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Documents authored by Gąsieniec, Leszek

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Gasieniec, Leszek

Document
DOI: 10.4230/LIPIcs.FUN.2024.15
Polyamorous Scheduling

Authors: Leszek Gąsieniec, Benjamin Smith, and Sebastian Wild

Published in: LIPIcs, Volume 291, 12th International Conference on Fun with Algorithms (FUN 2024)

Abstract
Finding schedules for pairwise meetings between the members of a complex social group without creating interpersonal conflict is challenging, especially when different relationships have different needs. We formally define and study the underlying optimisation problem: Polyamorous Scheduling. In Polyamorous Scheduling, we are given an edge-weighted graph and try to find a periodic schedule of matchings in this graph such that the maximal weighted waiting time between consecutive occurrences of the same edge is minimised. We show that the problem is NP-hard and that there is no efficient approximation algorithm with a better ratio than 4/3 unless P = NP. On the positive side, we obtain an O(log n)-approximation algorithm; indeed, an O(log Δ)-approximation for Δ the maximum degree, i.e., the largest number of relationships of any individual. We also define a generalisation of density from the Pinwheel Scheduling Problem, "poly density", and ask whether there exists a poly-density threshold similar to the 5/6-density threshold for Pinwheel Scheduling [Kawamura, STOC 2024]. Polyamorous Scheduling is a natural generalisation of Pinwheel Scheduling with respect to its optimisation variant, Bamboo Garden Trimming. Our work contributes the first nontrivial hardness-of-approximation reduction for any periodic scheduling problem, and opens up numerous avenues for further study of Polyamorous Scheduling.
Cite as

Leszek Gąsieniec, Benjamin Smith, and Sebastian Wild. Polyamorous Scheduling. In 12th International Conference on Fun with Algorithms (FUN 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 291, pp. 15:1-15:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{gasieniec_et_al:LIPIcs.FUN.2024.15,
  author =	{G\k{a}sieniec, Leszek and Smith, Benjamin and Wild, Sebastian},
  title =	{{Polyamorous Scheduling}},
  booktitle =	{12th International Conference on Fun with Algorithms (FUN 2024)},
  pages =	{15:1--15:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-314-0},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{291},
  editor =	{Broder, Andrei Z. and Tamir, Tami},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2024.15},
  URN =		{urn:nbn:de:0030-drops-199234},
  doi =		{10.4230/LIPIcs.FUN.2024.15},
  annote =	{Keywords: Periodic scheduling, Pinwheel scheduling, Edge-coloring, Chromatic index, Approximation algorithms, Hardness of approximation}
}
Document
DOI: 10.4230/LIPIcs.STACS.2023.33
New Clocks, Optimal Line Formation and Self-Replication Population Protocols

Authors: Leszek Gąsieniec, Paul G. Spirakis, and Grzegorz Stachowiak

Published in: LIPIcs, Volume 254, 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)

Abstract
In this paper we consider a known variant of the standard population protocol model in which agents are allowed to be connected by edges, referred to as the network constructor model. During an interaction between two agents the relevant connecting edge can be formed, maintained or eliminated by the transition function. Since pairs of agents are chosen uniformly at random the status of each edge is updated every Θ(n²) interactions in expectation which coincides with Θ(n) parallel time. This phenomenon provides a natural lower bound on the time complexity for any non-trivial network construction designed for this variant. This is in contrast with the standard population protocol model in which efficient protocols operate in O(polylog n) parallel time. The main focus of this paper is on efficient manipulation of linear structures including formation, self-replication and distribution (including pipelining) of complex information in the adopted model. - We propose and analyze a novel edge based phase clock counting parallel time Θ(nlog n) in the network constructor model, showing also that its leader based counterpart provides the same time guarantees in the standard population protocol model. Note that all currently known phase clocks can count parallel time not exceeding O(polylog n). - We prove that any spanning line formation protocol requires Ω(nlog n) parallel time if high probability guaranty is imposed. We also show that the new clock enables an optimal O(nlog n) parallel time spanning line construction, which improves dramatically on the best currently known O(n²) parallel time protocol, solving the main open problem in the considered model [O. Michail and P. Spirakis, 2016]. - We propose a new probabilistic bubble-sort algorithm in which random comparisons and transfers are limited to the adjacent positions in the sequence. Utilising a novel potential function reasoning we show that rather surprisingly this probabilistic sorting procedure requires O(n²) comparisons in expectation and whp, and is on par with its deterministic counterpart. - We propose the first population protocol allowing self-replication of a strand of an arbitrary length k (carrying k-bit message of size independent of the state space) in parallel time O(n(k+log n)). The bit pipelining mechanism and the time complexity analysis of self-replication process mimic those used in the probabilistic bubble-sort argument. The new protocol permits also simultaneous self-replication, where l copies of the strand can be created in parallel in time O(n(k+log n)log l). We also discuss application of the strand self-replication protocol to pattern matching. All protocols are always correct and provide time guarantees with high probability defined as 1-n^(-η), for a constant η > 0.
Cite as

