Algorithm - Big O Notation

About

The big-Oh notation is a vocabulary for the design and analysis of algorithms.

See:

<math>6n log_2 n + 6</math> is just <math>n log n</math> .
Terminology: running time is O(n log n) [“big-Oh” of n log n] where n = input size (e.g. length of input array).

Does array A contain the integer t?

for i=1 to n do
    if A[i] == t then
       Return TRUE
Return FALSE

The running time of this algorithm is linear in the input length n: O(n).

In the worse case, this code will do an unsuccessful search.

Whatever the constant (2, 3, 4…) is due some initial and final operation, it gets conveniently suppressed by the Big O notation.

Given A, B (arrays of length n) and t (an integer). [Does A or B contain t?]

for i=1 to n do
    if A[i] == t then
       Return TRUE
for i=1 to n do
    if B[i] == t then
       Return TRUE
Return FALSE

The running time will be roughly twice as many operations. as the previous piece of code : <math>2n</math>

But the coefficient two, being a constant independent of the input length n, is going to get suppressed once we passed a big O notation.

The running time is <math>O(n)</math>

Do arrays A, B have a number in common? Given arrays A, B of length n.

for i=1 to n do
   for j=1 to n do    
      if A[i] == B[j] then
         Return TRUE
Return FALSE

The running time is <math>O(n^2)</math>

This a quadratic time algorithm. because the running time is quadratic in the input length n.

Does array A have duplicate entries? Given arrays A of length n.

for i=1 to n do
   for j=i+1 to n do    
      if A[i] == A[j] then
         Return TRUE
Return FALSE

There's n choose 2 such choices of distinct i and j.

Again, suppressing lower-order terms and the constant factor, we still get a quadratic dependence on the length of the input array A.

The running time is <math>O(n^2)</math>