Are There Any Problems That Cannot Be Solved in Linear Time?

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In summary, linear-time-hard problems are a class of computational problems that require a time complexity of at least linear time. They are distinct from other complexity classes because they have a lower bound on the time complexity and cannot be solved any faster. Examples of linear-time-hard problems include the shortest path problem, travelling salesman problem, and longest common subsequence problem. These problems are challenging to solve because they require efficient algorithms in terms of both time and space complexity. Linear-time-hard problems have many real-world applications in various fields such as computer science, operations research, and engineering, where they are used to solve optimization, scheduling, and routing problems.
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Dragonfall
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Can I have a list of problems suspected of not having linear-time solutions? Like multiplication and sorting.
 
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  • #3
I meant a poly-time problem.
 
  • #4
Fast Fourier transform (2d or 3d).
Most trigonometric functions to any specified precision.
Finding the first "n" primes.
Defragmenting a disk with "n" fragments.

Technically, counting to "n" because eventually you would need longer and longer word-lengths.
 
  • #5


I am unable to provide a definitive list of problems that are suspected of not having linear-time solutions. The classification of a problem as "linear-time-hard" is a complex and ongoing topic of research in computer science and mathematics.

However, I can provide some examples of problems that are commonly believed to not have linear-time solutions, such as the traveling salesman problem, the knapsack problem, and the longest common subsequence problem. These problems have been extensively studied and have been shown to require exponential or super-polynomial time to solve.

It is important to note that the classification of a problem as linear-time-hard does not necessarily mean that it is impossible to find a linear-time solution. It simply means that, based on our current understanding and algorithms, a linear-time solution has not yet been found.

In the field of computer science, there is ongoing research and development in finding efficient algorithms for solving difficult problems. As technology and techniques advance, it is possible that some problems currently believed to be linear-time-hard may eventually have linear-time solutions. However, it is also likely that new, even more difficult problems will be discovered. This is the nature of scientific inquiry and progress.
 

FAQ: Are There Any Problems That Cannot Be Solved in Linear Time?

What are linear-time-hard problems?

Linear-time-hard problems refer to a class of computational problems that require a time complexity of at least linear time, meaning the running time of the algorithm must be directly proportional to the size of the input.

How do linear-time-hard problems differ from other complexity classes?

Linear-time-hard problems are distinct from other complexity classes, such as polynomial-time problems, because they require a lower bound on the time complexity of the algorithm, meaning they cannot be solved any faster.

What are some examples of linear-time-hard problems?

Some examples of linear-time-hard problems include the shortest path problem, the travelling salesman problem, and the longest common subsequence problem.

What makes linear-time-hard problems challenging to solve?

Linear-time-hard problems are challenging to solve because they require algorithms that are not only efficient in terms of time complexity, but also in terms of space complexity. In other words, the algorithm must not only run in linear time, but also use linear space.

How are linear-time-hard problems relevant to real-world applications?

Linear-time-hard problems have many real-world applications, particularly in fields such as computer science, operations research, and engineering. They are used to solve optimization problems, scheduling problems, and routing problems, among others.

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