**Time and location:** Tuesday/Thursday 12:30PM-1:45PM
**Instructor:**
Sofya Raskhodnikova

**Office hours:** Wednesday 1PM-2:30PM (on zoom)

CS 537, proficiency in understanding and writing mathematical proofs

This course will cover the design and analysis of algorithms that are restricted to run in sublinear time. Such algorithms are typically randomized and produce only approximate answers. A characteristic feature of sublinear algorithms is that they do not have time to access the entire input. Therefore, input representation and the model for accessing the input play an important role. We will study different models appropriate for sublinear algorithms. The course will cover sublinear algorithms discovered in a variety of areas, including graph theory, algebra, geometry, and discrete mathematics, and introduce many techniques that are applied to analyzing sublinear algorithms.

Students will be evaluated based on class participation, solutions to about 4-5 homework assignments, taking lecture notes about 1-2 times per person, and the final project.

Lec | Date | Topics | References | Reading/Homework |
---|---|---|---|---|

1 | Th, Sep 3 | Introduction. Basic models for sublinear-time computation. Simple examples of sublinear algorithms. (Slides) | RS96, GGR98, Ras03, EKKRV00, Fis04 | Ras03, Ras15; Course Information; Probabilistic Inequalities Review; HW1 out |

2 | Tu, Sep 8 | Properties of lists and functions. Testing if a list is sorted/Lipschitz and if a function is monotone. (Slides) | DGLRRS99, BGJRW09, Ras10, JR13 | |

3 | Th, Sep 10 | Testing a bounded-degree graph is connected. Approximating the number of connected components and MST weight. (Slides) | GR02, CRT05 | HW1 due; HW2 out |

4 | Tu, Sep 15 | Methods for proving lower bounds: Yao's principle. (Slides) (covered slides 1-16) | ||

5 | Th, Sep 17 | Methods for proving lower bounds: Yao's principle and communication complexity. (Slides) | FLNRRS02, BBM11 | HW2 due |

6 | Tu, Sep 22 | Finish communication complexity. Other models of sublinear time/space computation. Project discussion. (Slides) | BBM11 | |

7 | Th, Sep 24 | Streaming: Distinct Elements; k-wise independence (Slides) | AMS99, BJKST02 | |

8 | Tu, Sep 29 | Streaming: approximate counting; linear sketching; estimating second frequency moment (Slides) | Mor78, AMS99 | |

9 | Th, Oct 1 | Multi-purpose sketches: Count-Min and Count-Sketch; range queries, quantiles, heavy hitters (Slides) | CM05, GM07 | Project proposal due; HW3 out |

10 | Tu, Oct 6 | Streaming lower bounds via communication complexity (Slides) | BJKS04 | Rou16 (Chapter 1) |

11 | Th, Oct 8 | Graph streams; linear sketching for connected components; L0 sampling (Slides) | AGM12 | HW3 due |

Tu, Oct 13 | Monday Schedule |
|||

12 | Th, Oct 15 | Testing properties of dense graphs; bipartiteness (Slides) | GGR98 | HW3 due (extended deadline) |

13 | Tu, Oct 20 | Approximate Max-Cut (Slides) | GGR98 | |

14 | Th, Oct 22 | Testing triangle-freeness; Regularity Lemma (Slides) | AFKS00 | |

15 | Tu, Oct 27 | Testing triangle-freeness. Triangle-removal lemma. Testing other properties of dense graphs. Behrend's construction of progression-free sets. (Slides) | Alon02 | |

16 | Th, Oct 29 | Lower bound for testing triangle-freeness. Canonical testers in the dense-graph model (in class exercise). (Slides) | Alon02, GT03 | Project progress report due |

17 | Tu, Nov 3 | Approximating the average degree of a graph (Slides) | ||

18 | Th, Nov 5 | Testing linearity of Boolean functions (Slides) | BLR93 | HW4 out |

19 | Tu, Nov 10 | Finish linearity testing. Tolerant testing and distance approximation. (Slides) | ||

