Algorithms Lecture: Introduction and Basic Concepts
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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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Questions and Answers

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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