ā” š š”
Optimizing computational efficiency for better performance
š A faster algorithm is an improved way of solving a problem that reduces the time complexity (speed of execution) or space complexity (memory used) compared to basic methods.
How execution time grows as input size increases.
How much memory is used as input size increases.
š Example: Sorting
Normal bubble sort takes O(nĀ²).
Faster quicksort/merge sort takes O(n log n).
So, faster algorithms let us solve larger problems more efficiently.
Faster algorithms are vital in:
Handling millions/billions of records efficiently
Training models efficiently with large datasets
Secure communication depends on efficient math
Critical in robotics, trading, AI assistants
š” Faster algorithms enable technologies that would otherwise be impossible or impractical
Break into subproblems ā solve recursively ā combine results.
Examples: QuickSort, MergeSort, Strassen's Matrix Multiplication.
Always take the locally best choice ā hope it leads to a global solution.
Examples: Dijkstra (shortest path), Prim's (minimum spanning tree).
Break problem into overlapping subproblems ā store results ā avoid recalculation.
Example: Fibonacci using DP instead of recursion.
Use randomness to get good average-case performance.
Example: Randomized QuickSort.
Give near-optimal solutions for problems where exact solutions are too expensive.
Example: Traveling Salesman Problem approximations.
Multiplication is basic, but naive multiplication is slow for very large numbers. Faster algorithms optimize it:
Uses bitwise operations.
Example: Booth's Algorithm ā reduces unnecessary additions.
Divide-and-conquer method.
Reduces multiplications from O(nĀ²) to about O(n^1.58).
Extends Karatsuba using evaluation & interpolation.
More efficient for very large numbers (used in cryptography).
Uses Fast Fourier Transform.
Complexity: O(n log n).
Extremely fast for large numbers/polynomials ā used in signal processing.
Division is slower than multiplication, so optimized algorithms are essential:
Uses bitwise shifts and subtraction.
Example: Non-Restoring Division ā skips unnecessary "restoring" steps.
Old method ā repeatedly subtract divisor.
Slower, but exact.
Uses digit selection + refinement.
Faster for large operands.
Uses iterative approximation of reciprocal ā multiply to get division.
Very fast for real numbers.
Common in scientific computing & math libraries.
Encryption/decryption (RSA, ECC) needs fast big-number multiplication/division
FFT-based multiplication in audio/video compression
Numerical simulations, weather models, AI workloads
Faster algorithms are the backbone of modern computing