Leszek Gąsieniec, Paul G. Spirakis, and Grzegorz Stachowiak. New Clocks, Optimal Line Formation and Self-Replication Population Protocols. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 33:1-33:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{gasieniec_et_al:LIPIcs.STACS.2023.33,
  author =	{G\k{a}sieniec, Leszek and Spirakis, Paul G. and Stachowiak, Grzegorz},
  title =	{{New Clocks, Optimal Line Formation and Self-Replication Population Protocols}},
  booktitle =	{40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)},
  pages =	{33:1--33:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-266-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{254},
  editor =	{Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2023.33},
  URN =		{urn:nbn:de:0030-drops-176857},
  doi =		{10.4230/LIPIcs.STACS.2023.33},
  annote =	{Keywords: Population protocols, constructors, probabilistic bubble-sort, self-replication}
}
Document
Brief Announcement
DOI: 10.4230/LIPIcs.DISC.2022.44
Brief Announcement: New Clocks, Fast Line Formation and Self-Replication Population Protocols

Authors: Leszek Gąsieniec, Paul Spirakis, and Grzegorz Stachowiak

Published in: LIPIcs, Volume 246, 36th International Symposium on Distributed Computing (DISC 2022)

Abstract
In this paper we consider a known variant of the standard population protocol model in which agents can be connected by edges, referred to as the network constructor model. During an interaction between two agents the relevant connecting edge can be formed, maintained or eliminated by the transition function. The state space of agents is fixed (constant size) and the size n of the population is not known, i.e., not hard-coded in the transition function. Since pairs of agents are chosen uniformly at random the status of each edge is updated every Θ(n²) interactions in expectation which coincides with Θ(n) parallel time. This phenomenon provides a natural lower bound on the time complexity for any non-trivial network construction designed for this variant. This is in contrast with the standard population protocol model in which efficient protocols operate in O(polylog n) parallel time. The main focus in this paper is on efficient manipulation of linear structures including formation, self-replication and distribution (including pipelining) of complex information in the adopted model. - We propose and analyse a novel edge based phase clock counting parallel time Θ(nlog n) in the network constructor model, showing also that its leader based counterpart provides the same time guaranties in the standard population protocol model. Note that all currently known phase clocks can count parallel time not exceeding O(polylog n). - The new clock enables a nearly optimal O(nlog n) parallel time spanning line construction (a key component of universal network construction), which improves dramatically on the best currently known O(n²) parallel time protocol, solving the main open problem in the considered model [O. Michail and P. Spirakis, 2016]. - We propose a new probabilistic bubble-sort algorithm in which random comparisons and transfers are allowed only between the adjacent positions in the sequence. Utilising a novel potential function reasoning we show that rather surprisingly this probabilistic sorting (via conditional pipelining) procedure requires O(n²) comparisons in expectation and whp, and is on par with its deterministic counterpart. - We propose the first population protocol allowing self-replication of a strand of an arbitrary length k (carrying a k-bit message of size independent of the state space) in parallel time O(n(k+log n)). The pipelining mechanism and the time complexity analysis of the strand self-replication protocol mimic those used in the probabilistic bubble-sort. The new protocol permits also simultaneous self-replication, where l copies of the strand can be created in time O(n(k+log n)log l). Finally, we discuss application of the strand self-replication protocol to pattern matching. Our protocols are always correct and provide time guaranties with high probability defined as 1-n^{-η}, for a constant η > 0.
Cite as

Leszek Gąsieniec, Paul Spirakis, and Grzegorz Stachowiak. Brief Announcement: New Clocks, Fast Line Formation and Self-Replication Population Protocols. In 36th International Symposium on Distributed Computing (DISC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 246, pp. 44:1-44:3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{gasieniec_et_al:LIPIcs.DISC.2022.44,
  author =	{G\k{a}sieniec, Leszek and Spirakis, Paul and Stachowiak, Grzegorz},
  title =	{{Brief Announcement: New Clocks, Fast Line Formation and Self-Replication Population Protocols}},
  booktitle =	{36th International Symposium on Distributed Computing (DISC 2022)},
  pages =	{44:1--44:3},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-255-6},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{246},
  editor =	{Scheideler, Christian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2022.44},
  URN =		{urn:nbn:de:0030-drops-172351},
  doi =		{10.4230/LIPIcs.DISC.2022.44},
  annote =	{Keywords: Population protocols, network constructors, probabilistic bubble-sort, self-replication}
}
Document
Invited Paper
DOI: 10.4230/LIPIcs.SWAT.2022.2
Time, Clocks and Efficiency of Population Protocols (Invited Paper)

Authors: Leszek Gąsieniec and Grzegorz Stachowiak

Published in: LIPIcs, Volume 227, 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)

