Algorithm

Getting Started with AlgorithmWhat is an Algorithm?

Characteristics of Algorithm1 Topic

Analysis Framework

Performance Analysis3 Topics

Mathematical Analysis2 Topics

Sorting AlgorithmSorting Algorithm10 Topics

Searching Algorithm6 Topics

Fundamental of Data StructuresStacks

Queues

Graphs

Trees

Sets

Dictionaries

Divide and ConquerGeneral Method

Binary Search

Recurrence Equation for Divide and Conquer

Finding the Maximum and Minimum

Merge Sort

Quick Sort

Stassen’s Matrix Multiplication

Advantages and Disadvantages of Divide and Conquer

Decrease and ConquerInsertion Sort

Topological Sort

Greedy MethodGeneral Method

Coin Change Problem

Knapsack Problem

Job Sequencing with Deadlines

Minimum Cost Spanning Trees2 Topics

Single Source Shortest Paths1 Topic

Optimal Tree Problem1 Topic

Transform and Conquer Approach1 Topic

Dynamic ProgrammingGeneral Method with Examples

Multistage Graphs

Transitive Closure1 Topic

All Pairs Shortest Paths6 Topics

BacktrackingGeneral Method

NQueens Problem

Sum of Subsets problem

Graph Coloring

Hamiltonian Cycles

Branch and Bound2 Topics

0/1 Knapsack problem2 Topics

NPComplete and NPHard Problems1 Topic
Let ai, 1<= i <= n, be a list of elements that are sorted in non decreasing order. Consider the problem of determining whether a given element x is present in the list.If x is present,we are to determine a value j such that aj = x. If x is not in the list, then j is to be set to zero. Let P = (n,ai,…, al,x) denote an arbitrary instance of this search problem(n is the number of elements in the list ai,…, al is the list of elements,and x is the element searched for)
In this example,any given problem P gets divided (reduced)into one new subproblem. This division takes only theta(1) time. After a comparison with aq, the instance remaining to be solved (if any) can be solved by using this divideandconquer scheme again.If q is always chosen such that aq is the middle element(that is,q = [(n +l)/2)], then the resulting search algorithm is known as binary search.Note that the answer to the new subproblem is also the answer to the original problem P; there is no need for any combining. Algorithm 2 describes this binary search method, where BinSrch has four inputs a[], i, l and x. It is initially invoked as BinSrch(a,l,n,x).
A non recursive version of BinSrch is given in Algorithm 3. Bin Search has three inputs a,n,and x. The while loop continues processing as long as there are more elements left to check.At the conclusion of the procedure 0 is returned if x is not present, or j is returned, such that a[j] = x.
Is BinSearch an algorithm? We must be sure that all of the operations such as comparisons between x and a[mid] are well defined. The relational operators carry out the comparisons among elements of a correctly if these operators are appropriately defined. Does BinSearch terminate? We observe that low and high are integer variables such that each time through the loop either x is found or low is increased by at least one or high is decreased by at least one.Thus we have two sequences of integers approaching each other and eventually low becomes greater than high and causes termination in a finite number of steps if x is not present.