these applications as it finds an
erroneous consistent global state with the least number of events possible. The
total number of executed events in a global state is called its rank. BFS also
allows enumeration of all global states of a given rank or within a range of
ranks. If a computation on n processes has m events per process on average,
then the traditional BFS (Cooper-Marzullo and its variants) requires
space in the worst case, whereas ou r
algorithm performs the BFS requires space. Thus, we
reduce the space complexity for BFS enumeration of consistent global states
exponentially. and give the first polynomial space algorithm for this task. In
our experimental evaluation of seven benchmarks, traditional BFS fails in many
cases by exhausting the 2 GB heap space allowed to the JVM. In contrast, our
implementation uses less than 60 MB memory and is also faster in many cases
The on-line width of various classes of posets.
An on-line chain partitioning algorithm receives a poset, one element at a time, and irrevocably assigns the element to one of the chains. Over 30 years ago, Szemer\\u27edi proved that any on-line algorithm could be forced to use chains to partition a poset of width . The maximum number of chains that can be forced on any on-line algorithm remains unknown. In the survey paper by Bosek et al., variants of the problem were studied where the class is restricted to posets of bounded dimension or where the poset is presented via a realizer of size . We prove two results for this problem. First, we prove that any on-line algorithm can be forced to use chains to partition a -dimensional poset of width . Second, we prove that any on-line algorithm can be forced to use chains to partition a poset of width presented via a realizer of size . Chrobak and \\u27Slusarek considered variants of the on-line chain partitioning problem in which the elements are presented as intervals and intersecting intervals are incomparable. They constructed an on-line algorithm which uses at most chains, where is the width of the interval order, and showed that this algorithm is optimal. They also considered the problem restricted to intervals of unit-length and while they showed that first-fit needs at most chains, over years later, it remains unknown whether a more optimal algorithm exists. We improve upon previously known bounds and show that any on-line algorithm can be forced to use chains to partition a semi-order presented in the form of its unit-interval representation. As a consequence, we completely solve the problem for . We also consider entirely new variants and present the results for those
Private Isotonic Regression
In this paper, we consider the problem of differentially private (DP)
algorithms for isotonic regression. For the most general problem of isotonic
regression over a partially ordered set (poset) and for any
Lipschitz loss function, we obtain a pure-DP algorithm that, given input
points, has an expected excess empirical risk of roughly
, where
is the width of the poset. In contrast, we also
obtain a near-matching lower bound of roughly , that holds even for approximate-DP algorithms.
Moreover, we show that the above bounds are essentially the best that can be
obtained without utilizing any further structure of the poset.
In the special case of a totally ordered set and for and
losses, our algorithm can be implemented in near-linear running time; we also
provide extensions of this algorithm to the problem of private isotonic
regression with additional structural constraints on the output function.Comment: Neural Information Processing Systems (NeurIPS), 202
Recoloring Interval Graphs with Limited Recourse Budget
We consider the problem of coloring an interval graph dynamically. Intervals arrive one after the other and have to be colored immediately such that no two intervals of the same color overlap. In each step only a limited number of intervals may be recolored to maintain a proper coloring (thus interpolating between the well-studied online and offline settings). The number of allowed recolorings per step is the so-called recourse budget. Our main aim is to prove both upper and lower bounds on the required recourse budget for interval graphs, given a bound on the allowed number of colors. For general interval graphs with n vertices and chromatic number k it is known that some recoloring is needed even if we have 2k colors available. We give an algorithm that maintains a 2k-coloring with an amortized recourse budget of . For maintaining a k-coloring with k ≤ n, we give an amortized upper bound of \u1d4aa(k⋅ k! ⋅ √n), and a lower bound of , which can be as large as ). For unit interval graphs it is known that some recoloring is needed even if we have k+1 colors available. We give an algorithm that maintains a (k+1)-coloring with at most recolorings per step in the worst case. We also give a lower bound of on the amortized recourse budget needed to maintain a k-coloring. Additionally, for general interval graphs we show that if one does not insist on maintaining an explicit coloring, one can have a k-coloring algorithm which does not incur a factor of in the running time. For this we provide a data structure, which allows for adding intervals in amortized time per update and querying for the color of a particular interval in . Between any two updates, the data structure answers consistently with some optimal coloring. The data structure maintains the coloring implicitly, so the notion of recourse budget does not apply to it
Online algorithms for partially ordered sets