Abstract
The model of population protocols is used to study distributed processes based on pairwise interactions between simple anonymous agents drawn from a large population of size n. The order in which agents meet in pairs is determined by the random scheduler, s.t., each consecutive pair is chosen uniformly at random. After each interaction the state of the relevant agents are amended according to the predefined transition function (the actual code of the algorithm) which governs the considered process. The state space of agents is often fixed and the size n is not known in advance, i.e., not hard-coded in the transition function. We assume that a population protocol starts in the predefined initial configuration of agents' states representing the input. And if successful, the protocol stabilises in a final configuration of states forming the output representing the solution to the considered problem. The time complexity of a population protocol refers to the number of interactions required to stabilise this protocol in a final configuration. We also define parallel time as the time complexity divided by n. Note that the parallel time of the system and the expected local time of each agent, i.e., the number of interactions observed by each agent, are correlated. Several mechanisms, known as phase clocks, have been developed to measure parallel time more accurately than counting local interactions. Most of the clocks target counting Θ(log n) parallel time required to fully synchronise all agents in the population. There are leader (and junta) based phase clocks which utilise a fixed number of states [D. Angluin et al., 2008; L. Gąsieniec and G. Stachowiak, 2021]. This type of clocks allows also counting any poly-logarithmic time while preserving fix state utilisation. The other type refers to leaderless clocks utilising Θ(log n) states [D. Alistarh et al., 2018; D. Doty et al., 2021]. This type allows approximate counting of parallel time as fixed resolution clocks [D. Doty et al., 2021] or oscillators [D. Alistarh et al., 2018]. Another clock type introduced recently in [L. Gąsieniec et al., 2021] enables counting Θ(nlog n) parallel time utilising a fixed number of states and either leaders or connections in the network constructor model. We also discuss parallel efficiency of population protocols referring to protocols operating in Θ(polylog n) parallel time, we propose extensions of the population protocol model leading to further improvement in state and time utilisation, and we state some open problems.
Cite as

Leszek Gąsieniec and Grzegorz Stachowiak. Time, Clocks and Efficiency of Population Protocols (Invited Paper). In 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 227, pp. 2:1-2:2, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{gasieniec_et_al:LIPIcs.SWAT.2022.2,
  author =	{G\k{a}sieniec, Leszek and Stachowiak, Grzegorz},
  title =	{{Time, Clocks and Efficiency of Population Protocols}},
  booktitle =	{18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)},
  pages =	{2:1--2:2},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-236-5},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{227},
  editor =	{Czumaj, Artur and Xin, Qin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2022.2},
  URN =		{urn:nbn:de:0030-drops-161624},
  doi =		{10.4230/LIPIcs.SWAT.2022.2},
  annote =	{Keywords: Population protocols, phase clocks, oscillators, parallel time and efficiency}
}
Document
DOI: 10.4230/LIPIcs.OPODIS.2021.12
Efficient Assignment of Identities in Anonymous Populations

Authors: Leszek Gąsieniec, Jesper Jansson, Christos Levcopoulos, and Andrzej Lingas

Published in: LIPIcs, Volume 217, 25th International Conference on Principles of Distributed Systems (OPODIS 2021)

Abstract
We consider the fundamental problem of assigning distinct labels to agents in the probabilistic model of population protocols. Our protocols operate under the assumption that the size n of the population is embedded in the transition function. Their efficiency is expressed in terms of the number of states utilized by agents, the size of the range from which the labels are drawn, and the expected number of interactions required by our solutions. Our primary goal is to provide efficient protocols for this fundamental problem complemented with tight lower bounds in all the three aspects. W.h.p. (with high probability), our labeling protocols are silent, i.e., eventually each agent reaches its final state and remains in it forever, and they are safe, i.e., never update the label assigned to any single agent. We first present a silent w.h.p. and safe labeling protocol that draws labels from the range [1,2n]. Both the number of interactions required and the number of states used by the protocol are asymptotically optimal, i.e., O(n log n) w.h.p. and O(n), respectively. Next, we present a generalization of the protocol, where the range of assigned labels is [1,(1+ε) n]. The generalized protocol requires O(n log n / ε) interactions in order to complete the assignment of distinct labels from [1,(1+ε) n] to the n agents, w.h.p. It is also silent w.h.p. and safe, and uses (2+ε)n+O(n^c) states, for any positive c < 1. On the other hand, we consider the so-called pool labeling protocols that include our fast protocols. We show that the expected number of interactions required by any pool protocol is ≥ (n²)/(r+1), when the labels range is 1,… , n+r < 2n. Furthermore, we provide a protocol which uses only n+5√ n +O(n^c) states, for any c < 1, and draws labels from the range 1,… ,n. The expected number of interactions required by the protocol is O(n³). Once a unique leader is elected it produces a valid labeling and it is silent and safe. On the other hand, we show that (even if a unique leader is given in advance) any silent protocol that produces a valid labeling and is safe with probability > 1-(1/n), uses ≥ n+√{(n-1)/2}-1 states. Hence, our protocol is almost state-optimal. We also present a generalization of the protocol to include a trade-off between the number of states and the expected number of interactions. Finally, we show that for any silent and safe labeling protocol utilizing n+t < 2n states, the expected number of interactions required to achieve a valid labeling is ≥ (n²)/(t+1).
Cite as

