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Big O Notation in JavaScript

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Big O Notation, collectively called Bachmann-Landau notation or asymptotic notation, is a way to describe the performance of an algorithm. It is used to describe the worst-case scenario of an algorithm. It is used to compare the performance of different algorithms. It describes the implementation of an algorithm in terms of the input size.

Big O notation characterizes functions according to their growth rates: tasks with the same growth rate are considered to be of the same order. It is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. It is used to classify algorithms according to how their run time or space requirements grow as the input size grows. The letter O is used because the growth rate of a function is also called its order.

Iteration

For loop

for (let i = 0; i < n; i++) {
  console.log(i)
}

The above code will run n times. The time complexity of this code is O(n) and the space complexity is O(1).

While loop

let i = 0
while (i < n) {
  console.log(i)
  i++
}

The above code will run n times. The time complexity of this code is O(n) and the space complexity is O(1).

Do while loop

let i = 0
do {
  console.log(i)
  i++
} while (i < n)

The above code will run n times. The time complexity of this code is O(n).

Fibonacci (Iterative)

function fibonacci(n) {
  let arr = [0, 1]
  for (let i = 2; i <= n; i++) {
    arr.push(arr[i - 2] + arr[i - 1])
  }
  return arr[n]
}

The above code will run n times. The time complexity of this code is O(n) and the space complexity is O(n). This is the second best fibonacci in terms of performance benchmark. The best one is using behold binet's formula, and even better when we precalculate the formula first.

Recursion

Factorial

function factorial(n) {
  if (n === 0) {
    return 1
  }
  return n * factorial(n - 1)
}

The above code will run n times. The time complexity of this code is O(n) and the space complexity is O(n).

Fibonacci (Recursive)

function fibonacci(n) {
  if (n < 2) return n

  return fibonacci(n - 1) + fibonacci(n - 2)
}

The above code will run n times. The time complexity of this code is O(2^n) and the space complexity is O(n). This things is called exponential time complexity. This is the worst fibonacci in terms of performance benchmark.

Fibonacci (Recursive with Memoization)

function fibonacci(n, cache = []) {
  if (n === 0) return 0
  if (n === 1) return 1
  if (cache[n]) return cache[n]

  cache[n] = fibonacci(n - 2, cache) + fibonacci(n - 1, cache)
  return cache[n]
}

The above code will run n times. The time complexity of this code is O(n) and the space complexity is O(n). This is the second worst fibonacci in terms of performance benchmark, but a bunch a lot better than without memoization.

Searching

function linearSearch(arr, value) {
  for (let i = 0; i < arr.length; i++) {
    if (arr[i] === value) {
      return i
    }
  }
  return -1
}

The above code will run n times. The time complexity of this code is O(n).

function binarySearch(arr, value) {
  let start = 0
  let end = arr.length - 1
  let middle = Math.floor((start + end) / 2)
  while (arr[middle] !== value && start <= end) {
    if (value < arr[middle]) {
      end = middle - 1
    } else {
      start = middle + 1
    }
    middle = Math.floor((start + end) / 2)
  }
  if (arr[middle] === value) {
    return middle
  }
  return -1
}

The above code will run log(n) times. The time complexity of this code is O(log(n)).

Sorting

Bubble sort

function bubbleSort(arr) {
  for (let i = arr.length; i > 0; i--) {
    for (let j = 0; j < i - 1; j++) {
      if (arr[j] > arr[j + 1]) {
        let temp = arr[j]
        arr[j] = arr[j + 1]
        arr[j + 1] = temp
      }
    }
  }
  return arr
}

The above code will run n^2 times. The time complexity of this code is O(n^2).

Selection sort

function selectionSort(arr) {
  for (let i = 0; i < arr.length; i++) {
    let lowest = i
    for (let j = i + 1; j < arr.length; j++) {
      if (arr[j] < arr[lowest]) {
        lowest = j
      }
    }
    if (i !== lowest) {
      let temp = arr[i]
      arr[i] = arr[lowest]
      arr[lowest] = temp
    }
  }
  return arr
}

The above code will run n^2 times. The time complexity of this code is O(n^2).

Insertion sort

function insertionSort(arr) {
  for (let i = 1; i < arr.length; i++) {
    let currentVal = arr[i]
    for (var j = i - 1; j >= 0 && arr[j] > currentVal; j--) {
      arr[j + 1] = arr[j]
    }
    arr[j + 1] = currentVal
  }
  return arr
}

The above code will run n^2 times. The time complexity of this code is O(n^2).

