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Lower Bound Theory

In this article, we will learn about the concept of Lower Bound Theory and the proofing techniques that are useful for obtaining lower bounds.
Submitted by Shivangi Jain, on July 25, 2018

Lower Bound Theory

Lower bound (L(n)) is a property of the specific problem i.e. the sorting problem, matrix multiplication not of any particular algorithm solving that problem.

Lower bound theory says that no algorithm can do the job in fewer than that of (L (n)) times the units for arbitrary inputs i.e. that for every comparison based sorting algorithm must take at least L(n) time in the worst case.

L(n) is the minimum over all possible algorithm which is maximum complete.

Trivial lower bounds are used to yield the bound best option is to count the number of item in the problems input that must be processed and a number of output items that need to be produced.

The lower bound theory is the technique that has been used to establish the given algorithm in the most efficient way which is possible. This is done by discovering a function g(n) that is a lower bound on the time that any algorithm must take to solve the given problem. Now if we have an algorithm whose computing time is the same order as g(n), then we know that asymptotically we cannot do better.

If f(n) is the time for some algorithm, then we write f(n) = Ω(g(n)) to mean that g(n) is the lower bound of f(n). This equation can be formally written, if there exists positive constants c and n0 such that |f(n)| >= c|g(n)| for all n > n0. In addition for developing lower bounds within the constant factor, we are more conscious of the fact to determine more exact bounds whenever this is possible.

Deriving good lower bounds is more difficult than devising efficient algorithms. This happens because a lower bound states a fact about all possible algorithms for solving a problem. Generally, we cannot enumerate and analyze all these algorithms, so lower bound proofs are often hard to obtain.

The proofing techniques that are useful for obtaining lower bounds are:

1. Comparison trees:
   Comparison trees are the computational model useful for determining lower bounds for sorting and searching problems.
2. Oracles and adversary arguments:
   One of the techniques that are important for obtaining lower bounds consists of making the use of an oracle
3. Lower bounds through reduction:
   This is a very important technique of lower bound, This technique calls for reducing the given problem for which a lower bound is already known.
4. Techniques for the algebraic problem:
   Substitution and linear independence are two methods used for deriving lower bounds on algebraic and arithmetic problems. The algebraic problems are operation on integers, polynomials, and rational functions.