---
_id: '10416'
abstract:
- lang: eng
text: 'A fundamental algorithmic problem at the heart of static analysis is Dyck
reachability. The input is a graph where the edges are labeled with different
types of opening and closing parentheses, and the reachability information is
computed via paths whose parentheses are properly matched. We present new results
for Dyck reachability problems with applications to alias analysis and data-dependence
analysis. Our main contributions, that include improved upper bounds as well as
lower bounds that establish optimality guarantees, are as follows: First, we consider
Dyck reachability on bidirected graphs, which is the standard way of performing
field-sensitive points-to analysis. Given a bidirected graph with n nodes and
m edges, we present: (i) an algorithm with worst-case running time O(m + n · α(n)),
where α(n) is the inverse Ackermann function, improving the previously known O(n2)
time bound; (ii) a matching lower bound that shows that our algorithm is optimal
wrt to worst-case complexity; and (iii) an optimal average-case upper bound of
O(m) time, improving the previously known O(m · logn) bound. Second, we consider
the problem of context-sensitive data-dependence analysis, where the task is to
obtain analysis summaries of library code in the presence of callbacks. Our algorithm
preprocesses libraries in almost linear time, after which the contribution of
the library in the complexity of the client analysis is only linear, and only
wrt the number of call sites. Third, we prove that combinatorial algorithms for
Dyck reachability on general graphs with truly sub-cubic bounds cannot be obtained
without obtaining sub-cubic combinatorial algorithms for Boolean Matrix Multiplication,
which is a long-standing open problem. Thus we establish that the existing combinatorial
algorithms for Dyck reachability are (conditionally) optimal for general graphs.
We also show that the same hardness holds for graphs of constant treewidth. Finally,
we provide a prototype implementation of our algorithms for both alias analysis
and data-dependence analysis. Our experimental evaluation demonstrates that the
new algorithms significantly outperform all existing methods on the two problems,
over real-world benchmarks.'
acknowledgement: "The research was partly supported by Austrian Science Fund (FWF)
Grant No P23499-N23, FWF NFN Grant No S11407-N23 (RiSE/SHiNE), and ERC Start grant
(279307: Graph Games).\r\n"
article_number: '30'
article_processing_charge: No
article_type: original
author:
- first_name: Krishnendu
full_name: Chatterjee, Krishnendu
id: 2E5DCA20-F248-11E8-B48F-1D18A9856A87
last_name: Chatterjee
orcid: 0000-0002-4561-241X
- first_name: Bhavya
full_name: Choudhary, Bhavya
last_name: Choudhary
- first_name: Andreas
full_name: Pavlogiannis, Andreas
id: 49704004-F248-11E8-B48F-1D18A9856A87
last_name: Pavlogiannis
orcid: 0000-0002-8943-0722
citation:
ama: Chatterjee K, Choudhary B, Pavlogiannis A. Optimal Dyck reachability for data-dependence
and Alias analysis. Proceedings of the ACM on Programming Languages. 2017;2(POPL).
doi:10.1145/3158118
apa: 'Chatterjee, K., Choudhary, B., & Pavlogiannis, A. (2017). Optimal Dyck
reachability for data-dependence and Alias analysis. Proceedings of the ACM
on Programming Languages. Los Angeles, CA, United States: Association for
Computing Machinery. https://doi.org/10.1145/3158118'
chicago: Chatterjee, Krishnendu, Bhavya Choudhary, and Andreas Pavlogiannis. “Optimal
Dyck Reachability for Data-Dependence and Alias Analysis.” Proceedings of the
ACM on Programming Languages. Association for Computing Machinery, 2017. https://doi.org/10.1145/3158118.
ieee: K. Chatterjee, B. Choudhary, and A. Pavlogiannis, “Optimal Dyck reachability
for data-dependence and Alias analysis,” Proceedings of the ACM on Programming
Languages, vol. 2, no. POPL. Association for Computing Machinery, 2017.
ista: Chatterjee K, Choudhary B, Pavlogiannis A. 2017. Optimal Dyck reachability
for data-dependence and Alias analysis. Proceedings of the ACM on Programming
Languages. 2(POPL), 30.
mla: Chatterjee, Krishnendu, et al. “Optimal Dyck Reachability for Data-Dependence
and Alias Analysis.” Proceedings of the ACM on Programming Languages, vol.
