Data Types, Structures, and Algorithms

Data Types

• Integers are whole numbers.
• Floats are numbers with decimal points.
• Strings are textual data.
• Booleans represent True or False values.
• Lists are ordered and mutable collections of items.
• Tuples are ordered and immutable collections of items.
• Sets are unordered collections of unique items.
• Dictionaries are collections of key-value pairs.

Data Structures

• Arrays consist of homogeneous data and are efficient for numerical operations.
• Linked Lists are sequences of nodes, each containing data and a link to the next node.
• Trees are hierarchical data structures with a root and child nodes.
• Graphs consist of nodes connected by edges, representing relationships.
• Hash Tables implement an associative array data type for mapping keys to values.

Algorithms

• Sorting algorithms arrange items in a specific order.
• Searching algorithms find specific items in a dataset.
• Dynamic Programming is an optimization technique for problems with overlapping subproblems.
• Greedy Algorithms make locally optimal choices at each step to find a global optimum.
• Graph Algorithms traverse and analyze graphs.

Libraries

• NumPy is a Python library focusing on numerical computing with arrays, linear algebra, and random number generation.
• Pandas is a Python library used for data manipulation and analysis through DataFrames, data cleaning, and data aggregation.
• Scikit-learn provides machine learning algorithms, including classification, regression, clustering, and model selection.
• Matplotlib is used for plotting and visualization, including line plots, scatter plots, histograms, and bar charts.
• Seaborn enhances statistical data visualization, offering improved plots, color palettes, and plot themes.

Machine Learning

• Supervised Learning involves learning from labeled data for Regression and Classification tasks.
• Linear Regression predicts continuous values using the equation y = mx + b.
• Polynomial Regression uses the equation y = a₀ + a₁x + a₂x² +...+ aₙxⁿ to predict continuous values.
• Logistic Regression models the probability of a binary outcome.
• Support Vector Machines (SVM) aims to maximize margin $\frac{2}{|w|}$ subject to $y_i(w^Tx_i - b) \geq 1$, for all i.
• Common Supervised Learning Algorithms include Decision Trees and Random Forests
• Unsupervised Learning occurs with unlabeled data through Clustering and Dimensionality Reduction.
• K-Means Clustering minimizes the sum of squared distances within each cluster.
• Principal Component Analysis (PCA) maximizes variance $var(Xw) = w^TSw$ subject to $|w| = 1$.
• Reinforcement Learning involves learning through interactions with an environment using algorithms like Q-Learning.
• Q-Learning updates Q-values using the equation $Q(s, a) \leftarrow Q(s, a) + \alpha[R(s, a) + \gamma \max_{a'} Q(s', a') - Q(s, a)]$.

Statistics

• Measures of Central Tendency include Mean, Median and Mode.
• The mean provides the average of a dataset: $\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i$
• The median is the middle value of a sorted dataset.
• The mode indicates the most frequent value in a dataset.
• Measures of Dispersion include Variance, Standard Deviation and Range.
• Variance measures the spread of data points around the mean: $\sigma^2 = \frac{1}{n} \sum_{i=1}^{n} (x_i - \mu)^2$
• Standard Deviation is the square root of the variance.
• Range calculates the difference between the maximum and minimum values.
• Probability Distributions encompass Normal, Binomial, and Poisson Distributions.
• Normal Distribution: Is determined by the formula $f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2}$
• Binomial Distribution is determined by the formula $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$
• Poisson Distribution is determined by the formula $P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}$
• Hypothesis Testing uses T-tests, Chi-Square tests and ANOVA.
• Confidence Intervals can be calculated by the following formula: $\bar{x} \pm z * \frac{\sigma}{\sqrt{n}}$

Calculus

• Derivatives measure the rate of change of a function.
• Power Rule: $\frac{d}{dx} x^n = nx^{n-1}$
• Chain Rule: $\frac{d}{dx} f(g(x)) = f'(g(x)) * g'(x)$
• Integrals represent the area under a curve.
• Fundamental Theorem of Calculus is defined with the formula $\int_a^b f(x) dx = F(b) - F(a)$

Linear Algebra

• Vectors have magnitude and direction.
• Matrices are rectangular arrays of numbers.
• Eigenvalues and Eigenvectors satisfy the equation Av = λv.
• Matrix Operations include addition, multiplication, transpose, and inverse.

Big O Notation

• O(1) represents constant time complexity.
• O(log n) represents logarithmic time complexity.
• O(n) represents linear time complexity.
• O(n log n) represents linearithmic time complexity.
• O(n^2) represents quadratic time complexity.
• O(2^n) represents exponential time complexity.
• O(n!) represents factorial time complexity.

