arXiv subjects

Marek Chrobak

Publications and source records attributed to Marek Chrobak.

At least 37 records · Page 2Linked to original sources

arXiv2021

Optimal Search Trees with 2-Way Comparisons

In 1971, Knuth gave an $O(n^2)$-time algorithm for the classic problem of finding an optimal binary search tree. Knuth's algorithm works only for search trees based on 3-way comparisons, while most modern computers support only 2-way comparisons (e.g., $<, \le, =, \ge$, and $>$). Until this paper, the problem of finding an optimal search tree using 2-way comparisons remained open -- poly-time algorithms were known only for restricted variants. We solve the general case, giving (i) an $O(n^4)$-time algorithm and (ii) an $O(n \log n)$-time additive-3 approximation algorithm. Also, for finding optimal binary split trees, we (iii) obtain a linear speedup and (iv) prove some previous work incorrect.

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arXiv2021

On the Cost of Unsuccessful Searches in Search Trees with Two-way Comparisons

Search trees are commonly used to implement access operations to a set of stored keys. If this set is static and the probabilities of membership queries are known in advance, then one can precompute an optimal search tree, namely one that minimizes the expected access cost. For a non-key query, a search tree can determine its approximate location by returning the inter-key interval containing the query. This is in contrast to other dictionary data structures, like hash tables, that only report a failed search. We address the question "what is the additional cost of determining approximate locations for non-key queries"? We prove that for two-way comparison trees this additional cost is at most 1. Our proof is based on a novel probabilistic argument that involves converting a search tree that does not identify non-key queries into a random tree that does.

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arXiv2020

Scheduling with Gaps: New Models and Algorithms

We consider scheduling problems for unit jobs with release times, where the number or size of the gaps in the schedule is taken into consideration, either in the objective function or as a constraint. Except for a few papers on energy minimization, there is no work in the scheduling literature that uses performance metrics depending on the gap structure of a schedule. One of our objectives is to initiate the study of such scheduling problems with gaps. We show that such problems often lead to interesting algorithmic problems, with connections to other areas of algorithmics. We focus on the model with unit jobs. First we examine scheduling problems with deadlines, where we consider variants of minimum-gap scheduling, including maximizing throughput with a budget for gaps or minimizing the number of gaps with a throughput requirement. We then turn to other objective functions. For example, in some scenarios, gaps in a schedule may be actually desirable, leading to the problem of maximizing the number of gaps. Other versions we study include minimizing maximum gap or maximizing minimum gap. The second part of the paper examines the model without deadlines, where we focus on the tradeoff between the number of gaps and the total or maximum flow time. For all these problems we provide polynomial time algorithms, with running times ranging from $O(n \log n)$ for some problems, to $O(n^7)$ for other. The solutions involve a spectrum of algorithmic techniques, including different dynamic programming formulations, speed-up techniques based on searching Monge arrays, searching X + Y matrices, or implicit binary search.

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arXiv2020

Incremental Medians via Online Bidding

In the k-median problem we are given sets of facilities and customers, and distances between them. For a given set F of facilities, the cost of serving a customer u is the minimum distance between u and a facility in F. The goal is to find a set F of k facilities that minimizes the sum, over all customers, of their service costs. Following Mettu and Plaxton, we study the incremental medians problem, where k is not known in advance, and the algorithm produces a nested sequence of facility sets where the kth set has size k. The algorithm is c-cost-competitive if the cost of each set is at most c times the cost of the optimum set of size k. We give improved incremental algorithms for the metric version: an 8-cost-competitive deterministic algorithm, a 2e ~ 5.44-cost-competitive randomized algorithm, a (24+epsilon)-cost-competitive, poly-time deterministic algorithm, and a (6e+epsilon ~ .31)-cost-competitive, poly-time randomized algorithm. The algorithm is s-size-competitive if the cost of the kth set is at most the minimum cost of any set of size k, and has size at most s k. The optimal size-competitive ratios for this problem are 4 (deterministic) and e (randomized). We present the first poly-time O(log m)-size-approximation algorithm for the offline problem and first poly-time O(log m)-size-competitive algorithm for the incremental problem. Our proofs reduce incremental medians to the following online bidding problem: faced with an unknown threshold T, an algorithm submits "bids" until it submits a bid that is at least the threshold. It pays the sum of all its bids. We prove that folklore algorithms for online bidding are optimally competitive.

