Testing Connectedness of Images
We investigate algorithms for testing whether an image is connected. Given a proximity parameter ϵ ∈ ( 0 , 1 ) and query access to a black-and-white image represented by an n × n matrix of Boolean pixel values, a (1-sided error) connectedness tester accepts if the image is connected and rejects with...
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| Vydané v: | Algorithmica Ročník 86; číslo 11; s. 3496 - 3517 |
|---|---|
| Hlavní autori: | , , , |
| Médium: | Journal Article |
| Jazyk: | English |
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01.11.2024
Springer Nature B.V |
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| ISSN: | 0178-4617, 1432-0541 |
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| Abstract | We investigate algorithms for testing whether an image is connected. Given a proximity parameter
ϵ
∈
(
0
,
1
)
and query access to a black-and-white image represented by an
n
×
n
matrix of Boolean pixel values, a (1-sided error) connectedness tester accepts if the image is connected and rejects with probability at least 2/3 if the image is
ϵ
-far from connected. We show that connectedness can be tested nonadaptively with
O
(
1
ϵ
2
)
queries and adaptively with
O
(
1
ϵ
3
/
2
log
1
ϵ
)
queries. The best connectedness tester to date, by Berman, Raskhodnikova, and Yaroslavtsev (STOC 2014) had query complexity
O
(
1
ϵ
2
log
1
ϵ
)
and was adaptive. We also prove that every nonadaptive, 1-sided error tester for connectedness must make
Ω
(
1
ϵ
log
1
ϵ
)
queries. |
|---|---|
| AbstractList | We investigate algorithms for testing whether an image is connected. Given a proximity parameter ϵ∈(0,1) and query access to a black-and-white image represented by an n×n matrix of Boolean pixel values, a (1-sided error) connectedness tester accepts if the image is connected and rejects with probability at least 2/3 if the image is ϵ-far from connected. We show that connectedness can be tested nonadaptively with O(1ϵ2) queries and adaptively with O(1ϵ3/2log1ϵ) queries. The best connectedness tester to date, by Berman, Raskhodnikova, and Yaroslavtsev (STOC 2014) had query complexity O(1ϵ2log1ϵ) and was adaptive. We also prove that every nonadaptive, 1-sided error tester for connectedness must make Ω(1ϵlog1ϵ) queries. We investigate algorithms for testing whether an image is connected. Given a proximity parameter ϵ ∈ ( 0 , 1 ) and query access to a black-and-white image represented by an n × n matrix of Boolean pixel values, a (1-sided error) connectedness tester accepts if the image is connected and rejects with probability at least 2/3 if the image is ϵ -far from connected. We show that connectedness can be tested nonadaptively with O ( 1 ϵ 2 ) queries and adaptively with O ( 1 ϵ 3 / 2 log 1 ϵ ) queries. The best connectedness tester to date, by Berman, Raskhodnikova, and Yaroslavtsev (STOC 2014) had query complexity O ( 1 ϵ 2 log 1 ϵ ) and was adaptive. We also prove that every nonadaptive, 1-sided error tester for connectedness must make Ω ( 1 ϵ log 1 ϵ ) queries. |
| Author | Murzabulatov, Meiram Raskhodnikova, Sofya Ristache, Dragos-Florian Berman, Piotr |
| Author_xml | – sequence: 1 givenname: Piotr surname: Berman fullname: Berman, Piotr organization: Nazarbayev University, Boston University – sequence: 2 givenname: Meiram surname: Murzabulatov fullname: Murzabulatov, Meiram email: meiram.murzabulatov@nu.edu.kz organization: Nazarbayev University – sequence: 3 givenname: Sofya surname: Raskhodnikova fullname: Raskhodnikova, Sofya email: sofya@bu.edu organization: Boston University – sequence: 4 givenname: Dragos-Florian surname: Ristache fullname: Ristache, Dragos-Florian organization: Boston University |
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| ContentType | Journal Article |
| Copyright | The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2024. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. |
