Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Complexity & Induction: Algorithms & Proofs

Description

This lecture covers the concepts of worst-case complexity, complexity classes, mathematical induction, and brute-force algorithms. It delves into topics like linear search, binary search, bubble sort, insertion sort, and the classes P and NP.

Official source

Ontological neighbourhood

Theoretical computer science

Algorithms and data structures: Analysis of algorithms

Theory of computation: Computational complexity theory

Related concepts (32)

Sorting algorithm

In computer science, a sorting algorithm is an algorithm that puts elements of a list into an order. The most frequently used orders are numerical order and lexicographical order, and either ascending or descending. Efficient sorting is important for optimizing the efficiency of other algorithms (such as search and merge algorithms) that require input data to be in sorted lists. Sorting is also often useful for canonicalizing data and for producing human-readable output.

Bubble sort

Bubble sort, sometimes referred to as sinking sort, is a simple sorting algorithm that repeatedly steps through the input list element by element, comparing the current element with the one after it, swapping their values if needed. These passes through the list are repeated until no swaps had to be performed during a pass, meaning that the list has become fully sorted. The algorithm, which is a comparison sort, is named for the way the larger elements "bubble" up to the top of the list.

Insertion sort

Insertion sort is a simple sorting algorithm that builds the final sorted array (or list) one item at a time by comparisons. It is much less efficient on large lists than more advanced algorithms such as quicksort, heapsort, or merge sort. However, insertion sort provides several advantages: Simple implementation: Jon Bentley shows a three-line C/C++ version that is five lines when optimized. Efficient for (quite) small data sets, much like other quadratic (i.e.

Mathematical proof

A mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation".

Mathematical induction

Mathematical induction is a method for proving that a statement is true for every natural number , that is, that the infinitely many cases all hold. Informal metaphors help to explain this technique, such as falling dominoes or climbing a ladder: Mathematical induction proves that we can climb as high as we like on a ladder, by proving that we can climb onto the bottom rung (the basis) and that from each rung we can climb up to the next one (the step). A proof by induction consists of two cases.

Related lectures (27)

Complexity & Induction: Algorithms & Proofs

Covers worst-case complexity, algorithms, and proofs including mathematical induction and recursion.

Algorithm Complexity Analyses

Covers the complexity analyses of algorithms and their worst-case time complexities.

Complexity Analyses: Linear and Binary Search, Sorting Algorithms

Covers the complexity analyses of search and sorting algorithms.

Complexity of Algorithms: Quiz + Answers

Covers the time complexity of algorithms and includes a quiz.

Dynamic Programming: Palindromic Subsequences