Merge sort

function mergeSort(arr) {
  if (arr.length <= 1) return arr
  let mid = Math.floor(arr.length / 2)
  let left = mergeSort(arr.slice(0, mid))
  let right = mergeSort(arr.slice(mid))
  return merge(left, right)
}

function merge(left, right) {
  let results = []
  let i = 0
  let j = 0
  while (i < left.length && j < right.length) {
    if (left[i] < right[j]) {
      results.push(left[i])
      i++
    } else {
      results.push(right[j])
      j++
    }
  }
  while (i < left.length) {
    results.push(left[i])
    i++
  }
  while (j < right.length) {
    results.push(right[j])
    j++
  }
  return results
}

The above code will run n log(n) times. The time complexity of this code is O(n log(n)).

Quick sort

function pivot(arr, start = 0, end = arr.length + 1) {
  let pivot = arr[start]
  let swapIdx = start
  function swap(array, i, j) {
    let temp = array[i]
    array[i] = array[j]
    array[j] = temp
  }
  for (let i = start + 1; i < arr.length; i++) {
    if (pivot > arr[i]) {
      swapIdx++
      swap(arr, swapIdx, i)
    }
  }
  swap(arr, start, swapIdx)
  return swapIdx
}

function quickSort(arr, left = 0, right = arr.length - 1) {
  if (left < right) {
    let pivotIndex = pivot(arr, left, right)
    quickSort(arr, left, pivotIndex - 1)
    quickSort(arr, pivotIndex + 1, right)
  }
  return arr
}

The above code will run n log(n) times. The best case time complexity of quick sort is O(n log(n)) and the worst case time complexity of quick sort is O(n^2) with average case time complexity of O(n log(n)). The space complexity is O(n).

Common Data Structure Pperations

Data Structure Time Complexity Space Complexity
Average Worst Worst
Access Search Insertion Deletion Access Search Insertion Deletion
Array Θ(1) Θ(n) Θ(n) Θ(n) O(1) O(n) O(n) O(n) O(n)
Stack Θ(n) Θ(n) Θ(1) Θ(1) O(n) O(n) O(1) O(1) O(n)
Queue Θ(n) Θ(n) Θ(1) Θ(1) O(n) O(n) O(1) O(1) O(n)
Singly-Linked List Θ(n) Θ(n) Θ(1) Θ(1) O(n) O(n) O(1) O(1) O(n)
Doubly-Linked List Θ(n) Θ(n) Θ(1) Θ(1) O(n) O(n) O(1) O(1) O(n)
Skip List Θ(log(n)) Θ(log(n)) Θ(log(n)) Θ(log(n)) O(n) O(n) O(n) O(n) O(n log(n))
Hash Table N/A Θ(1) Θ(1) Θ(1) N/A O(n) O(n) O(n) O(n)
Binary Search Tree Θ(log(n)) Θ(log(n)) Θ(log(n)) Θ(log(n)) O(n) O(n) O(n) O(n) O(n)
Cartesian Tree N/A Θ(log(n)) Θ(log(n)) Θ(log(n)) N/A O(n) O(n) O(n) O(n)
B-Tree Θ(log(n)) Θ(log(n)) Θ(log(n)) Θ(log(n)) O(log(n)) O(log(n)) O(log(n)) O(log(n)) O(n)
Red-Black Tree Θ(log(n)) Θ(log(n)) Θ(log(n)) Θ(log(n)) O(log(n)) O(log(n)) O(log(n)) O(log(n)) O(n)
Splay Tree N/A Θ(log(n)) Θ(log(n)) Θ(log(n)) N/A O(log(n)) O(log(n)) O(log(n)) O(n)
AVL Tree Θ(log(n)) Θ(log(n)) Θ(log(n)) Θ(log(n)) O(log(n)) O(log(n)) O(log(n)) O(log(n)) O(n)
KD Tree Θ(log(n)) Θ(log(n)) Θ(log(n)) Θ(log(n)) O(n) O(n) O(n) O(n) O(n)

Tips for Big O

  • Arithmetic operations are constant
  • Variable assignment is constant
  • Accessing elements in an array (by index) or object (by key) is constant
  • In a loop, the complexity is the length of the loop times the complexity of whatever happens inside of the loop

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