Leszek Gąsieniec, Jesper Jansson, Christos Levcopoulos, and Andrzej Lingas. Efficient Assignment of Identities in Anonymous Populations. In 25th International Conference on Principles of Distributed Systems (OPODIS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 217, pp. 12:1-12:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{gasieniec_et_al:LIPIcs.OPODIS.2021.12,
  author =	{G\k{a}sieniec, Leszek and Jansson, Jesper and Levcopoulos, Christos and Lingas, Andrzej},
  title =	{{Efficient Assignment of Identities in Anonymous Populations}},
  booktitle =	{25th International Conference on Principles of Distributed Systems (OPODIS 2021)},
  pages =	{12:1--12:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-219-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{217},
  editor =	{Bramas, Quentin and Gramoli, Vincent and Milani, Alessia},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2021.12},
  URN =		{urn:nbn:de:0030-drops-157871},
  doi =		{10.4230/LIPIcs.OPODIS.2021.12},
  annote =	{Keywords: population protocol, state efficiency, time efficiency, one-way epidemics, leader election, agent identities}
}
Document
DOI: 10.4230/LIPIcs.OPODIS.2016.14
Deterministic Population Protocols for Exact Majority and Plurality

Authors: Leszek Gasieniec, David Hamilton, Russell Martin, Paul G. Spirakis, and Grzegorz Stachowiak

Published in: LIPIcs, Volume 70, 20th International Conference on Principles of Distributed Systems (OPODIS 2016)

Abstract
In this paper we study space-efficient deterministic population protocols for several variants of the majority problem including plurality consensus. We focus on space efficient majority protocols in populations with an arbitrary number of colours C represented by k-bit labels, where k = ceiling (log C). In particular, we present asymptotically space-optimal (with respect to the adopted k-bit representation of colours) protocols for (1) the absolute majority problem, i.e., a protocol which decides whether a single colour dominates all other colours considered together, and (2) the relative majority problem, also known in the literature as plurality consensus, in which colours declare their volume superiority versus other individual colours. The new population protocols proposed in this paper rely on a dynamic formulation of the majority problem in which the colours originally present in the population can be changed by an external force during the communication process. The considered dynamic formulation is based on the concepts studied by D. Angluin et al. and O. Michail et al. about stabilizing inputs and composition of population protocols. Also, the protocols presented in this paper use a composition of some known protocols for static and dynamic majority.
Cite as

Leszek Gasieniec, David Hamilton, Russell Martin, Paul G. Spirakis, and Grzegorz Stachowiak. Deterministic Population Protocols for Exact Majority and Plurality. In 20th International Conference on Principles of Distributed Systems (OPODIS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 70, pp. 14:1-14:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{gasieniec_et_al:LIPIcs.OPODIS.2016.14,
  author =	{Gasieniec, Leszek and Hamilton, David and Martin, Russell and Spirakis, Paul G. and Stachowiak, Grzegorz},
  title =	{{Deterministic Population Protocols for Exact Majority and Plurality}},
  booktitle =	{20th International Conference on Principles of Distributed Systems (OPODIS 2016)},
  pages =	{14:1--14:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-031-6},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{70},
  editor =	{Fatourou, Panagiota and Jim\'{e}nez, Ernesto and Pedone, Fernando},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2016.14},
  URN =		{urn:nbn:de:0030-drops-70837},
  doi =		{10.4230/LIPIcs.OPODIS.2016.14},
  annote =	{Keywords: Deterministic population protocols, majority, plurality consenus}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2011.423
Optimal Packed String Matching

Authors: Oren Ben-Kiki, Philip Bille, Dany Breslauer, Leszek Gasieniec, Roberto Grossi, and Oren Weimann

Published in: LIPIcs, Volume 13, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)

Abstract
In the packed string matching problem, each machine word accomodates alpha characters, thus an n-character text occupies n/alpha memory words. We extend the Crochemore-Perrin constant-space O(n)-time string matching algorithm to run in optimal O(n/alpha) time and even in real-time, achieving a factor alpha speedup over traditional algorithms that examine each character individually. Our solution can be efficiently implemented, unlike prior theoretical packed string matching work. We adapt the standard RAM model and only use its AC0 instructions (i.e. no multiplication) plus two specialized AC0 packed string instructions. The main string-matching instruction is available in commodity processors (i.e. Intel's SSE4.2 and AVX Advanced String Operations); the other maximal-suffix instruction is only required during pattern preprocessing. In the absence of these two specialized instructions, we propose theoretically-efficient emulation using integer multiplication (not AC0) and table lookup.
Cite as

Oren Ben-Kiki, Philip Bille, Dany Breslauer, Leszek Gasieniec, Roberto Grossi, and Oren Weimann. Optimal Packed String Matching. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 13, pp. 423-432, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{benkiki_et_al:LIPIcs.FSTTCS.2011.423,
  author =	{Ben-Kiki, Oren and Bille, Philip and Breslauer, Dany and Gasieniec, Leszek and Grossi, Roberto and Weimann, Oren},
  title =	{{Optimal Packed String Matching}},
  booktitle =	{IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)},
  pages =	{423--432},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-34-7},
  ISSN =	{1868-8969},
  year =	{2011},
  volume =	{13},
  editor =	{Chakraborty, Supratik and Kumar, Amit},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2011.423},
  URN =		{urn:nbn:de:0030-drops-33558},
  doi =		{10.4230/LIPIcs.FSTTCS.2011.423},
  annote =	{Keywords: String matching, bit parallelism, real time, space efficiency}
}

