We investigate dynamic algorithms for the interval scheduling problem. We focus on the case when the set of intervals is monotonic. This is when no interval properly contains another interval. We provide two data stru...
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We investigate dynamic algorithms for the interval scheduling problem. We focus on the case when the set of intervals is monotonic. This is when no interval properly contains another interval. We provide two data structures for representing the intervals that allow efficient insertion, removal and various query operations. The first dynamic algorithm, based on the data structure called compatibility forest, runs in amortised time O (log(2) n) for insertion and removal and O (logn) for query. The second dynamic algorithm, based on the data structure called linearised tree, runs in time O (logn) for insertion, removal and query. We discuss differences and similarities of these two data structures through theoretical and experimental results. (C) 2014 Elsevier B.V. All rights reserved.
We initiate the study of matroid problems in a new oracle model called dynamic oracle. Our algorithms in this model lead to new bounds for some classic problems, and a "unified" algorithm whose performance m...
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ISBN:
(纸本)9781450399135
We initiate the study of matroid problems in a new oracle model called dynamic oracle. Our algorithms in this model lead to new bounds for some classic problems, and a "unified" algorithm whose performance matches previous results developed in various papers for various problems. We also show a lower bound that answers some open problems from a few decades ago. Concretely, our results are as follows. Improved algorithms for matroid union and disjoint spanning trees. We show an algorithm with (O) over tilde (k)(n+r root r) dynamic-rank-query and time complexities for the matroid union problem over k matroids, where n is the input size, r is the output size, and (O) over tilde hides poly(k, log(n)). This implies the following consequences. (i) An improvement over the (O) over tilde (k)(n root r) bound implied by [Chakrabarty-Lee-Sidford-Singla-Wong FOCS'19] for matroid union in the traditional rank-query model. (ii) An (O) over tilde (k) (vertical bar E vertical bar + vertical bar V vertical bar root vertical bar V vertical bar)-time algorithm for the k-disjoint spanning tree problem. This is nearly linear for moderately dense input graphs and improves the (O) over tilde (k) (vertical bar V vertical bar + root vertical bar E vertical bar) bounds of Gabow-Westermann [STOC'88] and Gabow [STOC'91]. Consequently, this gives improved bounds for, e.g., Shannon Switching Game and Graph Irreducibility. Matroid intersection. We show a matroid intersection algorithm with (O) over tilde (n root r) dynamic-rank-query and time complexities. This implies new bounds for some problems (e.g. maximum forest with deadlines) and bounds that match the classic ones obtained in various papers for various problems, e.g. colorful spanning tree [Gabow-Stallmann ICALP'85], graphic matroid intersection [Gabow-Xu FOCS'89], simple job scheduling matroid intersection [Xu-Gabow ISAAC'94], and Hopcroft-Karp combinatorial bipartite matching. More importantly, this is done via a "unified" algorithm in
We present dynamic algorithms with polylogarithmic update time for estimating the size of the maximum matching of a graph undergoing edge insertions and deletions with approximation ratio strictly better than 2. Speci...
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We present dynamic algorithms with polylogarithmic update time for estimating the size of the maximum matching of a graph undergoing edge insertions and deletions with approximation ratio strictly better than 2. Specifically, we obtain a 1+ root 21 + & varepsilon;approximate to 1.707+& varepsilon;approximation in bipartite graphs and a 1.973+& varepsilon;approximation in general graphs. We thus answer in the affirmative the value version of the major open question repeatedly asked in the dynamic graph algorithms literature. Our randomized algorithms' approximation and worst-case update time bounds both hold w.h.p. against adaptive adversaries. Our algorithms are based on simulating new two-pass streaming matching algorithms in the dynamic setting. Our key new idea is to invoke the recent sublinear-time matching algorithm of Behnezhad (FOCS'21) in a white-box manner to efficiently simulate the second pass of our streaming algorithms, while bypassing the well-known vertex-update barrier.
AimsSuboptimal device programming is frequent in non-responders to cardiac resynchronization therapy (CRT). However, the role of device optimization and the most appropriate technique are still unknown. The aim of our...
