Show notes

What are some properties of a good algorithm?

  • Low time complexity
  • Low space complexity
  • Memoizability
  • Immutability & purity
  • Concurrent / async / parallel
  • General applicability (map reduce, actor model)
  • “Elegant”
    • Easy to understand/teach/talk about

Why is it interesting to formally define complexity?

  • Let you compare algorithms
  • Help you estimate how your system should behave in the real world

Asymptotic analysis

  • Always defined in terms of the size of the input (n), which can sometimes be unobvious (eg. multiplication)
  • Informally defined as:
    • For f(n) starting at some constant n, you will never cross the upper/lower bound function as n trends towards infinity
  • Omega(n): lower bound
    • Best case: in practice rarely considered:
      • It’s uncommon (eg. sorting a sorted list)
      • Often isn’t interesting when we consider scaling n
      • Could it represent our “real world” price?
  • Theta(n): bounded by two functions
    • Average case: in practice rarely considered:
      • Hard to quantify unless using probabilistic or randomized algorithms
      • Often the average case is very close to the worst case
  • O(n): Upper bound
    • Worst case: often considered
      • More interesting to analyze as n gets larger
      • Happens more often than we would think (Searching a list for an element that doesn’t exist)
  • Keep in mind that algorithmic complexity is defined in the context of what current computers can do. If processors had a SUM or SORT instruction, we’d count things differently
  • Since these are just regular functions, they exhibit cool mathematical properties too, like reflexivity and transitivity!

How do I calculate this?

  • Go through every statement of your algorithm, calculate how long that operation takes (in terms of n). Ignore all constant factors.
  • Once done, add everything up to get a polynomial function. BE CAREFUL OF NESTING THINGS
  • Note: This works for non-recursive algorithms. Recursive ones have their own way of calculating it and we should totally make a separate episode about it, there’s some cool shiet here!

A different kind of complexity: Kolmogorov complexity!

  • Aka. algorithmic entropy
  • The length of the shortest computer program needed to produce a specific output.
  • Example time!
  • TIL: Calculating the lower bound for finding the algorithm to describe the kolmogorov complexity is impossible


  • Sometimes, comparing asymptotic growth won’t cover the whole story.
    • On an infinitely large data set, a linear algorithm (f(n) = n + p) should always be better than a quadratic one (g(n) = n^2 + q), but in medium sized data set, the quadratic algorithm could be faster.
      • q could be much smaller than p