CSCE 222 Lecture 25

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Divide and Conquer

Example

Multiplication of Integers

In binary form, two number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} and $b$ consist of $2n$ bits. Each integer can be rewritten as a sum of two $n$-bit integers that represent the most and least significant bits.

${\begin{aligned}a&=\left(a_{2n-1},\ a_{2n-2},\ \ldots ,\ a_{1},\ a_{0}\right)\\&=2^{n}A_{1}+A_{0}=\left(a_{2n-1},\ \ldots ,\ a_{n}\right)+\left(a_{n-1},\ \ldots ,\ a_{0}\right)\\b&=\left(b_{2n-1},\ b_{2n-2},\ \ldots ,\ b_{1},\ b_{0}\right)\\&=2^{n}B_{1}+B_{0}=\left(b_{2n-1},\ \ldots ,\ b_{n}\right)+\left(b_{n-1},\ \ldots ,\ b_{0}\right)\\ab&=(2^{n}A_{1}+A_{0})(2^{n}B_{1}+B_{0})=(2^{2n}+2^{n})A_{1}B_{1}+2^{n}(A_{1}-A_{0})(B_{0}-B_{1})+(2^{n}+1)A_{0}B_{0}\end{aligned}}$

Let $f(n)$ be the total number of bit operations needed to multiply two $n$-bit integers. In this case, $f(2n)=3f(n)+Cn$.

Bitwise Shift Operator

a >> n shifts the binary digits of $a$ $n$ places to the right (essentially multiplying $a$ by $2^{n}$).

Master Theorem

Let $f$ be an increasing function satisfying a recurrence relation:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(n) = af\left(\frac{n}{b}\right) + cn^d}

Whenever

• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n=b^k} for some Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k\ge1} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \ge 1}
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} is an integer > 1
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d} are real numbers ≥ 0

Then

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(n) = af\left(\frac{n}{b}\right) + cn^d = \begin{cases} O\left(n^d\right) & a<b^d \\ O\left(n^d \log{n}\right) & a=b^d \\ O\left(n^{\log_b{a}}\right) & a>b^d \end{cases}}

Proof

Proposition. If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a=b^d} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} is a power of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} , then a function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} satisfying the recurrence relation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(n) = af(n/b) + cn^d} is of the form Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(n) = f(1)n^d+cn^d\log_b{n}}

Proof. Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k = \log_b{n}} . Then

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} f(n) &= cn^d + ac\left(\frac{n}{b}\right)^d+a^2c\left(\frac{n}{b^2}\right)^d+\dots+a^{k-1}c\left(\frac{n}{b^{k-1}}\right)^d+a^kf(1) \\ &= a^k f(1) + \sum_{j=0}^{k-1} a^jc\left(\frac{n}{b^j}\right)^d \\ &= a^k f(1) + \sum_{j=0}^{k-1} cn^d \\ &= a^k f(1) + kcn^d \\ &= a^{\log_b{n}} f(1) + c(\log_b{n})n^d \\ &= b^{d\log_b{n}} f(1) + c(\log_b{n})n^d \\ &= n^d f(1) + c(\log_b{n})n^d \\ &= O(n^d \log{n}) \end{align}}