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| References | Berman, P., Murzabulatov, M., Raskhodnikova, S., Ristache, D.: Testing connectedness of images. In: Megow, N., Smith, A.D. (eds.) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2023, September 11–13, 2023, Atlanta, Georgia, USA, volume 275 of LIPIcs, pp. 66:1–66:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023) MinskyMPapertSAPerceptrons: An Introduction to Computational Geometry2017CambridgeThe MIT Press10.7551/mitpress/11301.001.0001 Murzabulatov, M.: Testing Geometric Properties of Two-Dimensional Figures and Images. PhD thesis, The Pennsylvania State University (2017) BerenbrinkPKrayenhoffBMallmann-TrennFEstimating the number of connected components in sublinear timeInf. Process. Lett.201411411639642323091310.1016/j.ipl.2014.05.008 LeviAPallavoorRKSRaskhodnikovaSVarmaNErasure-resilient sublinear-time graph algorithmsACM Trans. Comput. Theory20221411:11:22441586510.1145/3488250 ParnasMRonDTesting the diameter of graphsRandom Struct. Algorithms2002202165183188443110.1002/rsa.10013 BermanPMurzabulatovMRaskhodnikovaSThe power and limitations of uniform samples in testing properties of figuresAlgorithmica201981312471266391798610.1007/s00453-018-0467-9 Raskhodnikova, S.: Property testing : theory and applications. PhD thesis, Massachusetts Institute of Technology (2003) Korman, S., Reichman, D., Tsur, G.: Tight approximation of image matching. CoRR, arXiv:1111.1713 (2011) RubinfeldRSudanMRobust characterizations of polynomials with applications to program testingSIAM J. Comput.1996252252271137930010.1137/S0097539793255151 Yao, A.C.: Probabilistic computation, towards a unified measure of complexity. In: Proceedings of the Eighteenth Annual Symposium on Foundations of Computer Science, pp. 222–227 (1977) BermanPMurzabulatovMRaskhodnikovaSTolerant testers of image propertiesACM Trans. Algorithms2022184139450981210.1145/3531527 ChazelleBRubinfeldRTrevisanLApproximating the minimum spanning tree weight in sublinear timeSIAM J. Comput.20053413701379216574510.1137/S0097539702403244 Korman, S., Reichman, D., Tsur, G., Avidan, S.: Fast-match: Fast affine template matching. In: CVPR, pp. 2331–2338. IEEE (2013) KleinerIKerenDNewmanIBen-ZwiOApplying property testing to an image partitioning problemIEEE Trans. Pattern Anal. Mach. Intell.201133225626510.1109/TPAMI.2010.165 Ben-Eliezer, O., Fischer, E.: Earthmover resilience and testing in ordered structures. In: Servedio, R.A. (ed.) 33rd Computational Complexity Conference, CCC 2018, June 22–24, 2018, San Diego, CA, USA, volume 102 of LIPIcs, pp. 18:1–18:35. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018) Berman, P., Raskhodnikova, S., Yaroslavtsev, G.: Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_p$$\end{document}-testing. In: Shmoys, D.B. (ed.) Symposium on Theory of Computing, STOC 2014, New York, NY, USA, May 31–June 03, 2014, pp. 164–173. ACM (2014) GoldreichORonDProperty testing in bounded degree graphsAlgorithmica2002322302343186951410.1007/s00453-001-0078-7 RonDTsurGTesting properties of sparse imagesACM Trans. Algorithms201410417:117:52325450510.1145/2635806 Ben-Eliezer, O., Korman, S., Reichman, D.: Deleting and testing forbidden patterns in multi-dimensional arrays. In: Chatzigiannakis, I., Indyk, P., Kuhn, F., Muscholl, A. (eds.) 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017, July 10–14, 2017, Warsaw, Poland, volume 80 of LIPIcs, pp. 9:1–9:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017) Alon, N., Ben-Eliezer, O., Fischer, E.: Testing hereditary properties of ordered graphs and matrices. In: Umans, C. (ed.) 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15–17, 2017, pp. 848–858. IEEE Computer Society (2017) GoldreichOGoldwasserSRonDProperty testing and its connection to learning and approximationJ. ACM1998454653750167509910.1145/285055.285060 FischerENewmanITesting of matrix-poset propertiesCombinatorica2007273293327234581210.1007/s00493-007-2154-3 ParalPChatterjeeARakshitAVision sensor-based shoe detection for human tracking in a human-robot coexisting environment: a photometric invariant approach using dbscan algorithmIEEE Sens. J.201919124549455910.1109/JSEN.2019.2897989 BermanPMurzabulatovMRaskhodnikovaSTesting figures under the uniform distributionRandom Struct. Algorithms2019543413443393877410.1002/rsa.20797 Raskhodnikova, S.: Approximate testing of visual properties. In: Arora, S., Jansen, K., Rolim, J.D.P., Sahai, A. (eds.) RANDOM-APPROX, volume 2764 of Lecture Notes in Computer Science, pp. 370–381. Springer (2003) 1248_CR8 D Ron (1248_CR25) 2014; 10 1248_CR4 1248_CR1 1248_CR2 1248_CR22 1248_CR23 1248_CR26 R Rubinfeld (1248_CR24) 1996; 25 P Berenbrink (1248_CR3) 2014; 114 P Berman (1248_CR5) 2019; 81 O Goldreich (1248_CR12) 1998; 45 M Parnas (1248_CR21) 2002; 20 P Paral (1248_CR20) 2019; 19 1248_CR9 E Fischer (1248_CR11) 2007; 27 B Chazelle (1248_CR10) 2005; 34 P Berman (1248_CR7) 2022; 18 I Kleiner (1248_CR14) 2011; 33 1248_CR15 M Minsky (1248_CR18) 2017 1248_CR16 1248_CR19 P Berman (1248_CR6) 2019; 54 O Goldreich (1248_CR13) 2002; 32 A Levi (1248_CR17) 2022; 14 |