20 | Th, Nov 12 | Approximating distance to sortedness for 0/1 sequences. (Slides) | HW4 due | |

21 | Tu, Nov 17 | Gap Edit Distance | ||

22 | Th, Nov 19 | L_p-Testing (Slides) | ||

23 | Tu, Nov 24 | L_p-Testing of monotonicity. Work investment strategy (Slides) | ||

Nov 25-29 | Thanksgiving Recess |
|||

24 | Tu, Dec 1 | Local Computation Algorithm for Maximal Independent Set (Slides) | ||

25 | Th, Dec 3 | Local Computation Algorithm for Maximal Independent Set (Slides) | Project final report due | |

26 | Tu, Dec 8 | Final project presentations | ||

27 | Th, Dec 10 | Final project presentations |

- (Optional) textbook
- Introduction to Property Testing by Oded Goldreich
- Latest in property testing
- Property testing review
- Open problems
- A list of open problems
- LaTeX
- Some LaTeX editors: TexShop for Mac, TexStudio for Windows, Overleaf on the web (no installation needed, allows for collaboration).
- Not so short intro to LaTeX and a LaTeX tutorial.
- Homework template files: tex, pdf, cls, jpg.

Most papers from the list below can be downloaded from the Princeton archive or my webpage.

RS96 | Ronitt Rubinfeld, Madhu Sudan, Robust Characterizations of Polynomials with Applications to Program Testing. SIAM Journal of Computing 1996. |

GGR98 | Oded Goldreich, Shafi Goldwasser, Dana Ron, Property Testing and its Connection to Learning and Approximation. Journal of ACM 1998, FOCS 1996. |

Ras03 | Sofya Raskhodnikova, Approximate Testing of Visual Properties. RANDOM-APPROX 2003. |

Ras15 | Sofya Raskhodnikova, Testing if an Array Is Sorted. Encyclopedia of Algorithms 2015. |

RS06 | Sofya Raskhodnikova, Adam Smith, A Note on Adaptivity in Testing Properties of Bounded Degree Graphs. Electronic Colloquium on Computational Complexity, Report No. 89, 2006. |

EKKRV00 | Funda Ergün, Sampath Kannan, Ravi Kumar, Ronitt Rubinfeld, Mahesh Viswanathan, Spot-Checkers. Journal of Computer System and Sciences 2000, STOC 1998. |

Fis04 | Eldar Fischer, On the strength of comparisons in property testing. Information and Computation 2004. |

DGLRRS99 | Yevgeniy Dodis, Oded Goldreich, Eric Lehman, Sofya Raskhodnikova, Dana Ron, Alex Samorodnitsky, Improved Testing Algorithms for Monotonicity. RANDOM-APPROX 1999. |

BFJRW09 | Arnab Bhattacharyya, Elena Grigorescu, Kyomin Jung, Sofya Raskhodnikova, David Woodruff, Transitive-Closure Spanners. SODA 2009. |

Ras10 | Sofya Raskhodnikova, Transitive-Closure Spanners: a Survey. In O. Goldreich, editor, Property Testing, LNCS 6390, LNCS State-of-the-Art Surveys, Springer, Heidelberg, 167--196, 2010. |

JR13 | Madhav Jha, Sofya Raskhodnikova, Testing and Reconstruction of Lipschitz Functions with Applications to Data Privacy. SIAM Journal on Computing 2013, STOC 11. |

GR02 | Oded Goldreich, Dana Ron, Property testing in bounded degree graphs. Algorithmica 2002, STOC 1997. |

CRT05 | Bernard Chazelle, Ronitt Rubinfeld, Luca Trevisan, Approximating the Minimum Spanning Tree Weight in Sublinear Time. SIAM Journal of Computing 2005, ICALP 2001. |

FLNRRS02 | Eldar Fischer, Eric Lehman, Ilan Newman, Sofya Raskhodnikova, Ronitt Rubinfeld, Alex Samorodnitsky Monotonicity Testing Over General Poset Domains. STOC 2002. |

