Understanding Algorithmic Complexity

Algorithmic Complexity

• Algorithmic complexity measures the time (time complexity) and memory (space complexity) an algorithm needs for a problem of size $n$.
• Big O notation is used, ignoring constant factors and lower order terms.
• Time complexity refers to the time an algorithm takes as a function of input size.
• Space complexity refers to the amount of memory space required as a function of input size.
• Algorithmic complexity helps estimate performance as input size grows.
• It enables comparison of different algorithms to choose the most efficient one.
• It aids in identifying and optimizing bottlenecks in code.

Common Complexities

• O(1) is constant time, such as accessing an array element.
• O(log n) is logarithmic time, like binary search.
• O(n) is linear time, like searching for an element in an array.
• O(n log n) is log-linear time, demonstrated by merge sort and quicksort.
• O($n^2$) is quadratic time, such as bubble sort and insertion sort.
• O($n^3$) is cubic time, as seen in matrix multiplication.
• O($2^n$) is exponential time, demonstrated by the Tower of Hanoi problem.
• O(n!) is factorial time, such as generating all permutations of an array.

Determining Complexity

• First, identify the input size.
• Next, count the basic operations, like comparisons, assignments, and arithmetic operations.
• Then, express operations as a function of $n$, where $n$ is the input size.
• Simplify using Big O notation by dropping constants and lower order terms.

Example: sum_array function

• The input size is $n$, the number of elements in the array.
• The basic operation sum += arr[i] is executed $n$ times.
• The number of operations is proportional to $n$.
• The time complexity is O(n).

Space Complexity

• For example, the space complexity of creating an array of size n is O(n).
• Algorithmic complexity provides a useful way to compare algorithm efficiency.
• It helps predict performance as input size increases.
• It's a theoretical measure, not accounting for hardware, language, or compiler optimizations.

Graphics Intro

• "Graphics" refers to computer-generated images other than text.
• Vector and raster graphics are the two main categories.
• Vector graphics are mathematical equations describing lines, curves, and shapes while raster graphics are grids of pixels with specific colors.
• Vector graphics are resolution-independent and generally for logos and illustrations while raster graphics are resolution-dependent and for photographs.

Coordinate Systems

• The Cartesian coordinate system is a 2D system with perpendicular x and y axes.
• The origin is where the axes intersect and any point is identified by its x and y coordinates.
• The screen coordinate system specifies pixel locations on a screen, with the origin at the top-left corner.
• The x-axis extends horizontally and the y-axis extends vertically; pixel coordinates are integers from the origin.

Drawing Shapes

• A line connects two points and is drawn by specifying endpoint coordinates.
• A rectangle has four sides with four right angles, specified by the top-left corner, width, and height.
• A circle's points are equidistant from a center.
• It's drawn by specifying the center coordinates and radius; its equation is $(x-h)^2 + (y-k)^2 = r^2$.
• An ellipse's points have constant sum of distances from two foci.
• It can be drawn with the center coordinates, major axis, and minor axis using equation $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$.
• A polygon is a closed shape of straight line segments, drawn by specifying vertex coordinates.

Color

• The RGB color model adds red, green, and blue light to reproduce colors
• The CMYK color model is used in color printing, utilizing cyan, magenta, yellow, and black inks.

Image File Formats

• JPEG (Joint Photographic Experts Group) is a lossy compression method for digital images.
• The degree of compression is adjustable enabling trade-offs between storage and quality.
• PNG (Portable Network Graphics) is a raster graphics format with lossless data compression.
• PNG is a non-patented replacement for GIF.
• GIF (Graphics Interchange Format) is a bitmap image, and supports static and animated images.
• TIFF (Tagged Image File Format) is a raster graphics format popular for graphic art, publishing, and photography.
• TIFF is widely supported by image-manipulation apps and other applications.

Graphics Summary

• Basic concepts of computer graphics include vector graphics, raster graphics, coordinate systems, drawing shapes, color models, and image file formats.

Bayes' Theorem Definition

• Describes the probability of an event based on prior knowledge of conditions related to the event.
• Expressed as $P(A|B) = \frac{P(B|A)P(A)}{P(B)}$.

Terms in Bayes' Theorem

• $P(A|B)$ is the conditional probability of A given that B is true.
• $P(B|A)$ is the conditional probability of B given that A is true.
• $P(A)$ and $P(B)$ are the probabilities of A and B occurring independently.

