MATH 302 Lecture 17

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Chapter 8.3: Divide and Conquer Algorithms

Examples:

• Binary Search on Sorted List: 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\left(\frac{n}{2}\right) + 2}
• Merge Sort: 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) = 2 f \left( \frac{n}{2} \right) + n}
• Recursive Max/Min of Unsorted List: $f(n)=2f\left({\frac {n}{2}}\right)+2$
• Multiplying 2n-bit Numbers: $f(2n)=3f(n)+c(n)$

In general, form is:

$f(n)=af\left({\frac {n}{b}}\right)+g(n)$

Note: the book restricts the last term to powers of n

Master Theorem

The master theorem provides a general boilerplate solution for solving recurrence relations (like divide and conquer algorithms) that satisfy a particular format.

Definition

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

$f(n)=af\left({\frac {n}{b}}\right)+cn^{d}$

Whenever

• $n=b^{k}$ for some $k\geq 1$, $a\geq 1$
• $b$ is an integer > 1
• $c$ and $d$ are real numbers ≥ 0

Then

$f(n)=af\left({\frac {n}{b}}\right)+cn^{d}={\begin{cases}\Theta \left(n^{d}\right)&a<b^{d}\\\Theta \left(n^{d}\log {n}\right)&a=b^{d}\\\Theta \left(n^{\log _{b}{a}}\right)&a>b^{d}\end{cases}}$

Proof

Proposition. If $a=b^{d}$ 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 $k=\log _{b}{n}$. 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}}

Counting

Product Rule

Suppose we have to make 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_1} choices followed by 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_2} choices. For example, selecting what to wear from 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_1} shirts 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_2} pants. For each shirt you choose, there are $n_{2}$ possible pants to choose from. Therefore, the total number of choices is their product: 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_1 n_2}

Addition Rule

When choices are independently selected. For example, when choosing a representative for a high school: there are 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_1} freshmen, 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_2} sophomores, 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_3} juniors, 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_4} seniors. There are $\sum _{i=1}^{4}n_{i}$ ways to choose a representative among all high school students.

Inclusion-Exclusion

Just like in statistics:

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 \left| A \cup B \right| = \left| A \right| + \left| B \right| - \left| A \cap B \right|}

For example: How many bit strings of length 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} either begin with 1 or end with 00?

• There are 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 2^{n-1}} bit strings that begin with 1
• There are 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 2^{n-2}} bit strings that end with 00
• There are $2^{n-3}$ bit strings thta begin with 1 and end with 00

Therefore, there are 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 2^{n-1} + 2^{n-2} - 2^{n-3}} bit strings that begin with 1 or end with 00.

Division Rule

How many 3-item subsets can be made from a set 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 n} elements?

There are $n$ ways to choose the first element, 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-1} ways to choose the second element, 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-2} ways to choose the last element. Since the three elements can be rearranged in 3! = 6 ways, and inclusion in a set does not rely on the order, we divide the total number of ways to choose the set by 6:

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 \frac{n(n-1)(n-2)}{3}= \binom{n}{3}}

Counting Functions

How many functions are there from {a, b, c, d} to {1, 2, 3}?

For each element in the domain (a, b, c, and d), there are 3 targets to choose from in the codomain. Therefore, the number of possible functions is 3 · 3 · 3 · 3 = 3⁴

In general, for a domain containing 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} items and a codomain containing $c$ items, the number of possible functions from the domain to the codomain is 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^d}

One-to-One Functions

For a one-to-one function, once an element in the codomain has been chosen, it cannot be pointed to by any element in the domain. Therefore, the number of possible one-to-one functions is <math>c(c-1)(c-2)\dots(c-d+1) = \frac{c!}{(c-d)!}