2, no. POPL, 30, Association for Computing Machinery, 2017, doi:10.1145/3158118.
short: K. Chatterjee, B. Choudhary, A. Pavlogiannis, Proceedings of the ACM on Programming
Languages 2 (2017).
conference:
end_date: 2018-01-13
location: Los Angeles, CA, United States
name: 'POPL: Programming Languages'
start_date: 2018-01-07
date_created: 2021-12-05T23:01:48Z
date_published: 2017-12-27T00:00:00Z
date_updated: 2023-02-23T12:27:13Z
day: '27'
ddc:
- '000'
department:
- _id: KrCh
doi: 10.1145/3158118
ec_funded: 1
external_id:
arxiv:
- '1910.00241'
file:
- access_level: open_access
checksum: faa3f7b3fe8aab84b50ed805c26a0ee5
content_type: application/pdf
creator: cchlebak
date_created: 2021-12-07T08:06:28Z
date_updated: 2021-12-07T08:06:28Z
file_id: '10421'
file_name: 2017_ACMProgLang_Chatterjee.pdf
file_size: 460188
relation: main_file
success: 1
file_date_updated: 2021-12-07T08:06:28Z
has_accepted_license: '1'
intvolume: ' 2'
issue: POPL
language:
- iso: eng
month: '12'
oa: 1
oa_version: Published Version
project:
- _id: 2581B60A-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '279307'
name: 'Quantitative Graph Games: Theory and Applications'
- _id: 2584A770-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: P 23499-N23
name: Modern Graph Algorithmic Techniques in Formal Verification
- _id: 25832EC2-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: S 11407_N23
name: Rigorous Systems Engineering
publication: Proceedings of the ACM on Programming Languages
publication_identifier:
eissn:
- 2475-1421
publication_status: published
publisher: Association for Computing Machinery
quality_controlled: '1'
related_material:
record:
- id: '5455'
relation: earlier_version
status: public
scopus_import: '1'
status: public
title: Optimal Dyck reachability for data-dependence and Alias analysis
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 8b945eb4-e2f2-11eb-945a-df72226e66a9
volume: 2
year: '2017'
...
---
_id: '5455'
abstract:
- lang: eng
text: 'A fundamental algorithmic problem at the heart of static analysis is Dyck
reachability. The input is a graphwhere the edges are labeled with different types
of opening and closing parentheses, and the reachabilityinformation is computed
via paths whose parentheses are properly matched. We present new results for Dyckreachability
problems with applications to alias analysis and data-dependence analysis. Our
main contributions,that include improved upper bounds as well as lower bounds
that establish optimality guarantees, are asfollows:First, we consider Dyck reachability
on bidirected graphs, which is the standard way of performing field-sensitive
points-to analysis. Given a bidirected graph withnnodes andmedges, we present:
(i) an algorithmwith worst-case running timeO(m+n·α(n)), whereα(n)is the inverse
Ackermann function, improving thepreviously knownO(n2)time bound; (ii) a matching
lower bound that shows that our algorithm is optimalwrt to worst-case complexity;
and (iii) an optimal average-case upper bound ofO(m)time, improving thepreviously
knownO(m·logn)bound.Second, we consider the problem of context-sensitive data-dependence
analysis, where the task is to obtainanalysis summaries of library code in the
presence of callbacks. Our algorithm preprocesses libraries in almostlinear time,
after which the contribution of the library in the complexity of the client analysis
is only linear,and only wrt the number of call sites.Third, we prove that combinatorial
algorithms for Dyck reachability on general graphs with truly sub-cubic bounds
cannot be obtained without obtaining sub-cubic combinatorial algorithms for Boolean
MatrixMultiplication, which is a long-standing open problem. Thus we establish
that the existing combinatorialalgorithms for Dyck reachability are (conditionally)
optimal for general graphs. We also show that the samehardness holds for graphs
of constant treewidth.Finally, we provide a prototype implementation of our algorithms
for both alias analysis and data-dependenceanalysis. Our experimental evaluation
demonstrates that the new algorithms significantly outperform allexisting methods
on the two problems, over real-world benchmarks.'