Component Probabilities

• $P(A)$ is the prior probability of A, before considering evidence B.
• $P(A|B)$ is the posterior probability of A given B.
• $P(B|A)$ is the likelihood of B given A.
• $P(B)$ is the prior probability of B, serving as a normalization factor.

Bayes' Theorem Derivation

• Derived from definitions of conditional probability: $P(A|B) = \frac{P(A \cap B)}{P(B)}$ and $P(B|A) = \frac{P(B \cap A)}{P(A)}$.
• Solving for $P(A \cap B)$, yields $P(A \cap B) = P(A|B)P(B) = P(B|A)P(A)$.
• Bayes' Theorem becomes: $P(A|B) = \frac{P(B|A)P(A)}{P(B)}$.

Disease Detection Test Example

• A certain disease detection test has 99% accuracy.
• If a person has the disease, the test is positive 99% of the time.
• If a person does not have the disease, the test is negative 99% of the time.
• 0.1% of the population has the disease.
• Problem to solve: If a randomly chosen person tests positive, what is the probability they have the disease?

Solution - Define Events and Probabilities

• Define A as the event that a person has the disease.
• Define B as the event that a person tests positive.
• $P(A) = 0.001$: probability of having the disease.
• $P(B|A) = 0.99$: probability of testing positive given having the disease.
• $P(B|¬A) = 0.01$: probability of testing positive given not having the disease.

Calculating $P(B)$

• $P(B) = P(B|A)P(A) + P(B|¬A)P(¬A)$.
• $P(B) = (0.99)(0.001) + (0.01)(0.999) = 0.00099 + 0.00999 = 0.01098$.

Bayes' Theorem Application

• Solving for $P(A|B) = \frac{P(B|A)P(A)}{P(B)} = \frac{(0.99)(0.001)}{0.01098} \approx 0.08998$.
• Therefore, even with a positive test, the probability of having the disease is about 9%.

Chemical Kinetics: Reaction Rate

• Reaction rate is the change in concentration of a reactant or product with respect to time.
• The rate law relates the rate to reactant concentrations: $Rate = k[A]^m[B]^n$.
• k is the rate constant, m is the order with respect to A, n is the order with respect to B, and m + n is the overall order.
• Order of reaction can be zero (Rate = k), first (Rate = k[A]), or second (Rate = k[A]^2 or Rate = k[A][B]).

Factors Affecting Reaction Rate

• Temperature: Rate generally increases with temperature.
• Catalyst: Speeds up reaction without being consumed.
• Concentration: Increased concentration generally increases rate.
• Surface Area: Increased surface area (for solids) increases rate.

Integrated Rate Laws

• In first-order reactions, $\ln[A]_t - \ln[A]_0 = -kt$.
  • $[A]_t$ is concentration at time t and $[A]_0$ is initial concentration.
• Half-life ($t_{1/2}$) is the time for concentration to halve with $t_{1/2} = \frac{0.693}{k}$ for first-order.

Activation Energy

• The Arrhenius equation relates rate constant to activation energy and temperature: $k = Ae^{-\frac{E_a}{RT}}$.
• $E_a$ is activation energy, R is the gas constant (8.314 J/mol·K), and T is temperature in Kelvin.
• Equation for graphical determination of $E_a$: $\ln(k) = -\frac{E_a}{R} \cdot \frac{1}{T} + \ln(A)$.
• By plotting $\ln(k)$ versus $\frac{1}{T}$, $E_a$ can be found from the slope, $Slope = -\frac{E_a}{R}$.

AWS CloudShell Definition

• AWS CloudShell is a browser-based, pre-authenticated, managed shell accessible directly from the AWS console.
• It includes the AWS Command Line Interface (AWS CLI) for managing AWS services through commands.
• The service is available at no additional charge.

Why Use AWS CloudShell?

• Facilitates development and management by enabling exploration of AWS services, command-line tasks, and script development through a browser.
• AWS CloudShell is pre-authenticated with console credentials.
• Offers persistent storage of 1 GB per AWS region for data preservation between sessions.
• AWS CloudShell is integrated directly into the AWS console.

Getting Started with AWS CloudShell

• To start, launch AWS CloudShell from the AWS console.
• AWS CloudShell opens within your browser.
• Verify the AWS CLI version with the command aws --version.
• List Amazon EC2 instances using aws ec2 describe-instances.

