Big O Notation: The Art of Writing Efficient Algorithms

When you’re building software—whether it’s a small utility or a full-scale application—performance plays a critical role. Efficient algorithms are the backbone of scalable systems, and understanding how they behave as the input size grows is essential. This is where Big O Notation comes in.

Big O is a mathematical notation that describes the worst-case growth rate of an algorithm in terms of time or space. It doesn’t measure exact speed but rather gives an upper bound on how an algorithm scales. Understanding it helps you compare solutions and make better design decisions.

Below is a breakdown of the most common time complexities, explained with simple examples and real-world use cases.

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O(1) – Constant Time

An operation that takes the same amount of time regardless of the input size.

Example: Accessing an array element by index:

const item = arr[5];

Use Cases:

• Hash table lookups
• Boolean checks
• Retrieving values from dictionaries

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O(n) – Linear Time

The runtime grows directly in proportion to the input size.

Example: Finding the maximum value in an unsorted array:

let max = arr[0];
for (let i = 1; i < arr.length; i++) {
  if (arr[i] > max) max = arr[i];
}

Use Cases:

• Looping through arrays or linked lists
• Filtering or mapping operations

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O(log n) – Logarithmic Time

The input size is repeatedly divided, which keeps the growth rate very low—even for large inputs.

Example: Binary search:

function binarySearch(arr, target) {
  let left = 0, right = arr.length - 1;
  while (left <= right) {
    const mid = Math.floor((left + right) / 2);
    if (arr[mid] === target) return mid;
    if (arr[mid] < target) left = mid + 1;
    else right = mid - 1;
  }
  return -1;
}

Use Cases:

• Search in sorted arrays
• Tree-based operations
• Efficient indexing systems

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O(n²) – Quadratic Time

The time grows with the square of the input size—usually the result of nested loops.

Example: Bubble sort:

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

Use Cases:

• Simple comparison-based sorts
• Graph adjacency matrix processing

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O(n³) – Cubic Time

Each layer of nesting adds another power to the complexity, making these algorithms slow for even modest inputs.

Example: Naive matrix multiplication:

for (let i = 0; i < n; i++) {
  for (let j = 0; j < n; j++) {
    for (let k = 0; k < n; k++) {
      result[i][j] += A[i][k] * B[k][j];
    }
  }
}

Use Cases:

• 3D simulations
• Dynamic programming over 3 variables

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O(n log n) – Linearithmic Time

A combination of linear and logarithmic growth—common in efficient sorting and divide-and-conquer algorithms.

Example: Merge sort:

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

Use Cases:

• Merge sort, heapsort, quicksort
• Balanced tree insertions and deletions

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O(2ⁿ) – Exponential Time

Each new element doubles the work, making these algorithms impractical for large inputs.

Example: Recursive Fibonacci:

function fib(n) {
  if (n <= 1) return n;
  return fib(n - 1) + fib(n - 2);
}

Use Cases:

• Brute-force combinatorics
• Subset or partition problems

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O(n!) – Factorial Time

Performance degrades extremely quickly. These are typically brute-force algorithms for problems with many permutations.

Example: Generating all permutations:

function permute(arr) {
  if (arr.length <= 1) return [arr];
  const result = [];
  for (let i = 0; i < arr.length; i++) {
    const rest = [...arr.slice(0, i), ...arr.slice(i + 1)];
    for (const p of permute(rest)) {
      result.push([arr[i], ...p]);
    }
  }
  return result;
}

Use Cases:

• Traveling Salesman
• Scheduling with constraints

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O(√n) – Square Root Time

These algorithms operate on square root-sized chunks of the input.

Example: Optimized prime checking (Sieve of Eratosthenes):

for (let i = 2; i * i <= n; i++) {
  if (isPrime[i]) {
    for (let j = i * i; j <= n; j += i) {
      isPrime[j] = false;
    }
  }
}

Use Cases:

• Number theory optimizations
• Spatial indexing

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Conclusion

Understanding Big O Notation empowers you to write smarter code. It allows you to analyze your algorithm’s behavior under pressure, make performance-conscious decisions, and scale your applications with confidence.

When in doubt, choose simplicity—but when performance is critical, Big O will be your guide.