SQL

• SELECT column1, column2 FROM table_name WHERE condition; - Selects columns from a table based on a condition.
• INSERT INTO table_name (column1, column2) VALUES (value1, value2); - Inserts new rows into a table.
• UPDATE table_name SET column1 = value1 WHERE condition; - Modifies existing data in a table.
• DELETE FROM table_name WHERE condition; - Removes rows from a table.
• JOIN: Combine rows from two or more tables based on a related column.
• GROUP BY: Group rows that have the same values into summary rows.

Cloud Computing

• AWS stands for Amazon Web Services.
• Azure represents Microsoft Azure.
• GCP refers to Google Cloud Platform.

Ethics

• Bias emerges as prejudice in data or algorithms.
• Privacy refers to protecting sensitive information.
• Fairness involves ensuring equitable outcomes
• Transparency focuses on understanding how algorithms make decisions.
• Accountability focuses on taking responsibility for the impact of data science.

Convex Functions

• A function $f$ is convex on interval $I$ if for all $x, y \in I, \forall t \in[0,1]$, $f(t x+(1-t) y) \leq t f(x)+(1-t) f(y)$.
• If $-f$ is convex, then $f$ is concave.
• Geometrically, the chord connecting $(x, f(x))$ and $(y, f(y))$ is above the graph of $f$ on $[x, y]$.
• A function such as $x \longmapsto x^{2}$ is convex on $\mathbb{R}$.
• If $f$ is convex on $I$, then $\forall x_{1}, \ldots, x_{n} \in I, \forall t_{1}, \ldots, t_{n} \in[0,1]$ such that $\sum_{i=1}^{n} t_{i}=1$, $f\left(\sum_{i=1}^{n} t_{i} x_{i}\right) \leq \sum_{i=1}^{n} t_{i} f\left(x_{i}\right)$.
• A differentiable function $f$ on an interval $I$ is convex if and only if $f^{\prime}$ is increasing on $I$.
• For a convex function, tangents are never above the curve.
• A twice-differentiable function $f$ on an interval $I$ is convex if and only if $f^{\prime \prime} \geq 0$ on $I$.
• Exponential function is convex on $\mathbb{R}$ as its second derivative is positive on $\mathbb{R}$.

Jensen's Inequality

• Let $I$ be an interval of $\mathbb{R}$, $f: I \rightarrow \mathbb{R}$ be a convex function, and $X$ be a real random variable such that $X(\Omega) \subset I$ and $\mathbb{E}(|X|)

Hans Hoogervorst and Measurement Approaches

• In a speech at the IFRS Conference in Paris, June 29, 2015, Hans Hoogervorst, addressed debates between historical cost and fair value.
• Historical cost and fair value exist on opposite ends of the measurement spectrum.
• Fair value requires a full update of all inputs, while historical cost updates partially and less frequently.
• Historical cost is favored for perceived objectivity and stability, carries elements of subjectivity, particularly when assessing impairments.
• Historical cost's stability may be misleading.
• Fair value is criticized for introducing volatility, but supporters argue it reflects economic reality + provides a more accurate picture of an entity's financial position.
• Fair value measurements can become subjective if not available, forcing reliance on model-based valuations, which can introduce uncertainties.
• Historical cost begins and ends with values close to fair value.
• Subjective impairment assessments can mean that historical cost is just as open to manipulation fair value accounting.
• IASB has not designated fair value as the default measurement basis.
• Current market value of assets used to produce goods for sale that are not going to be sold may not be crucial information

Policy Recommendations

• To improve financial reporting measurement, reports say standard-setters should adopt an evidence-based approach.
• Cost-effectiveness, fitness for purpose, and institutional context should guide decisions on measurements.
• Public policy considerations should be included in the decision-making process.

Common Measurement Issues

• Fundamental issues affect the reliability and relevance of measurement bases.
• Prediction and Estimation: rely on subjective predictions, prone to error
• Joint Asset Use: It is often difficult to measure individual assets that generate value jointly with other assets .Individual asset valuation inherently complex
• Market availability issues reduce fair assets and liabilities which reduces their objectivity
• Double-Entry Bookkeeping Impact: Measurement of one item in the accounts affects the measurement of another. This interconnectedness can sometimes lead to less sensible measurements

Alternative Measurement Approaches

• Value to the Business Basis: Measures how much worse off a business would be if it were deprived of a particular asset. Typically, this value is the asset's replacement cost.
• Fair Value - defined as the price at which an asset could be sold or a liability settled in an orderly transaction between market participants..
• Realisable Value - represents the amount that can be obtained from selling an asset, net of selling costs. This basis is especially relevant for assets held for sale
• Value in Use - the present value of future cash flows expected to be generated from an asset. Used for assessing the long-term performance and contribution of an asset to the business.