Gąsieniec, Leszek

Document
DOI: 10.4230/LIPIcs.FUN.2024.15
Polyamorous Scheduling

Authors: Leszek Gąsieniec, Benjamin Smith, and Sebastian Wild

Published in: LIPIcs, Volume 291, 12th International Conference on Fun with Algorithms (FUN 2024)

Abstract
Finding schedules for pairwise meetings between the members of a complex social group without creating interpersonal conflict is challenging, especially when different relationships have different needs. We formally define and study the underlying optimisation problem: Polyamorous Scheduling. In Polyamorous Scheduling, we are given an edge-weighted graph and try to find a periodic schedule of matchings in this graph such that the maximal weighted waiting time between consecutive occurrences of the same edge is minimised. We show that the problem is NP-hard and that there is no efficient approximation algorithm with a better ratio than 4/3 unless P = NP. On the positive side, we obtain an O(log n)-approximation algorithm; indeed, an O(log Δ)-approximation for Δ the maximum degree, i.e., the largest number of relationships of any individual. We also define a generalisation of density from the Pinwheel Scheduling Problem, "poly density", and ask whether there exists a poly-density threshold similar to the 5/6-density threshold for Pinwheel Scheduling [Kawamura, STOC 2024]. Polyamorous Scheduling is a natural generalisation of Pinwheel Scheduling with respect to its optimisation variant, Bamboo Garden Trimming. Our work contributes the first nontrivial hardness-of-approximation reduction for any periodic scheduling problem, and opens up numerous avenues for further study of Polyamorous Scheduling.
Cite as

Leszek Gąsieniec, Benjamin Smith, and Sebastian Wild. Polyamorous Scheduling. In 12th International Conference on Fun with Algorithms (FUN 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 291, pp. 15:1-15:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{gasieniec_et_al:LIPIcs.FUN.2024.15,
  author =	{G\k{a}sieniec, Leszek and Smith, Benjamin and Wild, Sebastian},
  title =	{{Polyamorous Scheduling}},
  booktitle =	{12th International Conference on Fun with Algorithms (FUN 2024)},
  pages =	{15:1--15:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-314-0},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{291},
  editor =	{Broder, Andrei Z. and Tamir, Tami},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2024.15},
  URN =		{urn:nbn:de:0030-drops-199234},
  doi =		{10.4230/LIPIcs.FUN.2024.15},
  annote =	{Keywords: Periodic scheduling, Pinwheel scheduling, Edge-coloring, Chromatic index, Approximation algorithms, Hardness of approximation}
}
Document
DOI: 10.4230/LIPIcs.STACS.2023.33
New Clocks, Optimal Line Formation and Self-Replication Population Protocols

Authors: Leszek Gąsieniec, Paul G. Spirakis, and Grzegorz Stachowiak

Published in: LIPIcs, Volume 254, 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)

Abstract
In this paper we consider a known variant of the standard population protocol model in which agents are allowed to be connected by edges, referred to as the network constructor model. During an interaction between two agents the relevant connecting edge can be formed, maintained or eliminated by the transition function. Since pairs of agents are chosen uniformly at random the status of each edge is updated every Θ(n²) interactions in expectation which coincides with Θ(n) parallel time. This phenomenon provides a natural lower bound on the time complexity for any non-trivial network construction designed for this variant. This is in contrast with the standard population protocol model in which efficient protocols operate in O(polylog n) parallel time. The main focus of this paper is on efficient manipulation of linear structures including formation, self-replication and distribution (including pipelining) of complex information in the adopted model. - We propose and analyze a novel edge based phase clock counting parallel time Θ(nlog n) in the network constructor model, showing also that its leader based counterpart provides the same time guarantees in the standard population protocol model. Note that all currently known phase clocks can count parallel time not exceeding O(polylog n). - We prove that any spanning line formation protocol requires Ω(nlog n) parallel time if high probability guaranty is imposed. We also show that the new clock enables an optimal O(nlog n) parallel time spanning line construction, which improves dramatically on the best currently known O(n²) parallel time protocol, solving the main open problem in the considered model [O. Michail and P. Spirakis, 2016]. - We propose a new probabilistic bubble-sort algorithm in which random comparisons and transfers are limited to the adjacent positions in the sequence. Utilising a novel potential function reasoning we show that rather surprisingly this probabilistic sorting procedure requires O(n²) comparisons in expectation and whp, and is on par with its deterministic counterpart. - We propose the first population protocol allowing self-replication of a strand of an arbitrary length k (carrying k-bit message of size independent of the state space) in parallel time O(n(k+log n)). The bit pipelining mechanism and the time complexity analysis of self-replication process mimic those used in the probabilistic bubble-sort argument. The new protocol permits also simultaneous self-replication, where l copies of the strand can be created in parallel in time O(n(k+log n)log l). We also discuss application of the strand self-replication protocol to pattern matching. All protocols are always correct and provide time guarantees with high probability defined as 1-n^(-η), for a constant η > 0.
Cite as

