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 for class on Jan 24) 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 of cost $2\lceil \log n\rceil$. 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 ($D(R_{\oplus}) \ge \lceil 2 \log n \rceil$) due to Khrapchenkov.

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

Lec 27 (12 Mar, Wed) : Completed the details of lower bound argument for $R_{\oplus}$. Boolean circuits - size and depth. Connection between circuit depth and parallel computation. Most of the Boolean functions on $n$ bits needs depth at least $n-O(1)$ (statement only). Introduction to Karchmer-Wigderson relation for a Boolean function $f$. Connection between parallel computation of $f$ and circuit/formula depth computing $f$.

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

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

Lec 28 (17 Mar, Mon) : Depth of min depth formula computing a Boolean function $f$ and the depth of min depth circuit computing $f$ are the same. Obtaining a protocol for Karchmer Wigderson relation (KW) for $f$ from a formula computing $f$ - protocol and correctness. Obtaining a formula computing $f$ from a protocol solving the Karchmer Wigderson relation for $f$ - construction and correctness.

^{Reading : (NK) Section 10.2, Section 5.1; (RY) Chap 9, Boolean Circuits, Karchmer-Wigderson Games}

Lec 29 (19 Mar, Wed) : Monotone Boolean functions and Monotone Boolean circuits. A Boolean function is monotone if and only if it can be computed by a Boolean monotone circuit. Examples of monotone functions - AND, OR, Majority, CLIQUE, Perfect Matching. Reachability and the directed s-t connectivity problem - $DSTCONN_n$. Upper bound on circuit depth for $DSTCONN_n$ of $O(\log^2 n)$. Monotone variant of the Karchmer Wigderson (KW) game.

^{Reading : (NK) Section 10.1}

Lec 30 (21 Mar, Fri) : Monotone KW relation. Cost of monotone KW relation for $f$ and its connection to monotone circuit depth of $f$. The $FORK_{w,\ell}$ relation - definition. Thinking of $FORK$ relation as hard instances of directed graph reachability problem. A protocol for the $FORK$ relation of cost $O(\log \ell \log w)$ via binary search strategy.

^{Reading : (NK) Section 5.3, Section 10.3}

Lec 31 (24 Mar, Mon) : Any protocol for monotone KW game of $DSTCONN_n$ can be used to solve $FORK_{w,\ell}$ where $n=w(\ell+2)$. A lower bound of $\Omega(\log^2 n)$ on $DSTCONN$ from $FORK_{w,\ell}$ lower bound. The notion of an $(\alpha, \ell)$ protocol solving FORK game for $\alpha$ fraction of inputs. A weaker lower bound of $\Omega(\log w)$ for $D(FORK)$ via round elimination.

^{Reading : (NK) Section 5.3, Section 10.3}

Lec 32 (26 Mar, Wed) : A lower bound for the communication cost of $FORK_{w,\ell}$ relation. Statements of round elimination and amplification argument. Completed the proof of round elimination and a weaker lower bound of $\Omega(\log w)$. Overall lower bound argument using round elimination and amplification arguments. Overview of ideas behind the amplification argument.

^{Reading : (NK) Section 5.3, Section 10.3}

Lec 33 (28 Mar, Fri) : Recap and statement of amplification lemma. From an $(\alpha,\ell)$-protocol to $(\frac{\sqrt{\alpha}}{2}, \ell/2)$-protocol when $\alpha = 1/8$. Statement on the property of square matrix with at least a constant fraction of entries being one - there is either a row with “too many” ones or many rows with “moderately many” ones. Precise formulation and a proof. Using this to obtain an $(\frac{1}{4}, \ell/2)$-protocol and the mind blocks ahead.

^{Reading : (NK) Section 5.3, Section 10.3}

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

Lec 34 (02 Apr, Wed) : Recap of the amplification argument so far. Handling the case when there are many rows with moderately many ones. Naive attempt and why it fails. Improved attempt by independent repetition and bounding the failure probability.