Mixed Measurement

• Mixed Measurement Approach: report advocates for using different measurement bases that are more appropriate for different assets and liabilities. Focus should be on cost-effective basis and fit for purpose

Financial Statements

• assets/liabilities and entities, may compromise the quality and usefulness of information.
• Users of Financial Statements consist of investors and creditors
• Existing and potential investors: Interested in risk and return on investment, and decisions to buy, sell or hold their shares. Impact on measurement: look forward with a perspective on current values
• creditors, whether amounts owing will be paid when due. Focus upon net position, impact on measurement: current value of assets and liabilities, to indicate future viability of the entity

Decision Criteria

• No single factor determines which measurement basis must be selected.
• The information must be relevant and truthful for users of financial statements
• The information must be as comparable, verifiable, timely and understandable
• Cost constraints take into consideration how useful financial statements are likely to justify the costs of providing and using that information

Historical Cost

• Assets recorded at the paid time of acquisition and liabilities as amount received or expected to be paid.
• Basis is widely regarded as reliable when based on actual events but lacks unrealistic gains and relevance for current decision making.
• Despite limitation, historical cost is useful to lenders and creditors aligns w/ preference for prudence in financial reporting

Aspects of a Mixed Measurement System

• Comparability: Amounts are discounted to the present day, reflecting the same point in time and currency but estimates and/or discretion is required
• Understandability: There is complex estimations and formulas.

Plant Property and Equipment (IAS 16)

• Initial measurement is cost
• Subsequent measurement: Cost Model using the cost less accumulated depreciation
• Revaluation Model = fair value at date of revaluation, less accumulated impairment losses

Intangible Assets (IAS 38)

• If acquired separately: record at purchase price + directly attributable costs
• If acquired as part of a business, record at fair value at acquisition date
• If internally generated: record expenditure to date as intangible meets eligibility criteria
• Cost model = cost less accumulated amortisation, less accumulated impairment losses. Revaluation model = fair value at revaluation time, less accumulated amortisation, less accumulated impairment losses.

Present Value and Value in Use/ Fulfilment Value.

• Cash flows must be adjusted to reflect the current value.
• Estimate and identify appropriate discount rates and future cash flows which are related to the items.
• Assumptions by market participants are less relevant.
• Assets are carried at the present discounted value of the future net cash inflows
• Liabilities are carried at the present discounted value of the future net cash outflows

Current Cost

• Current cost is the cost of an equivalent asset at the measurement date, and plus transaction fees.
• Current cost of a liability is the consideration that would be received minus transaction fees.
• Must be able to reflect what would be paid.

Fair Value and Realisable Value

• are exit values
• fair valued: the amount an entity would receive for an item assuming recognition is aligned w/ the characteristic of the item.
• Realiable valued, amount in an orderly and open transaction.

Popularity

• IFRS 3 financial instrument, IFRS for leases, IAS 2.

Historical Cost

• Items are recorded at the amount they were purchased or the amount at which were received
• amounts are based on transactions that actually occurred in the past.
• the cost is for acquiring or creating the asset , plus transactions costs.
• liability value plus transaction costs.

Current Costs

• are entry values
• Value and record items at the amount that would be paid at the current time to
• Provide or replace the future economic benefits expected to be derived from the current item (an equivalent) equivalent to current cost.
• identical value equals replacement costs

Equity

• This is the residual interest in the assets of the entity after deducting all its liabilities

Collision-Resistant Hash Functions

• A collision-resistant hash function h is where it's computationally infeasible to find distinct inputs x and y for which h(x) = h(y).
• Implies the function must be preimage resistant.
• An n-bit hash function can be broken with $2^{n/2}$ effort using a "birthday attack". Output size n should be at least 256 bits.
• A hash function $h$ : ${0,1}^* \rightarrow {0,1}^n$ is collision-resistant if no probabilistic polynomial-time adversary can find a collision with non-negligible probability.
• An adversary A outputs x and y. An experiment's generated key k is used by A to calculate probability. Output is 1 if $x \ne y$ and $h_k(x) = h_k(y)$; else 0.
• The function is collision-resistant if for every probabilistic polynomial-time algorithm A, the probability of the experiment's output being 1 is negligible. This protects digital signatures by hashing contracts. Eve creates two contracts, and finds a collision - one fair, one not.
• Putatively collision-resistant functions include SHA-1, SHA-2, SHA-3, BLAKE2, and BLAKE3.