Partially ordered sets (posets) have various applications in computer science ranging from database systems to distributed computing. Content-based routing in publish/subscribe systems is a major poset use case. Content-based routing requires efficient poset online algorithms, including efficient insertion and deletion algorithms. We study the query and total complexities of online operations on posets and poset-like data structures. The main data structures considered are the incidence matrix, Siena poset, ChainMerge, and poset-derived forest. The contributions of this thesis are twofold: First, we present an online adaptation of the ChainMerge data structure as well as several novel poset-derived forest variants. We study the effectiveness of a first-fit-equivalent ChainMerge online insertion algorithm and show that it performs close to optimal query-wise while requiring less CPU processing in a benchmark setting. Second, we present the results of an empirical performance evaluation. In the evaluation we compare the data structures in terms of query complexity and total complexity. The results indicate ChainMerge as the best structure overall. The incidence matrix, although simple, excels in some benchmarks. Poset-derived forest is very fast overall if a 'true' poset data structure is not a requirement. Placing elements in smaller poset-derived forests and then merging them is an efficient way to construct poset-derived forests. Lazy evaluation for poset-derived forests shows some promise as well
Contributions on secretary problems, independent sets of rectangles and related problems
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 187-198).We study three problems arising from different areas of combinatorial optimization. We first study the matroid secretary problem, which is a generalization proposed by Babaioff, Immorlica and Kleinberg of the classical secretary problem. In this problem, the elements of a given matroid are revealed one by one. When an element is revealed, we learn information about its weight and decide to accept it or not, while keeping the accepted set independent in the matroid. The goal is to maximize the expected weight of our solution. We study different variants for this problem depending on how the elements are presented and on how the weights are assigned to the elements. Our main result is the first constant competitive algorithm for the random-assignment random-order model. In this model, a list of hidden nonnegative weights is randomly assigned to the elements of the matroid, which are later presented to us in uniform random order, independent of the assignment. The second problem studied is the jump number problem. Consider a linear extension L of a poset P. A jump is a pair of consecutive elements in L that are not comparable in P. Finding a linear extension minimizing the number of jumps is NP-hard even for chordal bipartite posets. For the class of posets having two directional orthogonal ray comparability graphs, we show that this problem is equivalent to finding a maximum independent set of a well-behaved family of rectangles. Using this, we devise combinatorial and LP-based algorithms for the jump number problem, extending the class of bipartite posets for which this problem is polynomially solvable and improving on the running time of existing algorithms for certain subclasses. The last problem studied is the one of finding nonempty minimizers of a symmetric submodular function over any family of sets closed under inclusion. We give an efficient O(ns)-time algorithm for this task, based on Queyranne's pendant pair technique for minimizing unconstrained symmetric submodular functions. We extend this algorithm to report all inclusion-wise nonempty minimal minimizers under hereditary constraints of slightly more general functions.by José Antonio Soto.Ph.D
Algorithmic and Combinatorial Results in Selection and Computational Geometry
This dissertation investigates two sets of algorithmic and combinatorial problems. Thefirst part focuses on the selection problem under the pairwise comparison model. For the classic “median of medians” scheme, contrary to the popular belief that smaller group sizes cause superlinear behavior, several new linear time algorithms that utilize small groups are introduced. Then the exact number of comparisons needed for an optimal selection algorithm is studied. In particular, the implications of a long standing conjecture known as Yao’s hypothesis are explored. For the multiparty model, we designed low communication complexity protocols for selecting an exact or an approximate median of data that is distributed among multiple players.
In the second part, three computational geometry problems are studied. For the longestspanning tree with neighborhoods, approximation algorithms are provided. For the stretch factor of polygonal chains, upper bounds are proved and almost matching lower bound constructions in \mathbb{R}^2 and higher dimensions are developed. For the piercing number τ and independence number ν of a family of axis-parallel rectangles in the plane, a lower bound construction for ν = 4 that matches Wegner’s conjecture is analyzed. The previous matching construction for ν = 3, due to Wegner himself, dates back to 1968
Beyond Worst-Case Analysis for Joins with Minesweeper
We describe a new algorithm, Minesweeper, that is able to satisfy stronger
runtime guarantees than previous join algorithms (colloquially, `beyond
worst-case guarantees') for data in indexed search trees. Our first
contribution is developing a framework to measure this stronger notion of
complexity, which we call {\it certificate complexity}, that extends notions of
Barbay et al. and Demaine et al.; a certificate is a set of propositional
formulae that certifies that the output is correct. This notion captures a
natural class of join algorithms. In addition, the certificate allows us to
define a strictly stronger notion of runtime complexity than traditional
worst-case guarantees. Our second contribution is to develop a dichotomy
theorem for the certificate-based notion of complexity. Roughly, we show that
Minesweeper evaluates -acyclic queries in time linear in the certificate
plus the output size, while for any -cyclic query there is some instance
that takes superlinear time in the certificate (and for which the output is no
larger than the certificate size). We also extend our certificate-complexity
analysis to queries with bounded treewidth and the triangle query.Comment: [This is the full version of our PODS'2014 paper.
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