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AimsSuboptimal device programming is frequent in non-responders to cardiac resynchronization therapy (CRT). However, the role of device optimization and the most appropriate technique are still unknown. The aim of our study was to analyse the effect of different CRT optimization techniques within a network *** systematic search was conducted on MEDLINE, Embase and CENTRAL for studies comparing outcomes with empirical device settings or optimization using echocardiography, static algorithms or dynamic algorithms. Studies investigating the effect of optimization in non-responders were also *** total of 17 studies with 4346 patients were included in the quantitative analysis. Of the treatments and outcomes examined, a significant difference was found only between dynamic algorithms and echocardiography, with the former leading to a higher echocardiographic response rate [odds ratio (OR): 2.02, 95% confidence interval (CI) 1.21-3.35], lower heart failure hospitalization rate (OR: 0.75, 95% CI 0.57-0.99) and greater improvement in 6-minute walk test [mean difference (MD): 45.52 m, 95% credible interval (CrI) 3.91-82.44 m]. We found no significant difference between empirical settings, static algorithms and dynamic algorithms. Seven studies with 228 patients reported response rates after optimization in non-responders. Altogether, 34.3%-66.7% of initial non-responders showed improvement after optimization, depending on response *** the time of CRT implantation, dynamic algorithms may serve as a resource-friendly alternative to echocardiographic optimization, with similar or better mid-term outcomes. However, their superiority over empirical device settings needs to be investigated in further trials. For non-responders, CRT optimization should be considered, as the majority of patients experience improvement. In our study, we analyzed data from 17 studies on different cardiac resynchronization therapy optimization techniq
Dyck reachability is a principled, graph-based formulation of a plethora of static analyses. Bidirected graphs are used for capturing dataflow through mutable heap data, and are usual formalisms of demand-driven point...
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Dyck reachability is a principled, graph-based formulation of a plethora of static analyses. Bidirected graphs are used for capturing dataflow through mutable heap data, and are usual formalisms of demand-driven points-to and alias analyses. The best (offline) algorithm runs in O(m + n . alpha(n)) time, where n is the number of nodes and m is the number of edges in the flow graph, which becomes O(n(2)) in the worst case. In the everyday practice of program analysis, the analyzed code is subject to continuous change, with source code being added and removed. On-the-fly static analysis under such continuous updates gives rise to dynamic Dyck reachability, where reachability queries run on a dynamically changing graph, following program updates. Naturally, executing the offline algorithm in this online setting is inadequate, as the time required to process a single update is prohibitively large. In this work we develop a novel dynamic algorithm for bidirected Dyck reachability that has O(n . alpha(n)) worst-case performance per update, thus beating the O(n(2)) bound, and is also optimal in certain settings. We also implement our algorithm and evaluate its performance on on-the-fly data-dependence and alias analyses, and compare it with two best known alternatives, namely (i) the optimal offline algorithm, and (ii) a fully dynamic Datalog solver. Our experiments show that our dynamic algorithm is consistently, and by far, the top performing algorithm, exhibiting speedups in the order of 1000X. The running time of each update is almost always unnoticeable to the human eye, making it ideal for the on-the-fly analysis setting.
We present deterministic algorithms for maintaining a (3/2 + epsilon) and (2 + epsilon)approximate maximum matching in a fully dynamic graph with worst-case update times (O) over cap (root n) and (O) over cap (1) resp...
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We present deterministic algorithms for maintaining a (3/2 + epsilon) and (2 + epsilon)approximate maximum matching in a fully dynamic graph with worst-case update times (O) over cap (root n) and (O) over cap (1) respectively. The fastest known deterministic worst-case update time algorithms for achieving approximation ratio (2 - delta) (for any delta > 0) and (2 + epsilon) were both shown by Roghani et al. (Beating the folklore algorithm for dynamic matching, 2021) with update times O(n(3/4)) and O-epsilon (root n) respectively. We close the gap between worst-case and amortized algorithms for the two approximation ratios as the best deterministic amortized update times for the problem are O-epsilon (root n) and (O) over cap (1) which were shown in Bernstein and Stein (in: Proceedings of the twenty-seventh annual ACM-SIAM symposium on discrete algorithms, 2016) and Bhattacharya and Kiss (in: 48th international colloquium on automata, languages, and programming, ICALP 2021, 12-16 July, Glasgow, 2021) respectively. The algorithm achieving (3/2 + epsilon) approximation builds on the EDCS concept introduced by the influential paper of Bernstein and Stein (in: International colloquium on automata, languages, and programming, Springer, Berlin, 2015). Say that H is a (alpha, delta)-approximate matching sparsifier if at all times H satisfies that mu( H) center dot alpha + delta center dot n >= mu(G) (define (alpha, delta)-approximation similarly for matchings). We show how to maintain a locally damaged version of the EDCS which is a (3/2 + epsilon, delta)-approximate matching sparsifier. We further show how to reduce the maintenance of an a-approximate maximum matching to the maintenance of an (alpha, delta)-approximate maximum matching building based on an observation of Assadi et al. (in: Proceedings of the twenty-seventh annual (ACM-SIAM) symposium on discrete algorithms, (SODA) 2016, Arlington, VA, USA, January 10-12, 2016). Our reduction requires an update time blow-up
Semi-online algorithms that are allowed to perform a bounded amount of repacking achieve guaranteed good worst-case behaviour in a more realistic setting. Most of the previous works focused on minimization problems th...