Chemical Kinetics

• Chemical kinetics, also known as reaction kinetics, is the study of reaction rates and reaction mechanisms.
• Factors Affecting Reaction Rates.
• Reactant Concentration's general rule is increase to increase rate .
• Reaction $aA + bB \longrightarrow cC + dD$ Rate Law: $rate = k[A]^m[B]^n$. Where k= rate constant
• Reaction $aA + bB \longrightarrow cC + dD$ Rate Law: $rate = k[A]^m[B]^n$. Overall Reaction Order $m + n$.
• A temperature increase usually increases reaction rates.
• Arrhenius Equation: $k = Ae^{-\frac{E_a}{RT}}$
• Catalysts are substances that increase reaction rates without being consumed.
• Increasing surface area of a solid reactant increases the reaction rate. by providing more contact points for the reaction
• Increase the pressure of gaseous reactants usually increases the reaction rate. by increasing the concentration of the reactants

Reaction Mechanisms

• Elementary Steps are a series of simple steps that represent the progress of a reaction at the molecular level. Rate Determining Step. The slowest step in the mechanism, which determines the overall reaction rate is the rate determining step
• Intermediates do not appear in the overall balanced equation
• Rate Laws. Expresses reaction rate to concentrations of reactants from rate expression. Experimentally determined
• Differential Rate Law expresses the rate of reaction as a function of change in concentration of reactants per unit time.
• Integrated zero, first and second order functions as $[A]_t$, $[A]_0$ and $k$.
• Half-Life: time required for concentration of a reactant to decrease to one-half initial value

Collision Theory

• Molecular collisions, activation energy and molecule orientation are basic steps to collision theory.
• Effective collisions and ineffective collision key concepts depend on insufficient energy or incorrect molecule orientation.

Energy Profiles

• Activation Energy ($E_a$): Energy difference between reactants and the transition state.
• Enthalpy Change ($\Delta H$): Energy difference between reactants and products.
• Endothermic reactions results in $\Delta H > 0$ and absorbs heat
• Diagram: A plot of energy versus reaction progress, showing the energy changes during the reaction includes the activation energy, transition state, and enthalpy change

Reaction Rate: Definition

The reaction rate indicates the speed at which reactants are converted into products in a chemical reaction, and can be expressed as the change in concentration of a reactant or product per unit time.

Rate Expression - For a reaction: aA + bB ⟶ cC + dD

Rate = $-\frac{1}{a}\frac{\Delta[A]}{\Delta t} = -\frac{1}{b}\frac{\Delta[B]}{\Delta t} = \frac{1}{c}\frac{\Delta[C]}{\Delta t} = \frac{1}{d}\frac{\Delta[D]}{\Delta t}$.

Rate Law

• Equation that links the reaction rate to the concentrations of reactants is experimentally determined.
• General Form : The rate law is: Rate = k[A]^m[B]^n, where: k is the rate constant
• [A] and [B] are the concentrations of reactants. m and n the reaction orders.

Integrated rate laws

• Relate the concentration of reactants to time with general equations.

zero law as [A]_t = -kt + [A]_0,

and first law has been expressed $ln[A]_t = -kt + ln[A]_0$

second law as 1/[A]_t = kt + [A]_0.

- Reaction Mechanisms

Step-by-step sequence of elementary reactions by which overall chemical change occurs.

Elementary Steps

• Catalysis and frequency are related.
• Activation energy is required for a reaction to occur. Homogeneous Catalysis.
• catalyst is in the same phase.
• Heterogeneous Catalysis.
• catalyst in different phase.

Pan Peng and Overview

• Pan Peng is the instructor.
• Office is in Cupples I, Room 209.
• Office Hours are Tuesday & Thursday 2:00 PM - 3:00 PM.
• Email: [email protected]
• Website: Canvas.

Course Description

• This course explores fundamental principles in algorithm design and analysis. Discussions will include advanced data structures, graph algorithms, approximation algorithms, randomized algorithms, and online algorithms.

Prerequisites

• CSE 247 (Data Structures and Algorithms).
• CSE 347 (Analysis of Algorithms) or equivalent.

Grading

Course Schedule (Tentative)

Definition of Vector Spaces

• A vector space over a field (\mathbb{K}) (often (\mathbb{R}) or (\mathbb{C})) is a set (E)

• Addition Axioms $u, v, w \in E$ must have Associativity of $(u + v) + w = u + (v + w)$, have Commutativity of $u + v = v + u$, have element of zero.

• Multiplication scalar has associativity as $(ab) \cdot u = a \cdot (b \cdot u)$

• Subspaces, linear space and independence. Vector, eigenvalue and euclidean related to vectors. and space

• Types of matrices

• Square Matrix:

A matrix with the same number of rows and columns.

• Identity Matrix: A square matrix with 1s on the main diagonal and 0s elsewhere, denoted as I
• Null Matrix: A matrix whose elements are 0s.
• Transpose Matrix:

The transpose of a matrix A, denoted as $A^T$, is obtained by swapping the rows and columns of $A$ Invert Matrix:

The inverse of a square matrix $A$, denoted as $A^{-1}$ , is a matrix such that $AA^{-1} = A^{-1}A = I$