Lectures
Following is an outline of lectures given along with references and links to additional reading. For abbreviations, NK and RY check the references page.
\( \def\PAL{\rm PAL} \def\HP{\rm HP} \def\MP{\rm MP} \def\EQ{\rm EQ} \def\USELESS{\rm USELESS} \def\EMPTY{\rm EMPTY} \def\DTIME{\sf DTIME} \def\NTIME{\sf NTIME} \def\DSPACE{\sf DSPACE} \def\NSPACE{\sf NSPACE} \def\P{\sf P} \def\co{\sf co} \def\NP{\sf NP} \def\LOG{\sf LOG} \def\NLOG{\sf NLOG} \def\PSPACE{\sf PSPACE} \def\NPSPACE{\sf NPSPACE} \def\EXP{\sf EXP} \)
Lec 1 (06 Jan, Mon) : Administrative information and course outline. Two motivational scenarios - (1) Computing the parity function by a special kind of Boolean circuits called Formulas, (2) Area-time trade-offs for VLSI design. The basic setup - the two party communication model. Goals of this course.
Reading : (NK) Chap 1; (RY) Introduction; Hromkovic, Section 4.2.2
Lec 2 (08 Jan, Wed) : One more motivational scenario - (3) Time lower bounds for single tape Turing machines. Functions and their computation using the two party communication model. Informal description of a protocol. Cost of a protocol. Cost of computing a function. Naive upper bounds on protocol cost. Protocols for (1) computing parity, (2) computing the minimum, maximum and average of a list of numbers (3) computing the median of a list of numbers.
Reading : (NK) Section 1.1, (RY) Introduction
Theme: (1) Basics of deterministic two-party communication protocols.
Lec 3 (10 Jan, Fri) : A protocol for median computation using $O(\log^2 n)$ bits. Correctness of the protocol and estimating its communication cost. The clique-versus-independent set (CIS) problem. Idea behind a protocol of cost $O(\log^2 n)$.
Reading : (NK) Section 1.1, (RY) Introduction
Lec 4 (11 Jan, Sat) : (Compensatory) Protocol for CIS - details of a single round and a worked out example of the run of the protocol. Proof of correctness and communication cost assuming
Reading : (NK) Section 1.1, Ex. 1.8 and pg. 169, (RY) Introduction, (Roughgarden) Sec. 4.2.2
Lec 5 (13 Jan, Mon) : Completed the proof of correctness of CIS protocol. Equality, disjointness and $k$-disjointness function - protocols and cost. Protocols as trees. Formal definition of a protocols. A counting argument to show that most Boolean functions of $2n$ bits cannot even be computed by protocols using $n-1$ bits of communication.
Reading : (RY) Chap. 1
Lec 6 (15 Jan, Wed) : Completed the counting based argument. Introduction to deterministic communication lower bound techniques. A simple setting - lower bounds for deterministic one-way communication protocols via pigeon-hole principle. One way communication lower bounds for disjointness.
Reading : (RY) Chap.1, (Roughgarden) Sec. 1.9.1
Lec () (17 Jan, Fri) : Non-instructional day due to Institute day
Lec () (20 Jan, Mon) : Non-instructional day
Theme: (2) Structural properties of protocols and basic lower techniques
Lec 7 (22 Jan, Wed) : A combinatorial view of protocols via rectangles. Communication matrix. Nice property. Rectangles are nice. Inputs reaching an internal node forms a rectangle. Leaf rectangles are monochromatic. Protocols induces partition of the communication matrix into monochromatic rectangles.
Reading : (NK) Section 1.2, (RY) Chap. 1.
Lec () (24 Jan, Fri) : Class cancelled. Instructor out of town.
Lec 8 (27 Jan, Mon) : Relationship between number of leaves of a protocol and its cost. Introduction to rectangle method. Worked out examples - Equality and Disjointness function. Gave arguments to bound the size of $1$ rectangles for both the functions.
Reading : (NK) Section 1.2-1.3, (RY) Chap. 1.
Lec 9 (29 Jan, Wed) : Deterministic communication lower bounds for inner product function. Review of basics of vector space over $\mathbb{F}_2$. Bound on the size of $0$-rectangles using arguments from linear algebra.
Reading : (NK) Section 1.3, (RY) Chap. 1
Lec 10 (31 Jan, Fri) : Fooling set method. Lower bound for Disjointness and Equality by constructing fooling sets. Introduction to the rank method.
Reading : (NK) Section 1.3, (RY) Chap. 1
Lec 11 (03 Feb, Mon) : Algebraic technique - rank method. Equivalent formulations on rank of a matrix. Communication complexity is lower bounded by log rank of communication matrix. Tensor product of matrices and rank computation. Worked out examples - Equality, Disjointness and inner-product.
Reading : (NK) Section 1.3, (RY) Chap. 2
Lec 12 (05 Feb, Wed) : Tensor product structure of the communication matrix of disjointness and inner product functions. Lower bound for $k$-disjointness via rank.
Reading : (NK) Section 1.3, Section 2.3, ex 2.12 (RY) Chap. 2.