| References_xml | – reference: BermanPMurzabulatovMRaskhodnikovaSThe power and limitations of uniform samples in testing properties of figuresAlgorithmica201981312471266391798610.1007/s00453-018-0467-9 – reference: BermanPMurzabulatovMRaskhodnikovaSTolerant testers of image propertiesACM Trans. Algorithms2022184139450981210.1145/3531527 – reference: Berman, P., Murzabulatov, M., Raskhodnikova, S., Ristache, D.: Testing connectedness of images. In: Megow, N., Smith, A.D. (eds.) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2023, September 11–13, 2023, Atlanta, Georgia, USA, volume 275 of LIPIcs, pp. 66:1–66:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023) – reference: Raskhodnikova, S.: Approximate testing of visual properties. In: Arora, S., Jansen, K., Rolim, J.D.P., Sahai, A. (eds.) RANDOM-APPROX, volume 2764 of Lecture Notes in Computer Science, pp. 370–381. Springer (2003) – reference: Raskhodnikova, S.: Property testing : theory and applications. PhD thesis, Massachusetts Institute of Technology (2003) – reference: Korman, S., Reichman, D., Tsur, G.: Tight approximation of image matching. CoRR, arXiv:1111.1713 (2011) – reference: GoldreichORonDProperty testing in bounded degree graphsAlgorithmica2002322302343186951410.1007/s00453-001-0078-7 – reference: BerenbrinkPKrayenhoffBMallmann-TrennFEstimating the number of connected components in sublinear timeInf. Process. Lett.201411411639642323091310.1016/j.ipl.2014.05.008 – reference: RonDTsurGTesting properties of sparse imagesACM Trans. Algorithms201410417:117:52325450510.1145/2635806 – reference: ChazelleBRubinfeldRTrevisanLApproximating the minimum spanning tree weight in sublinear timeSIAM J. Comput.20053413701379216574510.1137/S0097539702403244 – reference: Murzabulatov, M.: Testing Geometric Properties of Two-Dimensional Figures and Images. PhD thesis, The Pennsylvania State University (2017) – reference: Korman, S., Reichman, D., Tsur, G., Avidan, S.: Fast-match: Fast affine template matching. In: CVPR, pp. 2331–2338. IEEE (2013) – reference: Ben-Eliezer, O., Korman, S., Reichman, D.: Deleting and testing forbidden patterns in multi-dimensional arrays. In: Chatzigiannakis, I., Indyk, P., Kuhn, F., Muscholl, A. (eds.) 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017, July 10–14, 2017, Warsaw, Poland, volume 80 of LIPIcs, pp. 9:1–9:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017) – reference: BermanPMurzabulatovMRaskhodnikovaSTesting figures under the uniform distributionRandom Struct. Algorithms2019543413443393877410.1002/rsa.20797 – reference: Yao, A.C.: Probabilistic computation, towards a unified measure of complexity. In: Proceedings of the Eighteenth Annual Symposium on Foundations of Computer Science, pp. 222–227 (1977) – reference: RubinfeldRSudanMRobust characterizations of polynomials with applications to program testingSIAM J. Comput.1996252252271137930010.1137/S0097539793255151 – reference: ParnasMRonDTesting the diameter of graphsRandom Struct. Algorithms2002202165183188443110.1002/rsa.10013 – reference: FischerENewmanITesting of matrix-poset propertiesCombinatorica2007273293327234581210.1007/s00493-007-2154-3 – reference: Alon, N., Ben-Eliezer, O., Fischer, E.: Testing hereditary properties of ordered graphs and matrices. In: Umans, C. (ed.) 