alternative_title:
- IST Austria Technical Report
article_processing_charge: No
author:
- first_name: Krishnendu
full_name: Chatterjee, Krishnendu
id: 2E5DCA20-F248-11E8-B48F-1D18A9856A87
last_name: Chatterjee
orcid: 0000-0002-4561-241X
- first_name: Bhavya
full_name: Choudhary, Bhavya
last_name: Choudhary
- first_name: Andreas
full_name: Pavlogiannis, Andreas
id: 49704004-F248-11E8-B48F-1D18A9856A87
last_name: Pavlogiannis
orcid: 0000-0002-8943-0722
citation:
ama: Chatterjee K, Choudhary B, Pavlogiannis A. Optimal Dyck Reachability for
Data-Dependence and Alias Analysis. IST Austria; 2017. doi:10.15479/AT:IST-2017-870-v1-1
apa: Chatterjee, K., Choudhary, B., & Pavlogiannis, A. (2017). Optimal Dyck
reachability for data-dependence and alias analysis. IST Austria. https://doi.org/10.15479/AT:IST-2017-870-v1-1
chicago: Chatterjee, Krishnendu, Bhavya Choudhary, and Andreas Pavlogiannis. Optimal
Dyck Reachability for Data-Dependence and Alias Analysis. IST Austria, 2017.
https://doi.org/10.15479/AT:IST-2017-870-v1-1.
ieee: K. Chatterjee, B. Choudhary, and A. Pavlogiannis, Optimal Dyck reachability
for data-dependence and alias analysis. IST Austria, 2017.
ista: Chatterjee K, Choudhary B, Pavlogiannis A. 2017. Optimal Dyck reachability
for data-dependence and alias analysis, IST Austria, 37p.
mla: Chatterjee, Krishnendu, et al. Optimal Dyck Reachability for Data-Dependence
and Alias Analysis. IST Austria, 2017, doi:10.15479/AT:IST-2017-870-v1-1.
short: K. Chatterjee, B. Choudhary, A. Pavlogiannis, Optimal Dyck Reachability for
Data-Dependence and Alias Analysis, IST Austria, 2017.
date_created: 2018-12-12T11:39:26Z
date_published: 2017-10-23T00:00:00Z
date_updated: 2023-02-21T15:54:10Z
day: '23'
ddc:
- '000'
department:
- _id: KrCh
doi: 10.15479/AT:IST-2017-870-v1-1
file:
- access_level: open_access
checksum: 177a84a46e3ac17e87b31534ad16a4c9
content_type: application/pdf
creator: system
date_created: 2018-12-12T11:54:02Z
date_updated: 2020-07-14T12:46:59Z
file_id: '5524'
file_name: IST-2017-870-v1+1_main.pdf
file_size: 960491
relation: main_file
file_date_updated: 2020-07-14T12:46:59Z
has_accepted_license: '1'
language:
- iso: eng
month: '10'
oa: 1
oa_version: Published Version
page: '37'
publication_identifier:
issn:
- 2664-1690
publication_status: published
publisher: IST Austria
pubrep_id: '870'
related_material:
record:
- id: '10416'
relation: later_version
status: public
status: public
title: Optimal Dyck reachability for data-dependence and alias analysis
type: technical_report
user_id: 8b945eb4-e2f2-11eb-945a-df72226e66a9
year: '2017'
...
---
_id: '10417'
abstract:
- lang: eng
text: "We present a new dynamic partial-order reduction method for stateless model
checking of concurrent programs. A common approach for exploring program behaviors
relies on enumerating the traces of the program, without storing the visited states
(aka stateless exploration). As the number of distinct traces grows exponentially,
dynamic partial-order reduction (DPOR) techniques have been successfully used
to partition the space of traces into equivalence classes (Mazurkiewicz partitioning),
with the goal of exploring only few representative traces from each class.\r\n\r\nWe
introduce a new equivalence on traces under sequential consistency semantics,
which we call the observation equivalence. Two traces are observationally equivalent
if every read event observes the same write event in both traces. While the traditional
Mazurkiewicz equivalence is control-centric, our new definition is data-centric.
We show that our observation equivalence is coarser than the Mazurkiewicz equivalence,
and in many cases even exponentially coarser. We devise a DPOR exploration of
the trace space, called data-centric DPOR, based on the observation equivalence."
acknowledgement: "The research was partly supported by Austrian Science Fund (FWF)
Grant No P23499- N23, FWF\r\nNFN Grant No S11407-N23 (RiSE/SHiNE), ERC Start grant
(279307: Graph Games), and Czech\r\nScience Foundation grant GBP202/12/G061."
article_number: '31'
article_processing_charge: No
article_type: original
author:
- first_name: Marek
full_name: Chalupa, Marek
last_name: Chalupa
- first_name: Krishnendu
full_name: Chatterjee, Krishnendu
id: 2E5DCA20-F248-11E8-B48F-1D18A9856A87
last_name: Chatterjee
orcid: 0000-0002-4561-241X
- first_name: Andreas
full_name: Pavlogiannis, Andreas
id: 49704004-F248-11E8-B48F-1D18A9856A87
last_name: Pavlogiannis
orcid: 0000-0002-8943-0722
- first_name: Nishant
full_name: Sinha, Nishant
last_name: Sinha
- first_name: Kapil
full_name: Vaidya, Kapil
last_name: Vaidya
citation:
ama: Chalupa M, Chatterjee K, Pavlogiannis A, Sinha N, Vaidya K. Data-centric dynamic
partial order reduction. Proceedings of the ACM on Programming Languages.