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arXiv2020

Algorithmic approaches to selecting control clones in DNA array hybridization experiments

We study the problem of selecting control clones in DNA array hybridization experiments. The problem arises in the OFRG method for analyzing microbial communities. The OFRG method performs classification of rRNA gene clones using binary fingerprints created from a series of hybridization experiments, where each experiment consists of hybridizing a collection of arrayed clones with a single oligonucleotide probe. This experiment produces analog signals, one for each clone, which then need to be classified, that is, converted into binary values 1 and 0 that represent hybridization and non-hybridization events. In addition to the sample rRNA gene clones, the array contains a number of control clones needed to calibrate the classification procedure of the hybridization signals. These control clones must be selected with care to optimize the classification process. We formulate this as a combinatorial optimization problem called Balanced Covering. We prove that the problem is NP-hard, and we show some results on hardness of approximation. We propose approximation algorithms based on randomized rounding and we show that, with high probability, our algorithms approximate well the optimum solution. The experimental results confirm that the algorithms find high quality control clones. The algorithms have been implemented and are publicly available as part of the software package called CloneTools.

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arXiv2019

Modeling Fluid Mixing in Microfluidic Grids

We describe an approach for modeling fluid concentration profiles in grid-based microfluidic chips for fluid mixing. This approach provides an algorithm that predicts fluid concentrations at the chip outlets. Our algorithm significantly outperforms COMSOL finite element simulations in term of runtime while still producing results that closely approximate those of COMSOL.

cs.CE↗

arXiv2019

Information Gathering in Ad-Hoc Radio Networks

In the ad-hoc radio network model, nodes communicate with their neighbors via radio signals, without knowing the topology of the graph. We study the information gathering problem, where each node has a piece of information called a rumor, and the objective is to transmit all rumors to a designated target node. We provide an O(n^1.5*polylog(n)) deteministic protocol for information gathering in ad-hoc radio networks, significantly improving the trivial bound of O(n^2).

cs.DS↗

arXiv2019

A Waste-Efficient Algorithm for Single-Droplet Sample Preparation on Microfluidic Chips

We address the problem of designing micro-fluidic chips for sample preparation, which is a crucial step in many experimental processes in chemical and biological sciences. One of the objectives of sample preparation is to dilute the sample fluid, called reactant, using another fluid called buffer, to produce desired volumes of fluid with prespecified reactant concentrations. In the model we adopt, these fluids are manipulated in discrete volumes called droplets. The dilution process is represented by a mixing graph whose nodes represent 1-1 micro-mixers and edges represent channels for transporting fluids. In this work we focus on designing such mixing graphs when the given sample (also referred to as the target) consists of a single-droplet, and the objective is to minimize total fluid waste. Our main contribution is an efficient algorithm called RPRIS that guarantees a better provable worst-case bound on waste and significantly outperforms state-of-the-art algorithms in experimental comparison.

cs.DS↗

arXiv2019

Better Bounds for Online Line Chasing

We study online competitive algorithms for the \emph{line chasing problem} in Euclidean spaces $\reals^d$, where the input consists of an initial point $P_0$ and a sequence of lines $X_1,X_2,...,X_m$, revealed one at a time. At each step $t$, when the line $X_t$ is revealed, the algorithm must determine a point $P_t\in X_t$. An online algorithm is called $c$-competitive if for any input sequence the path $P_0, P_1,...,P_m$ it computes has length at most $c$ times the optimum path. The line chasing problem is a variant of a more general convex body chasing problem, where the sets $X_t$ are arbitrary convex sets. To date, the best competitive ratio for the line chasing problem was $28.1$, even in the plane. We significantly improve this bound, by providing a~$3$-competitive algorithm for any dimension $d$. We also improve the lower bound on the competitive ratio, from $1.412$ to $1.5358$.

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arXiv2019

An Omega(n^2) Lower Bound for Random Universal Sets for Planar Graphs

A set $U\subseteq \reals^2$ is $n$-universal if all $n$-vertex planar graphs have a planar straight-line embedding into $U$. We prove that if $Q \subseteq \reals^2$ consists of points chosen randomly and uniformly from the unit square then $Q$ must have cardinality $Ω(n^2)$ in order to be $n$-universal with high probability. This shows that the probabilistic method, at least in its basic form, cannot be used to establish an $o(n^2)$ upper bound on universal sets.

