Data Structures&Algorithms-Asymptotic Notation, Recurrences

Introduction to analyzing algorithm running time with asymptotic notation, defining the five notations big O, big Omega, Theta, little o, and little omega, and their growth as input size expands.

Big-O notation provides an upper bound on a function’s growth for large n, using constants to compare f(n) to g(n). Examples: 5n+20 is O(n) and 5n^2+10 is O(n^2).

Explore omega notation as the asymptotic lower bound for running time, with constants C and n0; for example, 5n+20 is Omega(n) and 5n^2+20 is Omega(n^2).

Explore theta notation as the tight bound for functions, and compare it with big-O and omega bounds to understand upper and lower limits in asymptotic analysis.

Explore little-o and little-omega notations for asymptotic upper bounds, contrast with big-O and Omega, and apply definitions and properties to analyze running time and resources, plus practice exercises.

Explore asymptotic notation through practice exercises, proving when f(n) is in O(g(n)) or Omega(g(n)), and compare complexities like n^2, n log n, and log n.

Explore recurrence relations for characterizing an algorithm's running time, including divide-and-conquer, balanced and unbalanced recurrences. Learn solutions via substitution, expansion, recursion-tree, and the master method.

Learn the iteration (back substitution) method to solve recurrences by expanding to the initial condition, then bound sums; apply it to T(n)=2T(n/2)+n for O(n log n) and T(n)=T(n-1)+n for O(n^2).

Apply the iteration method to recurrences from divide-and-conquer problems, expand t(n) by halving, identify a pattern, and sum the series to show t(n) = theta(n) for constant split cost.

Learn to solve recurrences using the iteration method, expanding and substituting to uncover patterns in quiz problems, and deduce time complexity from linear to exponential.

Learn the substitution method for solving recurrences by guessing a solution and proving it with induction, and compare it to the attrition method to establish upper bounds using asymptotic notation.

Explore the substitution method with more examples for recurrences like t(n)=t(n/2)+c, proving logarithmic bounds via induction. Practice two follow-up problems to sharpen divide-and-conquer reasoning.

Practice the substitution method with discussion of solving recurrences by induction, including a T(n)=T(n-1)+n problem and a T(n)=2T(n/2) case, with Big O and Omega bounds.

Explore the recursion tree method for solving divide-and-conquer recurrences, learning how to build the tree, estimate per-level costs, and verify with substitution, including balanced and unbalanced cases.

Explore recursion tree analysis through practice exercises, calculating costs at each level, summing across the tree, and validating results via substitution and induction for balanced and unbalanced recurrences.

Apply the master method to solve divide-and-conquer recurrences of the form T(n)=a T(n/b)+f(n), compare three cases, and determine theta solutions under a regularity condition.

Explore the master method with its three cases, extended case two, and regularity conditions, and compare substitution, recursion tree, and Krishan methods for solving divide-and-conquer recurrences.

Discuss the master method for recurrences, verify cases one to three, and prove the regularity condition to determine the asymptotic bound.