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# Algorithms Lecture: Introduction and Basic Concepts

Learn about algorithms, computational problems, general algorithms, the concept of a function in mathematics, proving algorithm correctness, asymptotic analysis, word RAM as a model of computation, and strategies for algorithm problem solving. Also covers data structures, sorting, shortest paths, graphs, and dynamic programming.

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@PlayfulBiography

### What is the main focus of the Introduction to Algorithms course?

Solving computational problems

### How is a computational problem defined?

As a relation between inputs and outputs with a specified set of correct outputs

### What does the proposed algorithm by Jason Ku aim to find?

Pairs of students with the same birthday

### What does the proposed algorithm do if no match is found after interviewing all students?

<p>Returns that no pair was found</p> Signup and view all the answers

### How does the class define problems?

<p>Using predicates to allow for larger input spaces</p> Signup and view all the answers

### According to Jason Ku, do algorithms need to be pure functions in the mathematical sense?

<p>No, they do not need to be pure functions</p> Signup and view all the answers

### What is used to prove the correctness of an algorithm by setting up a base case and an inductive hypothesis?

<p>Induction</p> Signup and view all the answers

### What measure is used to analyze the efficiency of an algorithm by determining the number of fundamental operations needed to solve a problem?

<p>Asymptotic analysis</p> Signup and view all the answers

### What is used as a model of computation to analyze algorithms, assuming a CPU can change and operate on information in memory?

<p>Word RAM</p> Signup and view all the answers

### What are collections of 8 bits in modern computers called?

<p>Bytes</p> Signup and view all the answers

### Which part of the class focuses on data structures and sorting?

<p>First half</p> Signup and view all the answers

### What are the two strategies used to solve algorithm problems?

<p>Recursive algorithm design and using a known solution</p> Signup and view all the answers

### What do the first eight lectures of the class focus on?

<p>Data structures and sorting</p> Signup and view all the answers

### What does the second quiz cover?

<p>Shortest paths and graphs</p> Signup and view all the answers

### What is assumed as the model of computation for analyzing algorithms?

<p>CPU can perform binary operations</p> Signup and view all the answers

### What does asymptotic analysis measure in relation to an algorithm's performance?

<p>Number of fundamental operations needed to solve a problem</p> Signup and view all the answers

## Study Notes

• Jason Ku is teaching Introduction to Algorithms with two other faculty members, Eric Demaine and Justin Solomon.
• The course is about teaching students to solve computational problems, communicate solutions, and prove correctness and efficiency of these solutions.
• A computational problem is defined as a binary relation between inputs and outputs, with each input having a specified set of correct outputs.
• Problems are typically defined using predicates, allowing for larger input spaces.
• The class focuses on creating general algorithms that can handle arbitrarily sized inputs.
• An algorithm is a function that takes inputs and generates a single output, with the output being correct based on the problem.
• Jason Ku proposes an algorithm for finding pairs of students with the same birthday, involving interviewing students in order and checking the record for matches.
• The proposed algorithm maintains a record and interviews students, checking if their birthday is already in the record and returning a pair if a match is found.
• If no match is found after interviewing all students, the algorithm returns that no pair was found.
• Jason Ku mentions that algorithms do not need to be pure functions in a mathematical sense, but focus is on functional programming concepts.- Jason Ku explains the concept of a function in mathematics, using the example of an algorithm as a plan with inputs and outputs.
• Algorithms are sequences of procedures to produce an output from an input.
• The birthday problem is used as an example to illustrate the need to prove an algorithm's correctness.
• Induction is used to prove the correctness of an algorithm by setting up a base case and an inductive hypothesis.
• The inductive hypothesis states that if the first K students contain a match, the algorithm will return a match.
• The size of the input is a factor in the performance of an algorithm, and asymptotic analysis is used to measure efficiency.
• Asymptotic analysis measures the number of fundamental operations an algorithm needs to perform to solve a problem.
• Common functions used to relate an algorithm's input size to its performance include constant time, logarithmic time, quadratic time, and exponential time.
• A word RAM is used as a model of computation to analyze algorithms, which assumes a CPU can change and operate on information in memory and has instructions to randomly access different addresses.
• Modern computers are addressed in bytes, which are collections of 8 bits, and the size of the word a CPU can operate on at a time is called the word size.
• The CPU can perform binary operations, integer arithmetic, logical operations, and bitwise operations, and write to and read from memory in constant time.
• In the first half of the class, data structures will be discussed to optimize operations on a large amount of data.- The class is about solving algorithm problems.
• Two strategies are used to solve algorithm problems: either use a known solution or design a new recursive algorithm.
• Strategies may involve reducing to a solved problem, such as sorting, or searching in a graph.
• Recursive algorithm design paradigms are covered.
• First eight lectures focus on data structures and sorting.
• Second quiz covers shortest paths, algorithms, and graphs.
• Last quiz covers dynamic programming.
• Lectures occur weekly and are numbered.
• First lecture has ended.

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