Asymptotic Analysis

   Introduction

• Often a choice of algorithms, data structures
• Must choose most appropriate
• Different time requirements
• Different space requirements
• Always tradeoffs between the two!
• Must have objective basis for making choices

    Asymptotic Analysis

• Rate at which usage of time or space grows with input size
• Dependent on many factors
             particular machine, particular compiler, etc.
• Want to describe algorithm apart from these concerns
• Analyze rate of growth, not absolute usage: asymptotic analysis
• Always assume some operations take constant time

    Big-Oh Notation

• Resource requirements grow proportionally with some known function
• T(n) = O(f(n)):  for some positive constants c, n₀  T(n) < cf(n) if n > n₀
• Read: "T(n) is big-oh of f(n)"
• Gives an upper bound on T's complexity
• There are other, similar notations for lower bounds, etc.
• May come up with different bounds for best, average, worst cases

    Using Big-Oh Notation

• Gives an upper bound, but not unique!
• n³ - 2n = O(n³) and = O(n¹⁵)
• Smaller bounds more helpful -- more precisely describe behavior
• Not symmetric: f(n) = O(g(n)) doesn't mean g(n) = O(f(n))

    Common Big-Oh Bounds

• O(1): constant time          example: pointer dereference
• O(lg n): logarithmic time   example: binary search
• O(n): linear time               example: linear search
• O(n lg n)                          example: optimal sorts
• O(n^k): polynomial time      example: insertion sort is O(n²⁾
• O(2^k): exponential time     example: badly-written LCS

    Choosing Between Alternative Algorithms

• May have same Big-Oh bound, but may be very different!
• Asymptotic analysis ignores constant factors
• 500n² + 85n + 10   and   2n² + 3  are both O(n²)
• Choose based on size of input (one may perform better on
  smaller input, one on larger), ease of coding, time-space tradeoffs, type of input