Lower Bound Theory

LOWER BOUND THEORY 

If two algorithm for solving the problem where discovered and their times differed by an order of magnitude, the one with the smaller order was generally regarded as superior. THE PURPOSE OF LOWER BOUND THEORY  Is to find some techniques that have been used to establish that a given alg is the most efficient possible. THE SOLUTION OR TECHNIQUE It is by discovering discovering a function g(n) that is lower bound on the time that any algorithm must take to solve the given problem. If we have an algorithm whose computing time is the same order as g(n) then we know that asymptotically asymptotically we can do no better (i.e. this alg is the most efficient one, no other alg can be derived more efficient then this) TRIVAL LOWER BOUNDS Lower bounds which are so easy to obtain is called trivial lower bounds. E.g All alg that finds the maximum of are unordered set of n integers. Clearly every integer must be examined at least once. So (n) is a lower bound for any alg that solves this problem. SOME TECHNIQUES OR COMPUTATIONAL MODEL FOR FINDING LOWER BOUNDS.

1. Comparison trees. 2. Oracle and adversary argument 3. Lower bound through reductions. 1. Comparison Trees a) Ordered Searching

• Only comparison – based algorithms are considered • The process of searching algorithms can be described by a path in a binary tree. • Each internal node in this tree represent a comparision between rc and A[i] • There are three possible out come of this comparison. X < a[I]| x = a[I]|x > a[I] • If x = a[i] the lag terminates. terminates. • If the alg terminates following a left or right branch, then a[I] has been found s.t x = a[I] & alg must declare the search unsuccessful. Theorem 10.1

Let a[1:n], n> 1 contains n distinct elements, elements, ordered that a[1] < . . . < A[n]. Let Find (n) be the minimum number of comparisons needed, in the worst case, by any