# 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:**

**Comparison trees:**

Comparison trees are the computational model useful for determining lower bounds for sorting and searching problems.**Oracles and adversary arguments:**

One of the techniques that are important for obtaining lower bounds consists of making the use of an oracle**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.**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.

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