Persistent Storage in AWS CloudShell

• AWS CloudShell provides 1 GB of free persistent storage per AWS region.
• Persistent storage is private and tied to AWS login credentials.
• Implemented in the home directory ($HOME) on an Amazon EBS volume.

Physics Concepts: Work

• Work ($W$) done on an object by a constant force is the product of the force component along the displacement and displacement magnitude.
• Work equation: $W = F \cdot d \cdot \cos(\theta)$.
• Units: $F$ is the force magnitude, $d$ is the displacement, and $\theta$ is the angle between them.

Units of Work

• The SI unit of work is the Joule (J).
• $1 \text{ Joule} = 1 \text{ Newton} \cdot \text{meter}$ or $1 \text{ J} = 1 \text{ N} \cdot \text{m}$.

Work of a Variable Force

• If the force isn't constant, work is calculated through integrating force component along the displacement path.
• Formula: $W = \int_{x_i}^{x_f} F(x) , dx$ where $x_i$ is the initial and $x_f$ the final position, $F(x)$ is force as a function of position.

Work-Energy Theorem

• The total work is the change in kinetic energy: $W_{total} = \Delta KE = KE_f - KE_i = \frac{1}{2} m v_f^2 - \frac{1}{2} m v_i^2$.
• $KE$ is kinetic energy, $m$ is mass, $v_i$ is initial and $v_f$ is final velocity.

Potential Energy

• Potential energy relates to a system's configuration.
• Gravitational Potential Energy is represented as $U = mgh$.
  • $m$ is the mass.
  • $g$ is the gravitational acceleration.
  • $h$ is the height above the reference point.
• Elastic Potential Energy is represented as $U = \frac{1}{2}kx^2$
  • $k$ is the spring constant.
  • $x$ is the displacement from equilibrium.

Energy Conservation

• If only conservative forces act, is conserved mechanical energy is constant. That is kinetic energy $KE_i + U_i = KE_f + U_f = E_f$.
• Conservative forces are path-independent gravity and elastic force while non-conservative are path-dependent, eg friction.

Power Definition

• Power is the rate or change of work and/or energy transfer which is $P = \frac{W}{\Delta t} = F \cdot v$.
• In the above euqation, $P$ is power, $W$ work, $\Delta t$ time, $F$ is force, and $v$ is velocity.
• The unit for power is the Watt with $1 \text{ Watt} = 1 \text{ Joule/second}$ or $1 \text{ W} = 1 \text{ J/s}$.
• Also, $1 \text{ hp} = 746 \text{ W}$ with hp representing Horsepower.

Chemical Kinetics defined

• Chemical kinetics / reaction kinetics looks at how reaction conditions influence reaction speed via exploring reaction mechanisms, transition states and creating mathematical models.

Impacting Reaction Rate Factors

• Increasing reactants' concentrations increases reaction rate because of contact and for gas increasing pressure does.
• Increasing temp also increases rate as it increases movement and energy of molecules.

Reaction Area (Surface)

• Reactions are confined to where reactants are able to contact, smaller particles react faster since area is maximized.
• The solvent can affect reaction rate by changing viscosity and motion which in turn affects collison ability.
• Light can promote reactions which is photochemistry when used as studying.

Reaction Rate Generic Formula

• For: $aA + bB \rightarrow cC + dD$
• Apply: $rate = -\frac{1}{a}\frac{\Delta[A]}{\Delta t} = -\frac{1}{b}\frac{\Delta[B]}{\Delta t} = \frac{1}{c}\frac{\Delta[C]}{\Delta t} = \frac{1}{d}\frac{\Delta[D]}{\Delta t}$
• Apply example: $2HI(g) \rightarrow H_2(g) + I_2(g)$ becomes $rate = -\frac{1}{2}\frac{\Delta[HI]}{\Delta t} = \frac{\Delta[H_2]}{\Delta t} = \frac{\Delta[I_2]}{\Delta t}$

Applying Rate Laws

• Using $aA + bB \rightarrow cC + dD$ and $rate = k[A]^m[B]^n$:
• Where the rate law MUST be determined empirically.

Rate Constant

• k is the rate constant
• m is the order with respect to the reaction A
• n is th order with respect to reaction B