StudyLabs

Lecture 02 - Jan 6

2 min read

Algorithm Properties

  1. Input (zero or more)
  2. Output (at least one)
  3. Correct
  4. Language Independent
  5. Unambiguous
  6. Efficiency
    • Time Complexity: Amount of time taken
    • Space Complexity: Amount of memory used

Time Complexity Analysis

Example 1: Sum of Two Numbers

(1) Read a, b          → Assignment Statement     (2 time units)
(2) S = a + b          → Constant time 'c'         (2 time units)
                         Logic Operation
(3) Print S            → Print operation           (1 time unit)

Total Time:

Example 2: Sum of n Numbers

S = 0                  → 1 time unit
for i = 0 to n         → 2n time units
  S = S + A[i]         → 2n time units

Analysis:

  • numbers input operations
  • Loop executes times
  • Each iteration: constant time operations

Example 3: Matrix Addition - 2D Array

for i = 0 to n              → Outer loop runs n times
for j = 0 to m              → Inner loop runs m times
  C[i,j] = A[i,j] + B[i,j]  → Executed n × m times

Time Complexity:

Note: For square matrices where , complexity is

Fibonacci Algorithm

Iterative Algorithm: Time Complexity 

Fib(n)
  f1 = 0
  f2 = 1
  
  for i = 2 to n
    temp = f1 + f2
    f1 = f2
    f2 = temp
  
  return f2

Recursive Definition:

Graph View