^{Reading : (NK) Section 5.3, Section 10.3}

Theme: (4) Randomized setting - protocols and lower bounds

Lec 35 (03 Apr, Thu) : (Compensatory class for April 14) Completed the proof of amplification lemma. Introduction to Randomized communication protocols- basic model. Public versus private randomness. A public coin randomized protocol for equality of cost (independent of the input length).

^{Reading : (NK) Section 3.1, 3.3. (RY) Chap 3, Randomized protocols}

Lec 36 (04 Apr, Fri) : Error reduction by independent repetitions. A public coin protocol of cost $2k$ for equality with success probability of at least $1-2^{-k}$. A private coin protocol for equality on $n$ bits using $O(\log n)$ bits communication with a success probability of at least $1-1/n$.

^{Reading : (NK) Section 3.1, 3.3. (RY) Chap 3, Randomized protocols}

Lec 37 (05 Apr, Sat) : Public coin randomized protocols for greater than function of cost $O(\log n\log\log n)$ - correctness and error bounds. Overview of a public coin randomized protocol for $k$-disjointness in expected $O(k)$ bits. Basic ideas - for a given set $S$ of size $k$, (1) a randomly picked subset of $[n]$ contains the given set $S$ with probability $1/2^k$ and (2) intersecting $S$ with a random set from $[n]$ will likely halve the size of $S$. Description of the protocol for $k$-disjointness.

^{Reading : (RY) Chap 3, Randomized protocols}

Lec 38 (07 Apr, Mon) : Recap and analysis of a randomized protocol for $k$-disjointness due to Hastad and Wigderson. Bound on the expected number of bits communicated.

^{Reading : (RY) Chap 3, Randomized protocols}

Lec 39 (09 Apr, Wed) : Argued the correctness and the error probability of the Hastad-Wigderson protocol. Estimation via communication - computing Hamming distance. An emperical estimation algorithm. Argued that the expected value of the emperical estimate is exactly the Hamming distance (meaning that the estimation is unbiased). Chernoff-Hoeffding concentration inequality to bound the deviations from the expectation.

^{Reading : (RY) Chap 3, Randomized protocols}

Lec 40 (11 Apr, Fri) : Variants of randomized protocols - one sided, two sided error and zero error randomized protocols. Formal definitions of the variants and their costs. Expected versus worst case cost and public coins versus private coins in the definitions. Relation between the cost of the variants - error reduction for one-sided error protocols by independent repetitions.

^{Reading : (NK) Section 3.1, 3.3. (RY) Chap 3, Randomized protocols}

Lec () (14 Apr, Mon) : Holiday due to Ambedkar Jayathi.

Lec 41 (16 Apr, Wed) : Completed the proof of error reduction for one-sided error protocols. Relation between zero error protocols and one-sided errror randomized protocols. Error reduction for two-sided error protocols.

^{Reading : (NK) Section 3.1, 3.3. (RY) Chap 3, Randomized protocols}

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

Lec 42 (21 Apr, Mon) : Completed the proof of error reduction for two-sided error protocols using Chernoff inequality. Connection to non-deterministic communication cost. Lower bounds for zero error and one-sided error randomized protocol cost from lower bounds on non-deterministic protocol cost. Making private-coin randomized protocols deterministic. Viewing randomized protocols as a inducing a distribution over the monochromatic rectangles.

^{Reading : (NK) Section 3.2, 3.2. (RY) Chap 3, Randomized protocols}

Lec 43 (23 Apr, Wed) : Completed the details of obtaining a deterministic protocol from a private-coin randomized protocol. Newman’s theorem - converting protocols using public randomness to protocols using private randomness - argument via Chernoff-Heffding inequality. Course wrap-up and summary.

^{Reading : (NK) Section 3.2, 3.3. (RY) Chap 3, Randomized protocols}