Big O Notation

Motivation for Big O Notation

There are dozens of algorithms and/or data structures that can solve a problem you're faced with, and each one can be fundamentally different in its approach.

How do we know which is best? Big O notation is about generalizing code and talking about its performance compared to other pieces of code.

More importantly, big O notation allows us to numerically represent the performance of our code (as opposed to using vague descriptions like "it's OK" or "pretty good").

Note: Big O notation isn't a black and white calculation that can tell you the best solution in all cases. Implementations can have trade-offs.

An example

Suppose that we want to get the sum of all numbers from 1 up to n. There are 2 working solutions we can think of:

// Iteration
const addUpTo = n => {
  let sum = 0;
  for (let i = 1; i <= n; i++) {
    sum += i;
  }
  return sum;
};

// Algorithm derived mathematically
const algorithmAddUpTo = n => {
  return (n * (n + 1)) / 2;
};

Which is better? What does better even mean?

  • Faster?

  • Measured against benchmarks? (e.g. counting up to 1 billion)

  • Less memory-intensive? (i.e. number of variables stored to complete calculation)

  • More readable? Brevity? (i.e. number of characters used)

Pro tip: We care the most about speed and memory intensiveness. At the same time, prioritizing these will often come at the cost of readability, so it's a balancing act.

The problem with time

One way we might want to measure speed is to use a timer.

const time1 = performance.now(); // built-in timestamp in browser
addUpTo(1000000000); // 1 billion
const time2 = performance.now();

console.log(`Time elapsed: ${(time2 - time1) / 1000} seconds.`);

const time3 = performance.now();
algorithmAddUpTo(1000000000);
const time4 = performance.now();

console.log(`Time elapsed: ${(time4 - time3) / 1000} seconds.`); // logs MUCH less

The above example does demonstrate that the algorithm is much more efficient. However, there are a few problems with the timer technique:

  1. The time results vary—even on the same computer.

  2. How do we label the speed of the function? Is it based on percentage of speed relative to the other? Is it the difference? This is unclear.

  3. Fast algorithms are sometimes so fast that you can't compare using time, since the time difference is so small or invisible due to the limitations of the computer.

    • Pro tip: Very often, you'll be comparing very fast algorithms. If time was your measurement, they'd all look identical.

  4. Some code takes hours to run. It would be nice to be able to talk about the speed of our code without relying on these long tests.

If not time, then what?

Instead of time, the better way is to measure the number of simple operations that the computer has to perform.

const addUpTo = n => {
  let sum = 0; // 1 assignment operation
  // 1 assignment, n comparisons, n addition + assignment
  for (let i = 1; i <= n; i++) {
    sum += i; // n addition + assignment
  }
  return sum;
};

const algorithmAddUpTo = n => {
  return (n * (n + 1)) / 2; // 3 operations: addition, multiplication, division!
};

In the examples above, we might say:

  • Iterative approach is 5n + 2

  • Formulaic approach is 3

However, with big O notation, we aren't really concerned with whether it's 5n, 20n, or 10000n. The important thing is that the number of operations grow proportionally with n.

What is Big O Notation?

Big O notation is just a way to formally talk about how the runtime of an algorithm grows as the inputs grow.

The syntax for big O is O(f(n)):

  • O() denotes that we're talking about big O

  • f() is the function or algorithm being measured

  • n is the input passed into f()

f(n), when run, generates an output, which tells us the runtime in formalized terms. Some classic patterns include:

  • Linear: f(n) = n

  • Quadratic: f(n) = n^2

  • Constant: f(n) = 1

  • Or something else entirely!

Note: Big O notation is always talking about the worst-case scenario: huge inputs. Small inputs usually will be too similar for it to matter.

Simplifying Big O Expressions

How to simplify big O expressions:

  • Multiples of n like 100n don't matter. We just say n.

    • Instead, we care about order of magnitude: the size of the runtime of an input relative to its previous smaller inputs. In other words, growth rate!

  • Constant numbers of operations, whether it's 100 or 1000, get simplified down to 1 for simplicity.

  • Smaller terms, like multipliers or constants, don't matter.

    • For example, 95n^2 + 150, from a big picture view, starts to look the same as n^2. So we drop 150 and 95 and just say n^2.

    • Another example: n^2 + 5n. Say n = 10000. That means 100,000,000 + 50000. 5n, from big picture, is miniscule, so we drop it.

Rules of thumb:

  • Arithmetic operations (BEDMAS) are constant.

    • If you have n + 2, the computer takes roughly the same amount of time to perform 2 + 2 compared to 1,000,000 + 2.

  • Variable assignment is also constant.

    • The computer takes roughly the same about of time to assign x = 2 compared to x = 1,000,000.

  • Accessing elements in an array or object is constant.

  • In a loop, complexity = number of iterations * complexity of whatever happens inside 1 iteration

Time Complexity and Space Complexity

Everything we spoke about above is called time complexity: the runtime of an algorithm. However, we also care about memory intensiveness, which is known as space complexity.

In particular, we're concerned with auxiliary space complexity: the amount of space required by the algorithm, not including space taken up by inputs. If we factored in inputs, then of course as n approaches infinity, the space complexity would get very large.

Rules of thumb:

  • Most primitives take up constant space.

  • Strings take up O(n) space where n is the length of the string.

  • Reference types like arrays and objects take up O(n) space where n is the number of key/value pairs or items.

Examples

The example below is O(1):

const sum = arr => {
  let total = 0 // `total` stored in memory
  for (let i = 0 /* `i` stored in memory */, i < arr.length; i++) {
    total += arr[i]
  }
  return total
}

// TOTAL SPACE: 2 items stored in memory
// Simplifies to O(1)

This next example is O(n):

const double = arr => {
  const doubledArr = [];
  for (let i = 0; i < arr.length; i++) {
    doubledArr.push(arr[i] * 2); // doubledArr's length is the same as arr
  }
  return doubledArr;
};

// TOTAL SPACE: proportional to n inputs
// Ignoring constants, simplifies to O(n)

Logarithms

Logarithms are the inverse of exponentiation. They are one of the classic big O patterns.

Formula: log base(value) = exponent. For example, log 2(8) = 3 means that to get the value 8 given a base of 2, we need a power of 3. Or 2^3 = 8.

Note: The most common bases in logarithms are 2, 10, and E.

Pro tip: For our purposes, we will assume log 2. So, for simplicity's sake, we can just say log. In any case, given that we care about trends and patterns in big O, it doesn't matter whether we're using log 2 or log 10 or log 1000. The trend is still the same. That's why we can just say log.

Rule of thumb: The log 2(x) of x is the number of times you can divide x by 2 before you get a value that's less than or equal to 1. Example: you can divide 8 by 2 a total of 3 times, so log 2(8) = 3.

O(log n)

Being the inverse of exponentiation like O(n^2), log n grows very slowly in time and/or space complexity. This makes it one of the best big O expressions other than O(1).

In general, from best to worst classic big O expressions, you have:

  • O(1)

  • O(log n)

  • O(n)

  • O(n * log n)

  • O(n^2)

Where we'll see logarithmic complexity

  1. Certain search algorithms have it.

  2. Some more effective sorting algorithms have it.

  3. Finally, some cases of recursion will have it (specifically for space complexity).

Summary

  1. To analyze the performance of an algorithm, we use big O notation.

  2. Big O notation gives us a high-level understanding of time and space complexity.

  3. As a result of (2), we don't care about how precise our big O expressions are; the goal is to discover general trends: linear? quadratic? constant?

  4. Time and space complexity depend on the algorithm itself, not on hardware.

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