# Computational complexity theory

In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and relating these classes to each other. A computational problem is a task solved by a computer. A computation problem is solvable by mechanical application of mathematical steps, such as an algorithm. A problem is regarded as inherently difficult if its solution requires significant resources, whatever the algorithm used. The theory formalizes this intuition, by introducing mathematical models of computation to study these problems and quantifying their computational complexity, i.e., the amount of resources needed to solve them, such as time and storage. Other measures of complexity are also used, such as the amount of communication (used in communication complexity), the number of gates in a circuit (used in circuit complexity) and the number of processors (used in parallel computing). One of the roles of computational complexity theory is to determine the practical limits on what computers can and cannot do. The P versus NP problem, one of the seven Millennium Prize Problems, is dedicated to the field of computational complexity. Closely related fields in theoretical computer science are analysis of algorithms and computability theory. A key distinction between analysis of algorithms and computational complexity theory is that the former is devoted to analyzing the amount of resources needed by a particular algorithm to solve a problem, whereas the latter asks a more general question about all possible algorithms that could be used to solve the same problem. More precisely, computational complexity theory tries to classify problems that can or cannot be solved with appropriately restricted resources. In turn, imposing restrictions on the available resources is what distinguishes computational complexity from computability theory: the latter theory asks what kinds of problems can, in principle, be solved algorithmically. (Wikipedia).

Algorithms Explained: Computational Complexity

An overview of computational complexity including the basics of big O notation and common time complexities with examples of each. Understanding computational complexity is vital to understanding algorithms and why certain constructions or implementations are better than others. Even if y

From playlist Algorithms Explained

Lower Bound on Complexity - Intro to Algorithms

This video is part of an online course, Intro to Algorithms. Check out the course here: https://www.udacity.com/course/cs215.

From playlist Introduction to Algorithms

11b Machine Learning: Computational Complexity

Short lecture on the concept of computational complexity.

From playlist Machine Learning

A Theory of Cryptographic Complexity - Manoj M. Prabhakaran

Manoj M. Prabhakaran University of Illinois at Urbana-Champaign March 1, 2010 In this talk, I shall describe an ongoing project to develop a complexity theory for cryptographic (multi-party computations. Different kinds of cryptographic computations involve different constraints on how in

From playlist Mathematics

Results and open problems in theory of quantum complexity - Anindya De

Andris Ambainis University of Latvia; Member, School of Mathematics April 22, 2014 I will survey recent results and open problems in several areas of quantum complexity theory, with emphasis on open problems which can be phrased in terms of classical complexity theory or mathematics but ha

From playlist Mathematics

Understanding quantum algorithms via query complexity – Andris Ambainis – ICM2018

Mathematical Aspects of Computer Science Invited Lecture 14.2 Understanding quantum algorithms via query complexity Andris Ambainis Abstract: Query complexity is a model of computation in which we have to compute a function f(x_1, …, x_N) of variables x_i which can be accessed via querie

From playlist Mathematical Aspects of Computer Science

The Computational Complexity of Geometric Topology Problems - Greg Kuperberg

Greg Kuperberg University of California, Davis September 24, 2012 This talk will be a partial survey of the first questions in the complexity theory of geometric topology problems. What is the complexity, or what are known complexity bounds, for distinguishing n-manifolds for various n? Fo

From playlist Mathematics

Depth complexity and communication games - Or Meir

Or Meir Institute for Advanced Study; Member, School of Mathematics September 30, 2013 For more videos, visit http://video.ias.edu

From playlist Mathematics

The Complexity of Distributions - Emanuele Viola

Emanuele Viola Northeastern University March 5, 2012 Complexity theory, with some notable exceptions, typically studies the complexity of computing a function h(x) of a *given* input x. We advocate the study of the complexity of generating -- or sampling -- the output distribution h(x) for

From playlist Mathematics

Ximena Fernández 7/20/22: Morse theory for group presentations and the persistent fundamental group

Discrete Morse theory is a combinatorial tool to simplify the structure of a given (regular) CW-complex up to homotopy equivalence, in terms of the critical cells of discrete Morse functions. In this talk, I will present a refinement of this theory that guarantees not only a homotopy equiv

From playlist AATRN 2022

Computational Complexity Classes, Homotopy Classes and N-machines

Examined herein is the possible correspondence between computational complexity classes in computational graphs and higher homotopy classes between computability paths via the application of two methods. The first method is the use of category theory for formalizing a model of (categorifie

From playlist Wolfram Technology Conference 2021

Topological Strings and String Dualities by Rajesh Gopakumar

J-Holomorphic Curves and Gromov-Witten Invariants DATE:25 December 2017 to 04 January 2018 VENUE:Madhava Lecture Hall, ICTS, Bangalore Holomorphic curves are a central object of study in complex algebraic geometry. Such curves are meaningful even when the target has an almost complex stru

From playlist J-Holomorphic Curves and Gromov-Witten Invariants

Claudia Landi (8/29/21): Discrete Morse Theory meets Multi-Parameter Persistence

Discrete Morse theory permits reducing a cell complex to the critical cells of a gradient vector field. Critical cells carry all the relevant homological information about the input data. Multiparameter persistence is a promising tool in topological data analysis of multivariate data that

Haldun Özgür Bayindir : Adjoining roots to ring spectra and algebraic 𝐾-theory

CONFERENCE Recording during the thematic meeting : « Chromatic Homotopy, K-Theory and Functors» the January 24, 2023 at the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Jean Petit Find this video and other talks given by worldwide mathematicians on CIR

From playlist Topology

Vijay Balasubramanian - Quantum Complexity, Dynamical Chaos...

Quantum Complexity, Dynamical Chaos, and the Interior Structure of Black Holes Vijay Balasubramanian (University of Pennsylvania)

Lenore Blum - Alan Turing and the other theory of computing and can a machine be conscious?

Abstract Most logicians and theoretical computer scientists are familiar with Alan Turing’s 1936 seminal paper setting the stage for the foundational (discrete) theory of computation. Most however remain unaware of Turing’s 1948 seminal paper which introduces the notion of condition, sett

From playlist Turing Lectures

The Abel lectures: László Lovász and Avi Wigderson

0:30 Introduction by the Abel Prize Committee Chair, Hans Munthe-Kaas 02:42 László Lovász: Continuous limits of finite structures 49:27 Questions and answers 1:00:31 Avi Wigderson: The Value of Errors in Proofs 1:41:24 Questions and answers 1:50:20 Final remarks by John Grue, Chair of the

From playlist Abel Lectures

Daniel Isaksen - 2/3 Motivic and Equivariant Stable Homotopy Groups

Notes: https://nextcloud.ihes.fr/index.php/s/EyZRRtDq965o6WC I will discuss a program for computing C2-equivariant, ℝ-motivic, ℂ-motivic, and classical stable homotopy groups, emphasizing the connections and relationships between the four homotopical contexts. The Adams spectral sequence

Yang-Lee Zeros of Integrable Field Theories by Giuseppe Mussardo

PROGRAM: INTEGRABLE SYSTEMS IN MATHEMATICS, CONDENSED MATTER AND STATISTICAL PHYSICS ORGANIZERS: Alexander Abanov, Rukmini Dey, Fabian Essler, Manas Kulkarni, Joel Moore, Vishal Vasan and Paul Wiegmann DATE : 16 July 2018 to 10 August 2018 VENUE: Ramanujan Lecture Hall, ICTS Bangalore

Complexity Theory, Quantified Boolean Formula

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From playlist [Shai Simonson]Theory of Computation