56 lines
2.1 KiB
Markdown
56 lines
2.1 KiB
Markdown
## Asymptotic Equivalence Classes (Big-O)
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The equivalence relation definition given in the task is asking you to find functions that grow at the exact same rate (also known as Big-Theta $\Theta$):
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$f asymp g <==> f in O(g) and g in O(f)$
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Notation of $f in O(g)$ means "function $f$ doesn't grow faster than $g$"
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### The "Dominant Term" Rule
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To find which class a function belongs to, simplify it to its core growth rate:
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1. **Drop all lower-order terms:** In $n^3 + n^2$, drop the $n^2$.
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2. **Drop all constant multipliers:** $5n^2$ and $1000n^2$ both become just $n^2$.
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3. **Identify the highest rank:** Factorials ($n!$) > Exponentials ($2^n$, $e^n$) > Polynomials ($n^3$, $n^2$) > Linear ($n$) > Logarithmic ($log n$) > Constant ($1$).
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## Euclidian Algorithm
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Purpose is to find the **GCD** (Greatest Common Divisor).
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### Core Rule
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Fill out this formula:
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$$"Dividend" = ("Quotient" * "Divisor") + "Remainder"$$
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1. **Divide** the bigger number by the smaller one
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2. **How many times** does it fit -> $"Quotient"$
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3. Find out **whats leftover** -> $"Remainder"$
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4. **Shift to left** and repeat ($"Old Divisor" -> "Dividend"$, $"Remainder" -> "Divisor"$)
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Result is the last $"Remainder"$ that is not $0$.
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### Bezout Coefficients
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**Goal:** find a $x$ and $y$ so that $"Divident" * x + "Divisor" * y = gcd("Dividend", "Divisor")$
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1. **Rewrite Euclid (above) equations** to solve for remainder ($"Remainder" = "Old Remainder" - "Dividend" * "Divisor"$)
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2. **Substitute remainders**
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3. **Simplify** (Final Form: $"GCD" = x * "Dividend" + "y * Divisor"$)
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### Extended Euclidian Algorithm
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Only valid for GCD = 1
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1. **Find GCD** using Euclidian Algorithm
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1. $A = q_1 * B + r_1$
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2. $B = q_2 * r_1 + r_2$
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3. $r_1 = q_3 * r_2 + 1$
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2. **Unwrap**
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1. $1 = r_1 - (q_3 * r_2)$ mit $r_2 = B - q_2 * r_1$
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2. $1 = r_1 - (q_3 * (B - q_2 * r_1))$ mit $q_3$ ausmultiplizieren
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3. $1 = (1 + q_3 * q_2) * r_1 - q_3 * B$ mit $r_1 = A - q_1 * B$
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4. $1 = (1 + q_3 * q_2) * (A - q_1 * B) - q_3 * B$
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3. **Simplify to** $x * A + y * B$
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4. Coefficient of $A$ is the modular inverse
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## Totient function
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Denoted by $phi(n) = n (1 - 1/p_1)(1 - 1/p_2)...(1-1/p_k)$ where $p_i$ is a prime factor of $n$.
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