2017;2(POPL). doi:10.1145/3158119
apa: 'Chalupa, M., Chatterjee, K., Pavlogiannis, A., Sinha, N., & Vaidya, K.
(2017). Data-centric dynamic partial order reduction. Proceedings of the ACM
on Programming Languages. Los Angeles, CA, United States: Association for
Computing Machinery. https://doi.org/10.1145/3158119'
chicago: Chalupa, Marek, Krishnendu Chatterjee, Andreas Pavlogiannis, Nishant Sinha,
and Kapil Vaidya. “Data-Centric Dynamic Partial Order Reduction.” Proceedings
of the ACM on Programming Languages. Association for Computing Machinery,
2017. https://doi.org/10.1145/3158119.
ieee: M. Chalupa, K. Chatterjee, A. Pavlogiannis, N. Sinha, and K. Vaidya, “Data-centric
dynamic partial order reduction,” Proceedings of the ACM on Programming Languages,
vol. 2, no. POPL. Association for Computing Machinery, 2017.
ista: Chalupa M, Chatterjee K, Pavlogiannis A, Sinha N, Vaidya K. 2017. Data-centric
dynamic partial order reduction. Proceedings of the ACM on Programming Languages.
2(POPL), 31.
mla: Chalupa, Marek, et al. “Data-Centric Dynamic Partial Order Reduction.” Proceedings
of the ACM on Programming Languages, vol. 2, no. POPL, 31, Association for
Computing Machinery, 2017, doi:10.1145/3158119.
short: M. Chalupa, K. Chatterjee, A. Pavlogiannis, N. Sinha, K. Vaidya, Proceedings
of the ACM on Programming Languages 2 (2017).
conference:
end_date: 2018-01-13
location: Los Angeles, CA, United States
name: 'POPL: Programming Languages'
start_date: 2018-01-07
date_created: 2021-12-05T23:01:49Z
date_published: 2017-12-27T00:00:00Z
date_updated: 2023-02-23T12:27:16Z
day: '27'
department:
- _id: KrCh
doi: 10.1145/3158119
ec_funded: 1
external_id:
arxiv:
- '1610.01188'
intvolume: ' 2'
issue: POPL
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://dl.acm.org/doi/10.1145/3158119
month: '12'
oa: 1
oa_version: Published Version
project:
- _id: 2584A770-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: P 23499-N23
name: Modern Graph Algorithmic Techniques in Formal Verification
- _id: 25832EC2-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: S 11407_N23
name: Rigorous Systems Engineering
- _id: 2581B60A-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '279307'
name: 'Quantitative Graph Games: Theory and Applications'
publication: Proceedings of the ACM on Programming Languages
publication_identifier:
eissn:
- 2475-1421
publication_status: published
publisher: Association for Computing Machinery
quality_controlled: '1'
related_material:
record:
- id: '5448'
relation: earlier_version
status: public
- id: '5456'
relation: earlier_version
status: public
scopus_import: '1'
status: public
title: Data-centric dynamic partial order reduction
type: journal_article
user_id: 8b945eb4-e2f2-11eb-945a-df72226e66a9
volume: 2
year: '2017'
...
---
_id: '5456'
abstract:
- lang: eng
text: "We present a new dynamic partial-order reduction method for stateless model
checking of concurrent programs. A common approach for exploring program behaviors
relies on enumerating the traces of the program, without storing the visited states
(aka stateless exploration). As the number of distinct traces grows exponentially,
dynamic partial-order reduction (DPOR) techniques have been successfully used
to partition the space of traces into equivalence classes (Mazurkiewicz partitioning),
with the goal of exploring only few representative traces from each class.\r\nWe
introduce a new equivalence on traces under sequential consistency semantics,
which we call the observation equivalence. Two traces are observationally equivalent
if every read event observes the same write event in both traces. While the traditional
Mazurkiewicz equivalence is control-centric, our new definition is data-centric.