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arXiv2019

Towards a Theory of Mixing Graphs: A Characterization of Perfect Mixability

Some microfluidic lab-on-chip devices contain modules whose function is to mix two fluids, called reactant and buffer, in desired proportions. In one of the technologies for fluid mixing the process can be represented by a directed acyclic graph whose nodes represent micro-mixers and edges represent micro-channels. A micro-mixer has two input channels and two output channels; it receives two fluid droplets, one from each input, mixes them perfectly, and produces two droplets of the mixed fluid on its output channels. Such a mixing graph converts a set I of input droplets into a set T of output droplets, where the droplets are specified by their reactant concentrations. The most fundamental algorithmic question related to mixing graphs is to determine, given an input set I and a target set T, whether there is a mixing graph that converts I into T. We refer to this decision problem as mix-reachability. While the complexity of this problem remains open, we provide a solution to its natural sub-problem, called perfect mixability, in which we ask whether, given a collection C of droplets, there is a mixing graph that mixes C perfectly, producing only droplets whose concentration is the average concentration of C. We provide a complete characterization of such perfectly mixable sets and an efficient algorithm for testing perfect mixability. Further, we prove that any perfectly mixable set has a perfect-mixing graph of polynomial size, and that this graph can be computed in polynomial time.

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arXiv2016

Online Algorithms for Multi-Level Aggregation

In the Multi-Level Aggregation Problem (MLAP), requests arrive at the nodes of an edge-weighted tree T, and have to be served eventually. A service is defined as a subtree X of T that contains its root. This subtree X serves all requests that are pending in the nodes of X, and the cost of this service is equal to the total weight of X. Each request also incurs waiting cost between its arrival and service times. The objective is to minimize the total waiting cost of all requests plus the total cost of all service subtrees. MLAP is a generalization of some well-studied optimization problems; for example, for trees of depth 1, MLAP is equivalent to the TCP Acknowledgment Problem, while for trees of depth 2, it is equivalent to the Joint Replenishment Problem. Aggregation problem for trees of arbitrary depth arise in multicasting, sensor networks, communication in organization hierarchies, and in supply-chain management. The instances of MLAP associated with these applications are naturally online, in the sense that aggregation decisions need to be made without information about future requests. Constant-competitive online algorithms are known for MLAP with one or two levels. However, it has been open whether there exist constant competitive online algorithms for trees of depth more than 2. Addressing this open problem, we give the first constant competitive online algorithm for networks of arbitrary (fixed) number of levels. The competitive ratio is O(D^4 2^D), where D is the depth of T. The algorithm works for arbitrary waiting cost functions, including the variant with deadlines. We also show several additional lower and upper bound results for some special cases of MLAP, including the Single-Phase variant and the case when the tree is a path.

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arXiv2016

Online Clique Clustering

Clique clustering is the problem of partitioning the vertices of a graph into disjoint clusters, where each cluster forms a clique in the graph, while optimizing some objective function. In online clustering, the input graph is given one vertex at a time, and any vertices that have previously been clustered together are not allowed to be separated. The goal is to maintain a clustering with an objective value close to the optimal solution. For the variant where we want to maximize the number of edges in the clusters, we propose an online strategy based on the doubling technique. It has an asymptotic competitive ratio at most 15.646 and an absolute competitive ratio at most 22.641. We also show that no deterministic strategy can have an asymptotic competitive ratio better than 6. For the variant where we want to minimize the number of edges between clusters, we show that the deterministic competitive ratio of the problem is $n-ω(1)$, where n is the number of vertices in the graph.

