# CSE 598A Sublinear Algorithms1 Spring 2012

## General Information

Time and location: Tuesday/Thursday 1:00PM-2:15PM, 167 Willard Bldg
Office hours: Tuesdays, 2:30PM-3:30PM, IST 343F

#### Prerequisites

CSE 565; STAT 318 or MATH 318

#### Course description

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, image analysis 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 homework assignments, taking lecture notes about 2-3 times per person, and the final project.

## Lectures

Lec Date Topics References Handouts/Homework
1 Tu, Jan 10 Introduction. Basic models for sublinear-time computation. Simple examples of sublinear algorithms. (slides for lectures 1 & 2 ) RS96, GGR98, Ras03, EKKRV00, Fis04 General information
2 Th, Jan 12 Properties of lists, functions, graphs. Testing if a list is sorted/Lipschitz, if a function is monotone, and if a bounded-degree graph is connected. (See slides posted for previous lecture.) DGLRRS99, BGJRW09, Ras10, JR11, GR02
3 Tu, Jan 17 Finished testing connectedness. Approximating the number of connected components and MST weight. (slides ) GR02, CRT05
4 Th, Jan 19 Background on probability Handouts on probability by Dana Ron and from MIT lecture and recitation Homework 1 out
5 Tu, Jan 24 Methods for proving lower bounds: Yao's principle. (slides ) FLNRRS02
6 Th, Jan 26 Methods for proving lower bounds: communication complexity. Other models of sublinear time/space computation. Discussion of course projects. (slides ) BBM11 HW1 due
7 Tu, Jan 31 Dense graphs: testing bipartiteness GGR98, AK02 First project meeting
8 Th, Feb 2 Dense graphs: approximating max-cut GGR98 Project proposal due; Homework 2 out
9 Tu, Feb 7 Dense graphs: Finish approximating max-cut. GGR98
10 Th, Feb 9 Testing dense graphs: discuss characterization. Testing triangle-freeness. Regularity lemma. AFKS00
11 Tu, Feb 14 Triangle-removal lemma
12 Th, Feb 16 Dense graphs: lower bound for testing triangle-freeness. Alon02 Homework 2 due
13 Th, Feb 23 In class problem solving: adaptivity in the dense-graph model GT03
14 Tu, Feb 28 Approximating graph parameters: average degree GR08
15 Th, Mar 1 Approximating Vertex Cover and distributed algorithms PaRo07
16 Tu, Mar 13 Approximating Vertex Cover and distributed algorithms PaRo07, NO08, ORRR12
17 Th, Mar 15 Testing monotonicity of functions: range reduction DGLRRS99, Ras99
18 Tu, Mar 20 Testing monotonicity of functions on hypergrids: dimension reduction DGLRRS99, Ras99 Project progress reports due
19 Th, Mar 22 Properties defined by 2CNF formulas and testing monotonicity of functions on posets FLNRRS02
20 Tu, Mar 27 Induced matchings and a lower bound for testing monotonicity of functions on posets FLNRRS02 Homework 3 out
21 Th, Mar 29 Finish the lower bound for testing monotonicity on posets: (s,t)-Rusza-Semeredi graphs FLNRRS02
22 Tu, April 3 Testing monotonicity of Boolean functions on hypergrids DGLRRS99
23 Th, April 5 Testing monotonicity of Boolean functions on hypergrids (part 2) DGLRRS99 Homework 3 due
24 Tu, April 10 Resolution of monotonicity (and Lipschitz) testing conjectures CS12
25 Th, April 12 Talk by Avrim Blum: Active (and Passive) Property Testing BBBY12 Homework 3 due
26 Tu, Apr 17 Testing linearity of functions. (slides ) BLR93, BCHKS96 Project final reports due
27 Tu, Apr 24 Final project presentations
28 Th, Apr 26 Final project presentations (in 222 IST)

## Lecture notes

Preamble and bib files; lecture notes wiki

#### Notes themselves, possibly in a rough shape:

Lecture 3. Testing connectedness. Approximating the number of connected components and MST weight. pdf, tex

## Miscellaneous

LaTeX
For tips on using latex to type homework, see these links. Homework template files: tex, pdf, cls, jpg.
Other Useful Programs
On Mac, emacs and Skim

## Bibliography

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. 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. JR11 Madhav Jha, Sofya Raskhodnikova, Testing and Reconstruction of Lipschitz Functions with Applications to Data Privacy. FOCS 2011. 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 Linearity testing in characteristic two. IEEE Transactions on Information Theory, Vol. 42, No. 6, pp. 1781--1795, 1996, FOCS 95.

1. The design of this course is partially supported by the National Science Foundation under Grant No. 0845701. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation (NSF).