We show that our observation equivalence is coarser than the Mazurkiewicz equivalence,
and in many cases even exponentially coarser. We devise a DPOR exploration of
the trace space, called data-centric DPOR, based on the observation equivalence.\r\n1.
For acyclic architectures, our algorithm is guaranteed to explore exactly one
representative trace from each observation class, while spending polynomial time
per class. Hence, our algorithm is optimal wrt the observation equivalence, and
in several cases explores exponentially fewer traces than any enumerative method
based on the Mazurkiewicz equivalence.\r\n2. For cyclic architectures, we consider
an equivalence between traces which is finer than the observation equivalence;
but coarser than the Mazurkiewicz equivalence, and in some cases is exponentially
coarser. Our data-centric DPOR algorithm remains optimal under this trace equivalence.
\r\nFinally, we perform a basic experimental comparison between the existing Mazurkiewicz-based
DPOR and our data-centric DPOR on a set of academic benchmarks. Our results show
a significant reduction in both running time and the number of explored equivalence
classes."
alternative_title:
- IST Austria Technical Report
author:
- first_name: Marek
full_name: Chalupa, Marek
last_name: Chalupa
- first_name: Krishnendu
full_name: Chatterjee, Krishnendu
id: 2E5DCA20-F248-11E8-B48F-1D18A9856A87
last_name: Chatterjee
orcid: 0000-0002-4561-241X
- first_name: Andreas
full_name: Pavlogiannis, Andreas
id: 49704004-F248-11E8-B48F-1D18A9856A87
last_name: Pavlogiannis
orcid: 0000-0002-8943-0722
- first_name: Nishant
full_name: Sinha, Nishant
last_name: Sinha
- first_name: Kapil
full_name: Vaidya, Kapil
last_name: Vaidya
citation:
ama: Chalupa M, Chatterjee K, Pavlogiannis A, Sinha N, Vaidya K. Data-Centric
Dynamic Partial Order Reduction. IST Austria; 2017. doi:10.15479/AT:IST-2017-872-v1-1
apa: Chalupa, M., Chatterjee, K., Pavlogiannis, A., Sinha, N., & Vaidya, K.
(2017). Data-centric dynamic partial order reduction. IST Austria. https://doi.org/10.15479/AT:IST-2017-872-v1-1
chicago: Chalupa, Marek, Krishnendu Chatterjee, Andreas Pavlogiannis, Nishant Sinha,
and Kapil Vaidya. Data-Centric Dynamic Partial Order Reduction. IST Austria,
2017. https://doi.org/10.15479/AT:IST-2017-872-v1-1.
ieee: M. Chalupa, K. Chatterjee, A. Pavlogiannis, N. Sinha, and K. Vaidya, Data-centric
dynamic partial order reduction. IST Austria, 2017.
ista: Chalupa M, Chatterjee K, Pavlogiannis A, Sinha N, Vaidya K. 2017. Data-centric
dynamic partial order reduction, IST Austria, 36p.
mla: Chalupa, Marek, et al. Data-Centric Dynamic Partial Order Reduction.
IST Austria, 2017, doi:10.15479/AT:IST-2017-872-v1-1.
short: M. Chalupa, K. Chatterjee, A. Pavlogiannis, N. Sinha, K. Vaidya, Data-Centric
Dynamic Partial Order Reduction, IST Austria, 2017.
date_created: 2018-12-12T11:39:26Z
date_published: 2017-10-23T00:00:00Z
date_updated: 2023-02-23T12:26:54Z
day: '23'
ddc:
- '000'
department:
- _id: KrCh
doi: 10.15479/AT:IST-2017-872-v1-1
file:
- access_level: open_access
checksum: d2635c4cf013000f0a1b09e80f9e4ab7
content_type: application/pdf
creator: system
date_created: 2018-12-12T11:53:26Z
date_updated: 2020-07-14T12:46:59Z
file_id: '5487'
file_name: IST-2017-872-v1+1_main.pdf
file_size: 910347
relation: main_file
file_date_updated: 2020-07-14T12:46:59Z
has_accepted_license: '1'
language:
- iso: eng
month: '10'
oa: 1
oa_version: Published Version
page: '36'
publication_identifier:
issn:
- 2664-1690
publication_status: published
publisher: IST Austria
pubrep_id: '872'
related_material:
record:
- id: '10417'
relation: later_version
status: public
- id: '5448'
relation: earlier_version
status: public
status: public
title: Data-centric dynamic partial order reduction
type: technical_report
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2017'
...
