CS 411 Fall 2025  >  Outline for September 3, 2025

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CS 411 Fall 2025
Outline
for September 3, 2025

Outline

Analyzing Nonrecursive Algorithms [L 2.3, Appendix A]

• Idea
  • Analysis of recursive vs. nonrecursive algorithms is usually very different.
  • Introductory example: analyze a simple loop.
• Evaluating Summations
  • See Appendix A.
  • Summing powers of the index of summation.
  • Linearity properties.
  • Close relationship between summation, definite integral.
• General Plan
  • Plan for analyzing nonrecursive 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?
       • If it depends on some property of the input, then best/worst/average cases may differ.
       • If it depends on something else (pseudorandom number, behavior of device), then perhaps we need to expand our notion of what the input is.
    4. Set up a summation.
    5. Evaluate the summation.
  • Notes:
    • We choose the size measurement (step 1) and the basic operation (step 2) so that our analysis gives us useful information.
    • For now, we do steps 4 & 5. In practice, we simplify and/or skip these when we can.
• Example: analyze Insertion Sort
  • Rule of Thumb. \(k\) nested nontrivial loops: \(\Theta(n^k)\).