Algebra & Algorithms

In this module we will study a mathematical model of hardware: the Boolean circuits. They provide the birds eye view of how one builds a complex logical circuit from elementary building blocks by linking them with wires. We will learn efficient ways to construct circuits for all four arithmetic operations on integers represented in binary form. Among other things we will see how to efficiently calculate all carries while adding two numbers (much faster than just computing these carries sequentially) and how Newton’s approximation method for solving equations helps with implementing integer division.

In this second and last module devoted to Boolean circuits we move from arithmetics to the problem of constructing efficient circuits for arbitrary Boolean functions. On the way, among other things, we will see how to estimate the number of trees via DFS traversals and how to construct efficiently a circuit that computes all functions of given small number of variables.

Multiplication of integers is among the first things people learn to do with integers at school, later moving on to higher spheres: multiplying matrices, polynomials, permutations etc. Multiplication is one of the central things in algebra. This week we focus on classical algorithms for efficient integer multiplication first, and then move on to matrix and polynomial multiplication. On the way we will see how closely integer and polynomial multiplication are really intertwined and what polynomial interpolation has to do with multiplication.

This module is the first one to feature application pf efficient algorithms for algebraic operations to something outside algebra. Currently we turn to distances in graphs. We also study a non-typical way of multiplying matrices motivated by applications to graph reachibiilty, namely, Boolean matrix multiplication, and consider a corresponding rather general speedup technique.