Leszek Gąsieniec, Paul G. Spirakis, and Grzegorz Stachowiak. New Clocks, Optimal Line Formation and Self-Replication Population Protocols. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 33:1-33:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{gasieniec_et_al:LIPIcs.STACS.2023.33,
  author =	{G\k{a}sieniec, Leszek and Spirakis, Paul G. and Stachowiak, Grzegorz},
  title =	{{New Clocks, Optimal Line Formation and Self-Replication Population Protocols}},
  booktitle =	{40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)},
  pages =	{33:1--33:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-266-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{254},
  editor =	{Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2023.33},
  URN =		{urn:nbn:de:0030-drops-176857},
  doi =		{10.4230/LIPIcs.STACS.2023.33},
  annote =	{Keywords: Population protocols, constructors, probabilistic bubble-sort, self-replication}
}
Document
Brief Announcement
DOI: 10.4230/LIPIcs.DISC.2022.44
Brief Announcement: New Clocks, Fast Line Formation and Self-Replication Population Protocols

Authors: Leszek Gąsieniec, Paul Spirakis, and Grzegorz Stachowiak

Published in: LIPIcs, Volume 246, 36th International Symposium on Distributed Computing (DISC 2022)

Abstract
In this paper we consider a known variant of the standard population protocol model in which agents can be connected by edges, referred to as the network constructor model. During an interaction between two agents the relevant connecting edge can be formed, maintained or eliminated by the transition function. The state space of agents is fixed (constant size) and the size n of the population is not known, i.e., not hard-coded in the transition function. Since pairs of agents are chosen uniformly at random the status of each edge is updated every Θ(n²) interactions in expectation which coincides with Θ(n) parallel time. This phenomenon provides a natural lower bound on the time complexity for any non-trivial network construction designed for this variant. This is in contrast with the standard population protocol model in which efficient protocols operate in O(polylog n) parallel time. The main focus in this paper is on efficient manipulation of linear structures including formation, self-replication and distribution (including pipelining) of complex information in the adopted model. - We propose and analyse a novel edge based phase clock counting parallel time Θ(nlog n) in the network constructor model, showing also that its leader based counterpart provides the same time guaranties in the standard population protocol model. Note that all currently known phase clocks can count parallel time not exceeding O(polylog n). - The new clock enables a nearly optimal O(nlog n) parallel time spanning line construction (a key component of universal network construction), which improves dramatically on the best currently known O(n²) parallel time protocol, solving the main open problem in the considered model [O. Michail and P. Spirakis, 2016]. - We propose a new probabilistic bubble-sort algorithm in which random comparisons and transfers are allowed only between the adjacent positions in the sequence. Utilising a novel potential function reasoning we show that rather surprisingly this probabilistic sorting (via conditional pipelining) procedure requires O(n²) comparisons in expectation and whp, and is on par with its deterministic counterpart. - We propose the first population protocol allowing self-replication of a strand of an arbitrary length k (carrying a k-bit message of size independent of the state space) in parallel time O(n(k+log n)). The pipelining mechanism and the time complexity analysis of the strand self-replication protocol mimic those used in the probabilistic bubble-sort. The new protocol permits also simultaneous self-replication, where l copies of the strand can be created in time O(n(k+log n)log l). Finally, we discuss application of the strand self-replication protocol to pattern matching. Our protocols are always correct and provide time guaranties with high probability defined as 1-n^{-η}, for a constant η > 0.
Cite as

Leszek Gąsieniec, Paul Spirakis, and Grzegorz Stachowiak. Brief Announcement: New Clocks, Fast Line Formation and Self-Replication Population Protocols. In 36th International Symposium on Distributed Computing (DISC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 246, pp. 44:1-44:3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{gasieniec_et_al:LIPIcs.DISC.2022.44,
  author =	{G\k{a}sieniec, Leszek and Spirakis, Paul and Stachowiak, Grzegorz},
  title =	{{Brief Announcement: New Clocks, Fast Line Formation and Self-Replication Population Protocols}},
  booktitle =	{36th International Symposium on Distributed Computing (DISC 2022)},
  pages =	{44:1--44:3},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-255-6},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{246},
  editor =	{Scheideler, Christian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2022.44},
  URN =		{urn:nbn:de:0030-drops-172351},
  doi =		{10.4230/LIPIcs.DISC.2022.44},
  annote =	{Keywords: Population protocols, network constructors, probabilistic bubble-sort, self-replication}
}
Document
Invited Paper
DOI: 10.4230/LIPIcs.SWAT.2022.2
Time, Clocks and Efficiency of Population Protocols (Invited Paper)

Authors: Leszek Gąsieniec and Grzegorz Stachowiak

Published in: LIPIcs, Volume 227, 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)