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15–17, 2017, pp. 848–858. IEEE Computer Society (2017) – reference: KleinerIKerenDNewmanIBen-ZwiOApplying property testing to an image partitioning problemIEEE Trans. Pattern Anal. Mach. Intell.201133225626510.1109/TPAMI.2010.165 – reference: LeviAPallavoorRKSRaskhodnikovaSVarmaNErasure-resilient sublinear-time graph algorithmsACM Trans. Comput. Theory20221411:11:22441586510.1145/3488250 – reference: ParalPChatterjeeARakshitAVision sensor-based shoe detection for human tracking in a human-robot coexisting environment: a photometric invariant approach using dbscan algorithmIEEE Sens. J.201919124549455910.1109/JSEN.2019.2897989 – reference: Ben-Eliezer, O., Fischer, E.: Earthmover resilience and testing in ordered structures. In: Servedio, R.A. (ed.) 33rd Computational Complexity Conference, CCC 2018, June 22–24, 2018, San Diego, CA, USA, volume 102 of LIPIcs, pp. 18:1–18:35. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018) – reference: GoldreichOGoldwasserSRonDProperty testing and its connection to learning and approximationJ. ACM1998454653750167509910.1145/285055.285060 – reference: Berman, P., Raskhodnikova, S., Yaroslavtsev, G.: Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_p$$\end{document}-testing. In: Shmoys, D.B. (ed.) Symposium on Theory of Computing, STOC 2014, New York, NY, USA, May 31–June 03, 2014, pp. 164–173. ACM (2014) – reference: MinskyMPapertSAPerceptrons: An Introduction to Computational Geometry2017CambridgeThe MIT Press10.7551/mitpress/11301.001.0001 – volume: 18 start-page: 1 issue: 4 year: 2022 ident: 1248_CR7 publication-title: ACM Trans. Algorithms doi: 10.1145/3531527 – ident: 1248_CR23 – ident: 1248_CR8 – volume: 33 start-page: 256 issue: 2 year: 2011 ident: 1248_CR14 publication-title: IEEE Trans. Pattern Anal. Mach. Intell. doi: 10.1109/TPAMI.2010.165 – volume: 54 start-page: 413 issue: 3 year: 2019 ident: 1248_CR6 publication-title: Random Struct. Algorithms doi: 10.1002/rsa.20797 – volume: 27 start-page: 293 issue: 3 year: 2007 ident: 1248_CR11 publication-title: Combinatorica doi: 10.1007/s00493-007-2154-3 – ident: 1248_CR1 doi: 10.1109/FOCS.2017.83 – ident: 1248_CR9 doi: 10.1145/2591796.2591887 – ident: 1248_CR16 doi: 10.1109/CVPR.2013.302 – ident: 1248_CR22 doi: 10.1007/978-3-540-45198-3_31 – volume: 81 start-page: 1247 issue: 3 year: 2019 ident: 1248_CR5 publication-title: Algorithmica doi: 10.1007/s00453-018-0467-9 – volume-title: Perceptrons: An Introduction to Computational Geometry year: 2017 ident: 1248_CR18 doi: 10.7551/mitpress/11301.001.0001 – volume: 10 start-page: 17:1 issue: 4 year: 2014 ident: 1248_CR25 publication-title: ACM Trans. Algorithms doi: 10.1145/2635806 – ident: 1248_CR26 doi: 10.1109/SFCS.1977.24 – volume: 114 start-page: 639 issue: 11 year: 2014 ident: 1248_CR3 publication-title: Inf. Process. Lett. doi: 10.1016/j.ipl.2014.05.008 – ident: 1248_CR4 – volume: 20 start-page: 165 issue: 2 year: 2002 ident: 1248_CR21 publication-title: Random Struct. Algorithms doi: 10.1002/rsa.10013 – ident: 1248_CR2 – ident: 1248_CR15 – volume: 14 start-page: 1:1 issue: 1 year: 2022 ident: 1248_CR17 publication-title: ACM Trans. Comput. Theory doi: 10.1145/3488250 – volume: 19 start-page: 4549 issue: 12 year: 2019 ident: 1248_CR20 publication-title: IEEE Sens. J. doi: 10.1109/JSEN.2019.2897989 – ident: 1248_CR19 – volume: 34 start-page: 1370 year: 2005 ident: 1248_CR10 publication-title: SIAM J. Comput. doi: 10.1137/S0097539702403244 – volume: 32 start-page: 302 issue: 2 year: 2002 ident: 1248_CR13 publication-title: Algorithmica doi: 10.1007/s00453-001-0078-7 – volume: 25 start-page: 252 issue: 2 year: 1996 ident: 1248_CR24 publication-title: SIAM J. Comput. doi: 10.1137/S0097539793255151 – volume: 45 start-page: 653 issue: 4 year: 1998 ident: 1248_CR12 publication-title: J. ACM doi: 10.1145/285055.285060 |
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| Snippet | We investigate algorithms for testing whether an image is connected. Given a proximity parameter
ϵ
∈
(
0
,
1
)
and query access to a black-and-white image... We investigate algorithms for testing whether an image is connected. Given a proximity parameter ϵ∈(0,1) and query access to a black-and-white image... |
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| SubjectTerms | Algorithm Analysis and Problem Complexity Algorithms Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Mathematics of Computing Queries Theory of Computation |
| Title | Testing Connectedness of Images |
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