BBM11 | Eric Blais, Joshua Brody, Kevin Matulef, Property Testing Lower Bounds via Communication Complexity. CCC 2011. |

AK02 | Noga Alon, Michael Krivelevich, Testing k-colorability. SIAM J. Discrete Math. 15 (2002), 211-227. |

Alon02 | Noga Alon, Testing subgraphs in large graphs. Random Structures and Algorithms 21 (2002), 359-370. |

AFKS02 | Noga Alon, Eldar Fischer, Michael Krivelevich, Mario Szegedy, Efficient testing of large graphs, Combinatorica 20 (2000), 451-476. |

GT03 | Oded Goldreich, Luca Trevisan, Three theorems regarding testing graph properties, Random Struct. Algorithms 23(1): 23-57 (2003) |

GR08 | Oded Goldreich, Dan Ron, Approximating average parameters of graphs, Random Struct. Algorithms 32(4): 473-493 (2008) |

PaRo07 | Michal Parnas, Dan Ron, Approximating the minimum vertex cover in sublinear time and a connection to distributed algorithms, Theor. Comput. Sci., 381(1-3):183--196 (2007) |

NO08 | Huy N. Nguyen and Krzysztof Onak, Constant-Time Approximation Algorithms via Local Improvements, FOCS (2008) |

ORRR12 | Krzysztof Onak, Dana Ron, Michal Rosen, and Ronitt Rubinfeld, A near-optimal sublinear-time algorithm for approximating the minimum vertex cover size, In Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1123--1131 (2012) |

Ras99 | Sofya Raskhodnikova, Monotonicity Testing, Master's Thesis, Massachusetts Institute of Technology, Cambridge, MA, 1999 |

CS12 | Deeparnab Chakrabarty, C. Seshadhri Optimal bounds for monotonicity and Lipschitz testing over the hypercube. ECCC, TR12-030, 2012. |

BBBY12 | Maria-Florina Balcan, Eric Blais, Avrim Blum, Liu Yang, Active Property Testing. Manuscript, 2012. |

BLR93 | Manuel Blum, Michael Luby, Ronitt Rubinfeld, Self-Testing/Correcting with Applications to Numerical Problems. Journal of Computer System and Sciences 1993, STOC 1990. |

BCHKS96 | M. Bellare, D. Coppersmith, J. Hastad, M. Kiwi, M. Sudan, Linearity testing in characteristic two. IEEE Transactions on Information Theory, Vol. 42, No. 6, pp. 1781--1795, 1996, FOCS 95. |

AMS99 | Noga Alon, Yossi Matias, Mario Szegedy, The Space Complexity of Approximating the Frequency Moments. J. Comput. Syst. Sci. 58(1): 137–147, 1999. |

BJKST02 | Ziv Bar-Yossef, T. S. Jayram, Ravi Kumar, D. Sivakumar, Luca Trevisan. Counting Distinct Elements in a Data Steam. RANDOM 02: 1-10, 2002. |

Mor78 | Robert Morris, Counting Large Numbers of Events in Small Registers. Commun. ACM, 21(10): 840-842, 1978. |

CM05 | Graham Cormode, S. Muthukrishnan, An improved data stream summary: the count-min sketch and its applications. J. Algorithms 55(1): 58-75, 2005 |

GM07 | Sumit Ganguly, Anirban Majumder, CR-precis: A Deterministic Summary Structure for Update Data Streams. ESCAPE 2007: 48-59, 2007 |

Rou16 | Tim Roughgarden, Communication Complexity (for Algorithm Designers). Found. Trends Theor. Comput. Sci. 11(3-4): 217-404, 2016 |

BJKS04 | Ziv Bar-Yossef, T.S. Jayram, Ravi Kumar, and D. Sivakumar, An information statistics approach to data stream and communication complexity. Journal of Computer and System Sciences 68, 702–732, 2004 |

AGM12 | Kook Jin Ahn, Sudipto Guha, Andrew McGregor, Analyzing graph structure via linear measurements. SODA 2012: 459-467 |