Abstract
The model of population protocols is used to study distributed processes based on pairwise interactions between simple anonymous agents drawn from a large population of size n. The order in which agents meet in pairs is determined by the random scheduler, s.t., each consecutive pair is chosen uniformly at random. After each interaction the state of the relevant agents are amended according to the predefined transition function (the actual code of the algorithm) which governs the considered process. The state space of agents is often fixed and the size n is not known in advance, i.e., not hard-coded in the transition function. We assume that a population protocol starts in the predefined initial configuration of agents' states representing the input. And if successful, the protocol stabilises in a final configuration of states forming the output representing the solution to the considered problem. The time complexity of a population protocol refers to the number of interactions required to stabilise this protocol in a final configuration. We also define parallel time as the time complexity divided by n. Note that the parallel time of the system and the expected local time of each agent, i.e., the number of interactions observed by each agent, are correlated. Several mechanisms, known as phase clocks, have been developed to measure parallel time more accurately than counting local interactions. Most of the clocks target counting Θ(log n) parallel time required to fully synchronise all agents in the population. There are leader (and junta) based phase clocks which utilise a fixed number of states [D. Angluin et al., 2008; L. Gąsieniec and G. Stachowiak, 2021]. This type of clocks allows also counting any poly-logarithmic time while preserving fix state utilisation. The other type refers to leaderless clocks utilising Θ(log n) states [D. Alistarh et al., 2018; D. Doty et al., 2021]. This type allows approximate counting of parallel time as fixed resolution clocks [D. Doty et al., 2021] or oscillators [D. Alistarh et al., 2018]. Another clock type introduced recently in [L. Gąsieniec et al., 2021] enables counting Θ(nlog n) parallel time utilising a fixed number of states and either leaders or connections in the network constructor model. We also discuss parallel efficiency of population protocols referring to protocols operating in Θ(polylog n) parallel time, we propose extensions of the population protocol model leading to further improvement in state and time utilisation, and we state some open problems.
Cite as

Leszek Gąsieniec and Grzegorz Stachowiak. Time, Clocks and Efficiency of Population Protocols (Invited Paper). In 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 227, pp. 2:1-2:2, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{gasieniec_et_al:LIPIcs.SWAT.2022.2,
  author =	{G\k{a}sieniec, Leszek and Stachowiak, Grzegorz},
  title =	{{Time, Clocks and Efficiency of Population Protocols}},
  booktitle =	{18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)},
  pages =	{2:1--2:2},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-236-5},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{227},
  editor =	{Czumaj, Artur and Xin, Qin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2022.2},
  URN =		{urn:nbn:de:0030-drops-161624},
  doi =		{10.4230/LIPIcs.SWAT.2022.2},
  annote =	{Keywords: Population protocols, phase clocks, oscillators, parallel time and efficiency}
}
Document
DOI: 10.4230/LIPIcs.OPODIS.2021.12
Efficient Assignment of Identities in Anonymous Populations

Authors: Leszek Gąsieniec, Jesper Jansson, Christos Levcopoulos, and Andrzej Lingas

Published in: LIPIcs, Volume 217, 25th International Conference on Principles of Distributed Systems (OPODIS 2021)

Abstract
We consider the fundamental problem of assigning distinct labels to agents in the probabilistic model of population protocols. Our protocols operate under the assumption that the size n of the population is embedded in the transition function. Their efficiency is expressed in terms of the number of states utilized by agents, the size of the range from which the labels are drawn, and the expected number of interactions required by our solutions. Our primary goal is to provide efficient protocols for this fundamental problem complemented with tight lower bounds in all the three aspects. W.h.p. (with high probability), our labeling protocols are silent, i.e., eventually each agent reaches its final state and remains in it forever, and they are safe, i.e., never update the label assigned to any single agent. We first present a silent w.h.p. and safe labeling protocol that draws labels from the range [1,2n]. Both the number of interactions required and the number of states used by the protocol are asymptotically optimal, i.e., O(n log n) w.h.p. and O(n), respectively. Next, we present a generalization of the protocol, where the range of assigned labels is [1,(1+ε) n]. The generalized protocol requires O(n log n / ε) interactions in order to complete the assignment of distinct labels from [1,(1+ε) n] to the n agents, w.h.p. It is also silent w.h.p. and safe, and uses (2+ε)n+O(n^c) states, for any positive c < 1. On the other hand, we consider the so-called pool labeling protocols that include our fast protocols. We show that the expected number of interactions required by any pool protocol is ≥ (n²)/(r+1), when the labels range is 1,… , n+r < 2n. Furthermore, we provide a protocol which uses only n+5√ n +O(n^c) states, for any c < 1, and draws labels from the range 1,… ,n. The expected number of interactions required by the protocol is O(n³). Once a unique leader is elected it produces a valid labeling and it is silent and safe. On the other hand, we show that (even if a unique leader is given in advance) any silent protocol that produces a valid labeling and is safe with probability > 1-(1/n), uses ≥ n+√{(n-1)/2}-1 states. Hence, our protocol is almost state-optimal. We also present a generalization of the protocol to include a trade-off between the number of states and the expected number of interactions. Finally, we show that for any silent and safe labeling protocol utilizing n+t < 2n states, the expected number of interactions required to achieve a valid labeling is ≥ (n²)/(t+1).
Cite as

