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 protocol. 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 bound 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 form a rectangle. Leaf rectangles are monochromatic. Protocols induce 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 the log rank of the 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.}

Lec 13 (07 Feb, Fri) : Application 0 - time-space lower bounds against deterministic multi-tape Turing machines accepting the Palindrome language. Obtaining a deterministic communication protocol from a multi-tape Turing machine.

^{Reading : (NK) Section 12.1, Section 8.3}

Lec 14 (10 Feb, Mon) : Argued correctness and communication cost. Time-space lower bounds for palindrome language. Application 1 - Area-time trade-offs in VLSI design - Introduction to best case partition communication complexity. Best case partition complexity - definition, examples.

^{Reading : (NK) Section 8.3, Section 7.2}

Lec 15 (12 Feb, Wed) : Need for the best case partition setting. Obtaining a communication protocol for an arbitrary partition from a VLSI circuit. Shifted equality function (SEQ). Towards showing a best case partition complexity lower bound for SEQ.

^{Reading : (NK) Section 8.3, Section 7.2}

Lec 16 (14 Feb, Fri) : Details of the communication lower bound for SEQ. Area-time trade-offs for VLSI design for SEQ. Application 2 - Lower bounds on static data structures. The basic setup of static data structure setting - words, cells and query time. Three example data structure schemes for set disjointness.

^{Reading : (NK) Section 8.3, Section 7.2; (RY) Chap 1, Asymmetric Computation}

Lec 17 (17 Feb, Mon) : Obtaining lower bounds for static data structure via communication complexity. Communication complexity with partitions of unequal length. Data structure lower bounds for the lopsided set intersection.

^{Reading : (RY) Chap 1, Asymmetric Computation}

Lec 18 (19 Feb, Wed) : Definition of rich sets. A general argument for obtaining large sized monochromatic rectangles from rich sets from communication protocols. Showing lower bound for lopsided disjointness – existence of rich sets. A combinatorial argument to show upper bound on the size of monochromatic rectangles.

^{Reading : (RY) Chap 1, Asymmetric Computation}

Lec 19 (21 Feb, Fri) : Completed the communication lower bound argument for lopsided disjointness. Span problem, definition and trivial protocols. Exhibiting rich sets for span problem. A combinatorial argument for upper bounding the size of a $1$-monochromatic rectangles. Completed the communication lower bound for span problem.

^{Reading : (RY) Chap 1, Asymmetric Computation}

Lec 20 (24 Feb, Mon) : Communication protocol from rank bounds. Statement of the log-rank conjecture. Best known gap between communication cost and rank. Recap of protocols inducing partitions. Not all partitions can correspond to protocols (illustrated by an example partition). A partial converse - given a rectangle partition, identifying the rectangle containing the input. Details of the communication protocol of cost $O(c^2)$ for partitions of size $2^{c}$.

^{Reading : (NK) Section 2.1; (RY) Chap 1, From rectangles to protocols.}

Lec 21 (26 Feb, Wed) : Relaxing the notion of partitions to covers. $0$-covers and $1$-covers. Interpretation of covers as non-deterministic protocols. Size of covers. Upper bounds on the size of $0$ covers and $1$ covers for equality. Obtaining protocols from a small sized partition of $f^{-1}(1)$ or $f^{-1}(0)$ by a reduction to the clique-versus-independent set problem.

^{Reading : (NK) Section 2.1; (RY) Chap 1, Rectangle covers}

Lec 22 (28 Feb, Fri) : Quick review of covers - a $0$-cover and $1$-cover for the disjointness function. From protocol covers to partitions - making a non-deterministic protocol deterministic efficiently. Given a $0$-cover and $1$-cover of size $2^{c_0}$ and $2^{c_1}$ respectively, obtaining a deterministic protocol of cost $O(c_0c_1)$ computing $f$ - details of the protocol, correctness and analysis of the communication cost.

^{Reading : (NK) Section 2.3; (RY) Chap 1, Rectangle covers, Class notes.}

Lec 23 (03 Mar, Mon) : Equivalence between non-determinism and covers. Lower bounds for covers - connection to linear programming and extension complexity (statements only, details to be covered later). Rectangle method also gives a lower bound for covers. Overview of an argument to obtain a $1$-cover for $k$-disjointness using the probabilistic method.

^{Reading : (NK) Section 2.3; (RY) Chap 1, Rectangle covers. Notes due to Lovett}

Lec 24 (05 Mar, Wed) : $1$-cover for $k$-disjointness of size $2^{2k+1}\ln \binom{n}{k}$ via a covering argument using probabilistic method. Showed that for $k=\log n$, the function $k$-Disjointness satisfied $D(f) = \Omega(N^0(f)\cdot N^1(f))$.

^{Reading : (NK) Section 2.3, Example 2.12; (RY) Chap 1, Rectangle covers.}

Theme: (3) Communication Complexity of relations and applications

Lec 25 (07 Mar, Fri) : Communication complexity of relation, examples. Monochromatic rectangles, leaves of a protocol forms a monochromatic rectangle. Communication cost of the Universal relation, lower bound via Not Equal function. Connection to formula size lower bound for computing the parity function.

^{Reading : (NK) Chap 5; (RY) Chap 1, Lower bounds for relations.}

Lec 26 (10 Mar, Mon) : The relation $R_{\oplus}$. A protocol for computing the relation $R_{\oplus}$ via binary search. Computing $R_{\oplus}$ from a Boolean formula computing parity with cost at most the depth of the formula. Structure of Boolean hypercube on $n$ bits. Overview of the lower bound argument due to Khrapchenkov.

^{Reading : (NK) Section 10.2, Section 5.1; (RY) Chap 1, Lower bounds for relations.}

Lec () (14 Mar, Fri) : Holiday due to Holi !

Theme: (4) Randomized communication protocols - design and lower bounds

Theme: (5) Information theoretic tools

Lec () (31 Mar, Mon) : Holiday due to Eid al-Fitr.

Theme: (6) The Lifting technique

Lec () (18 Apr, Fri) : Holiday due to Good Friday