CS 411 Fall 2025  >  Outline for September 5, 2025

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CS 411 Fall 2025
Outline
for September 5, 2025

Outline

Analyzing Recursive Algorithms [L 2.4, Appendix B]

• Recursion
  • Algorithm or function (code) calling itself, perhaps indirectly.
  • Must have base case, and must eventually reach base case.
  • Simple analysis: recursive factorial function.
• Introductory example: analyze recursive factorial
  • A recurrence relation—or recurrence—defines a (mathematical) function in terms of itself.
  • Often there are one or more associated initial conditions, giving specific values of the function.
  • Analysis of iterative algorithms usually involves summations; analysis of recursive algorithms usually involves recurrences.
  • The recursive call gives the recurrence; base cases give initial conditions.
  • To solve a recurrence is to find a (mathematical) function that makes the recurrence—and possibly initial conditions—true.
  • Some recurrences can be solved using the method of forward substitutions: compute values of function, guess a formula (guess right!), verify that it solves the recurrence.
• Example: analyze recursive factorial function
• General Plan
  • Plan for analyzing recursive algorithms
    1. Choose way of measuring size of input.
    2. Choose basic operation.
    3. Does # of basic ops depend only on the size of the input?
    4. Set up a recurrence.
    5. Solve the recurrence.
• Example: analyze Binary Search
• Applying the Smoothness Rule
  • Example: Binary Search
  • Smoothness Rule
    • Suppose \(T(n)\) is \(\Theta(f(n))\) for \(n\) a power of \(b ≥ 2\).
    • Three conditions:
      • \(T\) eventually nondecreasing.
      • \(f\) eventually nondecreasing.
      • \(f(2n)\) is \(\Theta(f(n))\).
    • Conclusion: \(T(n)\) is \(\Theta(f(n))\) for all \(n\).
    • The above also works for \(O\) and \(\Omega\).
• Example: number of bits in binary representation of a number