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ISBN:
(数字)9783030800499
ISBN:
(纸本)9783030800499;9783030800482
Semi-online algorithms that are allowed to perform a bounded amount of repacking achieve guaranteed good worst-case behaviour in a more realistic setting. Most of the previous works focused on minimization problems that aim to minimize some costs. In this work, we study maximization problems that aim to maximize their profit. We mostly focus on a class of problems that we call choosing problems, where a maximum profit subset of a set objects has to be maintained. Many known problems, such as KNAPSACK, MAXIMUMINDEPENDENTSET and variations of these, are part of this class. We present a framework for choosing problems that allows us to transfer offline ca-approximation algorithms into (alpha - epsilon)-competitive semi-online algorithms with amortized migration O(1/epsilon). Moreover we complement these positive results with lower bounds that show that our results are tight in the sense that no amortized migration of o(1/epsilon) is possible.
We study the fully dynamic maximum matching problem. In this problem, the goal is to efficiently maintain an approximate maximum matching of a graph that is subject to edge insertions and deletions. Our focus is parti...
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ISBN:
(纸本)9798331516758;9798331516741
We study the fully dynamic maximum matching problem. In this problem, the goal is to efficiently maintain an approximate maximum matching of a graph that is subject to edge insertions and deletions. Our focus is particularly on algorithms that maintain the edges of a (1 - epsilon)-approximate maximum matching for an arbitrarily small constant epsilon > 0. Until recently, the fastest known algorithm for this problem required Theta(n) time per update where n is the number of vertices. This bound was slightly improved to n/(log * n)(Omega(1)) by Assadi, Behnezhad, Khanna, and Li [STOC'23] and very recently to n/2(Omega(root log n)) by Liu [FOCS'24]. Whether this can be improved to n(1-Omega(1)) remains a major open problem. In this paper, we introduce Ordered Ruzsa-Szemeredi (ORS) graphs (a generalization of Ruzsa-Szemeredi graphs) and show that the complexity of dynamic matching is closely tied to them. For delta > 0, define ORS(delta n) to be the maximum number of matchings M-1, ... , M-t, each of size delta n, that one can pack in an n-vertex graph such that each matching M-i is an induced matching in subgraph M-1 boolean OR ... boolean OR M-i. We show that there is a randomized algorithm that maintains a (1 - epsilon)-approximate maximum matching of a fully dynamic graph in (O) over tilde(root n(1+epsilon) . ORS(Theta(epsilon)(n))) amortized update-time. While the value of ORS(Theta(n)) remains unknown and is only upper bounded by n(1-o(1)), the densest construction known from more than two decades ago only achieves ORS(Theta(n)) >= n(1/Theta(log log n)) = n(o(1)) [Fischer et al. STOC'02]. If this is close to the right bound, then our algorithm achieves an update-time of root n(1+O(epsilon)), resolving the aforementioned longstanding open problem in dynamic algorithms in a strong sense.
A recent work by Christiansen, Nowicki, and Rotenberg [STOC'23] provides dynamic algorithms for coloring sparse graphs, concretely as a function of the graph's arboricity alpha. They give two randomized algori...
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ISBN:
(纸本)9798400703836
A recent work by Christiansen, Nowicki, and Rotenberg [STOC'23] provides dynamic algorithms for coloring sparse graphs, concretely as a function of the graph's arboricity alpha. They give two randomized algorithms: $ (alpha log alpha) implicit coloring in poly(log n) worst-case update and query times, and O ( min{alpha log alpha, alpha log log log n}) implicit coloring in poly(log n) amortized update and query times (against an oblivious adversary). We improve these results in terms of the number of colors and the time guarantee: First, we present an extremely simple algorithm that computes an $ (alpha)-implicit coloring with poly( log n) amortized update and query times. Second, and as the main technical contribution of our work, we show that the time complexity guarantee can be strengthened from amortized to worst-case. That is, we give a dynamic algorithm for implicit O (alpha)-coloring with poly (log n) worst-case update and query times (against an oblivious adversary).
We study dynamic (1 - epsilon)-approximate rounding of fractional matchings-a key ingredient in numerous breakthroughs in the dynamic graph algorithms literature. Our first contribution is a surprisingly simple determ...
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ISBN:
(纸本)9798400703836
We study dynamic (1 - epsilon)-approximate rounding of fractional matchings-a key ingredient in numerous breakthroughs in the dynamic graph algorithms literature. Our first contribution is a surprisingly simple deterministic rounding algorithm in bipartite graphs with amortized update time O(epsilon(-1) log(2) (epsilon(-1) center dot n)), matching an (unconditional) recourse lower bound of Omega(epsilon(-1)) up to logarithmic factors. Moreover, this algorithm's update time improves provided the minimum (non-zero) weight in the fractional matching is lower bounded throughout. Combining this algorithm with novel dynamic partial rounding algorithms to increase this minimum weight, we obtain a number of algorithms that improve this dependence on n. For example, we give a high-probability randomized algorithm with (O) over tilde (epsilon(-1) center dot ( log log n)(2))-update time against adaptive adversaries. Using our rounding algorithms, we also round known (1 - epsilon)-decremental fractional bipartite matching algorithms with no asymptotic overhead, thus improving on state-of-the-art algorithms for the decremental bipartite matching problem. Further, we provide extensions of our results to general graphs and to maintaining almost-maximal matchings.
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