---
_id: '551'
abstract:
- lang: eng
text: 'Evolutionary graph theory studies the evolutionary dynamics in a population
structure given as a connected graph. Each node of the graph represents an individual
of the population, and edges determine how offspring are placed. We consider the
classical birth-death Moran process where there are two types of individuals,
namely, the residents with fitness 1 and mutants with fitness r. The fitness indicates
the reproductive strength. The evolutionary dynamics happens as follows: in the
initial step, in a population of all resident individuals a mutant is introduced,
and then at each step, an individual is chosen proportional to the fitness of
its type to reproduce, and the offspring replaces a neighbor uniformly at random.
The process stops when all individuals are either residents or mutants. The probability
that all individuals in the end are mutants is called the fixation probability,
which is a key factor in the rate of evolution. We consider the problem of approximating
the fixation probability. The class of algorithms that is extremely relevant for
approximation of the fixation probabilities is the Monte-Carlo simulation of the
process. Previous results present a polynomial-time Monte-Carlo algorithm for
undirected graphs when r is given in unary. First, we present a simple modification:
instead of simulating each step, we discard ineffective steps, where no node changes
type (i.e., either residents replace residents, or mutants replace mutants). Using
the above simple modification and our result that the number of effective steps
is concentrated around the expected number of effective steps, we present faster
polynomial-time Monte-Carlo algorithms for undirected graphs. Our algorithms are
always at least a factor O(n2/ log n) faster as compared to the previous algorithms,
where n is the number of nodes, and is polynomial even if r is given in binary.
We also present lower bounds showing that the upper bound on the expected number
of effective steps we present is asymptotically tight for undirected graphs. '
alternative_title:
- LIPIcs
article_number: '61'
author:
- first_name: Krishnendu
full_name: Chatterjee, Krishnendu
id: 2E5DCA20-F248-11E8-B48F-1D18A9856A87
last_name: Chatterjee
orcid: 0000-0002-4561-241X
- first_name: Rasmus
full_name: Ibsen-Jensen, Rasmus
id: 3B699956-F248-11E8-B48F-1D18A9856A87
last_name: Ibsen-Jensen
orcid: 0000-0003-4783-0389
- first_name: Martin
full_name: Nowak, Martin
last_name: Nowak
citation:
ama: 'Chatterjee K, Ibsen-Jensen R, Nowak M. Faster Monte Carlo algorithms for fixation
probability of the Moran process on undirected graphs. In: Leibniz International
Proceedings in Informatics. Vol 83. Schloss Dagstuhl - Leibniz-Zentrum für
Informatik; 2017. doi:10.4230/LIPIcs.MFCS.2017.61'
apa: 'Chatterjee, K., Ibsen-Jensen, R., & Nowak, M. (2017). Faster Monte Carlo
algorithms for fixation probability of the Moran process on undirected graphs.
In Leibniz International Proceedings in Informatics (Vol. 83). Aalborg,
Denmark: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.MFCS.2017.61'
chicago: Chatterjee, Krishnendu, Rasmus Ibsen-Jensen, and Martin Nowak. “Faster
Monte Carlo Algorithms for Fixation Probability of the Moran Process on Undirected
Graphs.” In Leibniz International Proceedings in Informatics, Vol. 83.
Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. https://doi.org/10.4230/LIPIcs.MFCS.2017.61.
ieee: K. Chatterjee, R. Ibsen-Jensen, and M. Nowak, “Faster Monte Carlo algorithms
for fixation probability of the Moran process on undirected graphs,” in Leibniz
International Proceedings in Informatics, Aalborg, Denmark, 2017, vol. 83.
ista: 'Chatterjee K, Ibsen-Jensen R, Nowak M. 2017. Faster Monte Carlo algorithms
for fixation probability of the Moran process on undirected graphs. Leibniz International
Proceedings in Informatics. MFCS: Mathematical Foundations of Computer Science
(SG), LIPIcs, vol. 83, 61.'
mla: Chatterjee, Krishnendu, et al. “Faster Monte Carlo Algorithms for Fixation
Probability of the Moran Process on Undirected Graphs.” Leibniz International
Proceedings in Informatics, vol. 83, 61, Schloss Dagstuhl - Leibniz-Zentrum
für Informatik, 2017, doi:10.4230/LIPIcs.MFCS.2017.61.
short: K. Chatterjee, R. Ibsen-Jensen, M. Nowak, in:, Leibniz International Proceedings
in Informatics, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017.
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