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arXiv2016

Online Packet Scheduling with Bounded Delay and Lookahead

We study the online bounded-delay packet scheduling problem (BDPS), where packets of unit size arrive at a router over time and need to be transmitted over a network link. Each packet has two attributes: a non-negative weight and a deadline for its transmission. The objective is to maximize the total weight of the transmitted packets. This problem has been well studied in the literature, yet its optimal competitive ratio remains unknown: the best upper bound is $1.828$, still quite far from the best lower bound of $ϕ\approx 1.618$. In the variant of BDPS with $s$-bounded instances, each packet can be scheduled in at most $s$ consecutive slots, starting at its release time. The lower bound of $ϕ$ applies even to the special case of $2$-bounded instances, and a $ϕ$-competitive algorithm for $3$-bounded instances was given in Chin et al. Improving that result, and addressing a question posed by Goldwasser, we present a $ϕ$-competitive algorithm for $4$-bounded instances. We also study a variant of BDPS where an online algorithm has the additional power of $1$-lookahead, knowing at time $t$ which packets will arrive at time $t+1$. For BDPS with $1$-lookahead restricted to $2$-bounded instances, we present an online algorithm with competitive ratio $(\sqrt{13} - 1)/2 \approx 1.303$ and we prove a nearly tight lower bound of $(1 + \sqrt{17})/4 \approx 1.281$.

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arXiv2015

Faster Information Gathering in Ad-Hoc Radio Tree Networks

We study information gathering in ad-hoc radio networks. Initially, each node of the network has a piece of information called a rumor, and the overall objective is to gather all these rumors in the designated target node. The ad-hoc property refers to the fact that the topology of the network is unknown when the computation starts. Aggregation of rumors is not allowed, which means that each node may transmit at most one rumor in one step. We focus on networks with tree topologies, that is we assume that the network is a tree with all edges directed towards the root, but, being ad-hoc, its actual topology is not known. We provide two deterministic algorithms for this problem. For the model that does not assume any collision detection nor acknowledgement mechanisms, we give an $O(n\log\log n)$-time algorithm, improving the previous upper bound of $O(n\log n)$. We also show that this running time can be further reduced to $O(n)$ if the model allows for acknowledgements of successful transmissions.

cs.DS↗

arXiv2015

Approximation Algorithms for the Joint Replenishment Problem with Deadlines

The Joint Replenishment Problem (JRP) is a fundamental optimization problem in supply-chain management, concerned with optimizing the flow of goods from a supplier to retailers. Over time, in response to demands at the retailers, the supplier ships orders, via a warehouse, to the retailers. The objective is to schedule these orders to minimize the sum of ordering costs and retailers' waiting costs. We study the approximability of JRP-D, the version of JRP with deadlines, where instead of waiting costs the retailers impose strict deadlines. We study the integrality gap of the standard linear-program (LP) relaxation, giving a lower bound of 1.207, a stronger, computer-assisted lower bound of 1.245, as well as an upper bound and approximation ratio of 1.574. The best previous upper bound and approximation ratio was 1.667; no lower bound was previously published. For the special case when all demand periods are of equal length we give an upper bound of 1.5, a lower bound of 1.2, and show APX-hardness.

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arXiv2014

Information Gathering in Ad-Hoc Radio Networks with Tree Topology

We study the problem of information gathering in ad-hoc radio networks without collision detection, focussing on the case when the network forms a tree, with edges directed towards the root. Initially, each node has a piece of information that we refer to as a rumor. Our goal is to design protocols that deliver all rumors to the root of the tree as quickly as possible. The protocol must complete this task within its allotted time even though the actual tree topology is unknown when the computation starts. In the deterministic case, assuming that the nodes are labeled with small integers, we give an O(n)-time protocol that uses unbounded messages, and an O(n log n)-time protocol using bounded messages, where any message can include only one rumor. We also consider fire-and-forward protocols, in which a node can only transmit its own rumor or the rumor received in the previous step. We give a deterministic fire-and- forward protocol with running time O(n^1.5), and we show that it is asymptotically optimal. We then study randomized algorithms where the nodes are not labelled. In this model, we give an O(n log n)-time protocol and we prove that this bound is asymptotically optimal.

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arXiv2014

A Note on NP-Hardness of Preemptive Mean Flow-Time Scheduling for Parallel Machines

In the paper "The complexity of mean flow time scheduling problems with release times", by Baptiste, Brucker, Chrobak, Dürr, Kravchenko and Sourd, the authors claimed to prove strong NP-hardness of the scheduling problem $P|pmtn,r_j|\sum C_j$, namely multiprocessor preemptive scheduling where the objective is to minimize the mean flow time. We point out a serious error in their proof and give a new proof of strong NP-hardness for this problem.

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