Leszek Gąsieniec, Jesper Jansson, Christos Levcopoulos, and Andrzej Lingas. Efficient Assignment of Identities in Anonymous Populations. In 25th International Conference on Principles of Distributed Systems (OPODIS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 217, pp. 12:1-12:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{gasieniec_et_al:LIPIcs.OPODIS.2021.12,
  author =	{G\k{a}sieniec, Leszek and Jansson, Jesper and Levcopoulos, Christos and Lingas, Andrzej},
  title =	{{Efficient Assignment of Identities in Anonymous Populations}},
  booktitle =	{25th International Conference on Principles of Distributed Systems (OPODIS 2021)},
  pages =	{12:1--12:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-219-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{217},
  editor =	{Bramas, Quentin and Gramoli, Vincent and Milani, Alessia},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2021.12},
  URN =		{urn:nbn:de:0030-drops-157871},
  doi =		{10.4230/LIPIcs.OPODIS.2021.12},
  annote =	{Keywords: population protocol, state efficiency, time efficiency, one-way epidemics, leader election, agent identities}
}
Document
DOI: 10.4230/LIPIcs.OPODIS.2016.14
Deterministic Population Protocols for Exact Majority and Plurality

Authors: Leszek Gasieniec, David Hamilton, Russell Martin, Paul G. Spirakis, and Grzegorz Stachowiak

Published in: LIPIcs, Volume 70, 20th International Conference on Principles of Distributed Systems (OPODIS 2016)

Abstract
In this paper we study space-efficient deterministic population protocols for several variants of the majority problem including plurality consensus. We focus on space efficient majority protocols in populations with an arbitrary number of colours C represented by k-bit labels, where k = ceiling (log C). In particular, we present asymptotically space-optimal (with respect to the adopted k-bit representation of colours) protocols for (1) the absolute majority problem, i.e., a protocol which decides whether a single colour dominates all other colours considered together, and (2) the relative majority problem, also known in the literature as plurality consensus, in which colours declare their volume superiority versus other individual colours. The new population protocols proposed in this paper rely on a dynamic formulation of the majority problem in which the colours originally present in the population can be changed by an external force during the communication process. The considered dynamic formulation is based on the concepts studied by D. Angluin et al. and O. Michail et al. about stabilizing inputs and composition of population protocols. Also, the protocols presented in this paper use a composition of some known protocols for static and dynamic majority.
Cite as

Leszek Gasieniec, David Hamilton, Russell Martin, Paul G. Spirakis, and Grzegorz Stachowiak. Deterministic Population Protocols for Exact Majority and Plurality. In 20th International Conference on Principles of Distributed Systems (OPODIS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 70, pp. 14:1-14:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{gasieniec_et_al:LIPIcs.OPODIS.2016.14,
  author =	{Gasieniec, Leszek and Hamilton, David and Martin, Russell and Spirakis, Paul G. and Stachowiak, Grzegorz},
  title =	{{Deterministic Population Protocols for Exact Majority and Plurality}},
  booktitle =	{20th International Conference on Principles of Distributed Systems (OPODIS 2016)},
  pages =	{14:1--14:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-031-6},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{70},
  editor =	{Fatourou, Panagiota and Jim\'{e}nez, Ernesto and Pedone, Fernando},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2016.14},
  URN =		{urn:nbn:de:0030-drops-70837},
  doi =		{10.4230/LIPIcs.OPODIS.2016.14},
  annote =	{Keywords: Deterministic population protocols, majority, plurality consenus}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2011.423
Optimal Packed String Matching

Authors: Oren Ben-Kiki, Philip Bille, Dany Breslauer, Leszek Gasieniec, Roberto Grossi, and Oren Weimann

Published in: LIPIcs, Volume 13, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)

Abstract
In the packed string matching problem, each machine word accomodates alpha characters, thus an n-character text occupies n/alpha memory words. We extend the Crochemore-Perrin constant-space O(n)-time string matching algorithm to run in optimal O(n/alpha) time and even in real-time, achieving a factor alpha speedup over traditional algorithms that examine each character individually. Our solution can be efficiently implemented, unlike prior theoretical packed string matching work. We adapt the standard RAM model and only use its AC0 instructions (i.e. no multiplication) plus two specialized AC0 packed string instructions. The main string-matching instruction is available in commodity processors (i.e. Intel's SSE4.2 and AVX Advanced String Operations); the other maximal-suffix instruction is only required during pattern preprocessing. In the absence of these two specialized instructions, we propose theoretically-efficient emulation using integer multiplication (not AC0) and table lookup.
Cite as

Oren Ben-Kiki, Philip Bille, Dany Breslauer, Leszek Gasieniec, Roberto Grossi, and Oren Weimann. Optimal Packed String Matching. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 13, pp. 423-432, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{benkiki_et_al:LIPIcs.FSTTCS.2011.423,
  author =	{Ben-Kiki, Oren and Bille, Philip and Breslauer, Dany and Gasieniec, Leszek and Grossi, Roberto and Weimann, Oren},
  title =	{{Optimal Packed String Matching}},
  booktitle =	{IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)},
  pages =	{423--432},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-34-7},
  ISSN =	{1868-8969},
  year =	{2011},
  volume =	{13},
  editor =	{Chakraborty, Supratik and Kumar, Amit},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2011.423},
  URN =		{urn:nbn:de:0030-drops-33558},
  doi =		{10.4230/LIPIcs.FSTTCS.2011.423},
  annote =	{Keywords: String matching, bit parallelism, real time, space efficiency}
}
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