Theme: (3) Design and analysis of randomized protocols
Lec 13 (07 Feb, Fri) : Application 0 - time-space lower bounds against one-tape Turing machines. Application 1 - Lower bounds on static data structures for set intersection, lopsided set intersection and span checking. Lower bound for nearest neighbour search. Best case communication complexity. Application 2 - Area-time trade-offs in VLSI design. $k$-disjointness (Disjointness for $k$ sized sets). Recap of protocols inducing partitions. Relaxing the notion of partitions to covers. $0$-covers and $1$-covers. Interpretation as non-deterministic communication. Size of covers. Bounds on the size of covers for equality, inner product and $k$-disjointness. From protocol covers to partitions - making a non-deterministic protocol deterministic efficiently. Upper bound for communication cost of a Boolean function in terms of rank of its communication matrix and $1$-cover. Upper bounds for communication cost in terms of rank of the communication matrix. The log-rank conjecture.
Reading :Lec 14 (10 Feb, Mon) : Randomized communication protocols. Variants and formal definitions. Example protocol for equality, greater than, $k$-disjointness and Gap-Hamming distance. Error reduction by independent repetition. Public versus private randomness. Public coin protocol for equality. Newman’s theorem - converting public randomness to private randomness.
Reading :Theme: (4) Communication Complexity of relations and applications
Lec 15 (12 Feb, Wed) : Communication complexity of relation. Examples. Formula lower bound for computing parity function.
Reading :Lec 16 (14 Feb, Fri) : Karchmer-Wigderson games for a Boolean function $f$. Cost of $KW(f)$. Monotone variant of the KW game.
Reading :Lec 17 (17 Feb, Mon) : Monotone Boolean functions and Monotone circuits. Relation between monotone circuit depth and cost of monotone KW games.
Reading :Lec 18 (19 Feb, Wed) : Application - Circuit depth lower bounds for $s$-$t$ reachability problem for monotone circuits. The FORK relation. Details of the game and its connection to circuit depth.
Reading :Lec 19 (21 Feb, Fri) :
Reading :Lec 20 (24 Feb, Mon) : Lower bound for the communication cost of FORK relation.
Reading :Lec 21 (26 Feb, Wed) : Application - A lower bound on the size of regular expressions.
Reading :Lec 22 (28 Feb, Fri) : Depth lower bound for monotone circuits computing matching via reductions.
Reading :Theme: (4) Lower bounds for randomized protocols with applications
Lec 23 (03 Mar, Mon) : Back to randomized protocols. Randomized communication lower bounds. The basic technique - Yao’s mini-max theorem and its proof. Distributional complexity.
Reading :Lec 24 (05 Mar, Wed) : Application 1 - showing lower bounds for streaming algorithms. An example of a streaming algorithm. One-way communication protocols and its connection to streaming algorithms.
Reading :Lec 25 (07 Mar, Fri) : One-way communication lower bounds for index and disjointness function.
Reading :Lec 26 (10 Mar, Mon) : Application 2 - showing lower bounds for compressed sensing and sparse recovery. A toy version of the sparse recovery problem.
Reading :Lec 27 (12 Mar, Wed) : Lower bound for sparse recovery - an attempt. Reduction via index function.
Reading :Lec () (14 Mar, Fri) : Holiday due to Holi !
Lec 28 (17 Mar, Mon) : Showing randomized communication lower bounds for disjointness by carefully choosing hard distributions.
Reading :Lec 29 (19 Mar, Wed) : Introduction to discrepancy. Definition and motivation. Connection to randomized communication lower bounds.
Reading :Lec 30 (21 Mar, Fri) : Randomized communication lower bound for inner product via discrepancy.
Reading :Lec 31 (24 Mar, Mon) : Randomized communication lower bound for disjointness via discrepancy.
Reading :Lec 32 (26 Mar, Wed) : Randomized communication lower bound for gap-hamming problem via discrepancy.
Reading :Theme: (5) Information theoretic tools
Lec 33 (28 Mar, Fri) : Towards a tight lower bound for disjointness. Introduction to basics of information theory - axiomatic definition of entropy, chain rule, conditional entropy, mutual information. Two combinatorial applications.
Reading :Lec () (31 Mar, Mon) : Holiday due to Eid al-Fitr.
Lec 34 (02 Apr, Wed) : Lower bounds for index function using information theoretic tools.
Reading :Lec 35 (04 Apr, Fri) : Application of information theoretic tools 1 - Randomized lower bounds for round restricted protocols for greater than function.
Reading :Lec 36 (05 Apr, Sat) : (Instructional day)
Reading :Lec 37 (07 Apr, Mon) : Application of information theoretic tools 2 - Randomized lower bounds for round restricted protocols for pointer chasing.
Reading :Lec 38 (09 Apr, Wed) :
Reading :Lec 39 (11 Apr, Fri) : Application of information theoretic tools 3 - Randomized communication lower bound for computing disjointness.
Reading :Lec 40 (14 Apr, Mon) :
Reading :Theme: (6) The Lifting technique
Lec 41 (16 Apr, Wed) : (Friday’s timetable) Communication lower bounds via lifting technique.
Reading :Lec () (18 Apr, Fri) : Holiday due to Good Friday
Lec 42 (21 Apr, Mon) : Separating rank and communication complexity.
Reading :Lec 43 (23 Apr, Wed) :
Reading :