Growth Rate of Functions Calculator

Growth Rate of Functions Calculator

Compare how rapidly different mathematical functions increase as the input size (n) grows. Essential for understanding algorithm asymptotic complexity.

Enter a positive integer representing the dataset size or number of operations.

Results for Input Size n =

Function Name	Big O Notation	Calculated Value (Magnitude)
Logarithmic	O(log n)
Linear	O(n)
Linearithmic	O(n log n)
Quadratic	O(n²)
Cubic	O(n³)
Exponential (Base 2)	O(2ⁿ)

*Note: Logarithms are calculated using base 2, common in computer science analysis. Values are rounded for readability. Very large exponential values may exceed standard number limits.

function calculateGrowthRates() { var nInput = document.getElementById("inputSize").value; var n = parseFloat(nInput); var resultDiv = document.getElementById("growthResult"); if (isNaN(n) || n 0)."); resultDiv.style.display = "none"; return; } // Calculations // Using Math.log2 for base-2 logarithm typical in CS var logVal = Math.log2(n); var linearVal = n; var linearithmicVal = n * logVal; var quadraticVal = n * n; var cubicVal = n * n * n; // Math.pow(2, n) can get very large very quickly. var exponentialVal = Math.pow(2, n); // Formatting helpers function formatNumber(num) { if (num === Infinity) return "Overflow (Too Large)"; if (num 0) return num.toExponential(4); // For manageable numbers, use locale string with max 2 decimal places if (num < 1e12) { return num.toLocaleString(undefined, { minimumFractionDigits: 0, maximumFractionDigits: 2 }); } // For very large numbers, switch to exponential notation return num.toExponential(4); } // Update UI document.getElementById("displayN").textContent = n.toLocaleString(); document.getElementById("valLog").textContent = formatNumber(logVal); document.getElementById("valLinear").textContent = formatNumber(linearVal); document.getElementById("valLinearithmic").textContent = formatNumber(linearithmicVal); document.getElementById("valQuadratic").textContent = formatNumber(quadraticVal); document.getElementById("valCubic").textContent = formatNumber(cubicVal); document.getElementById("valExponential").textContent = formatNumber(exponentialVal); resultDiv.style.display = "block"; }

Understanding Function Growth Rates

In computer science and mathematics, understanding the growth rate of functions is crucial for analyzing algorithms. It helps us predict how the running time or space requirements of a process will increase as the input size, denoted as n, gets larger. This is often expressed using Big O notation.

Why Input Size (n) Matters

When dealing with small datasets (e.g., sorting 10 items), almost any algorithm will be fast enough. However, as n grows to millions or billions, the differences between growth rates become catastrophic. An algorithm with a quadratic growth rate O(n²) might take years to finish a task that a linear O(n) or linearithmic O(n log n) algorithm could complete in seconds.

Common Growth Classes

• O(1) – Constant: The ideal. The time taken doesn't change regardless of the input size (e.g., looking up an item in a hash map).
• O(log n) – Logarithmic: Very efficient. The required time increases slowly as n grows exponentially. Binary search is a classic example.
• O(n) – Linear: The time taken grows in direct proportion to the input size. Iterating through a list once is O(n).
• O(n log n) – Linearithmic: Slightly slower than linear, but much faster than quadratic. Most efficient sorting algorithms (like Merge Sort or Quick Sort) fall into this category.
• O(n²) – Quadratic: The time taken is proportional to the square of the input size. Often seen in algorithms with nested loops, like Bubble Sort.
• O(2ⁿ) – Exponential: The time taken doubles with every addition to the input size. These algorithms become impractical very quickly and are often associated with brute-force solutions to complex problems.

Using the Calculator

Enter a value for the input size n in the calculator above. This value represents the scale of your problem—for example, the number of elements to sort in an array. The calculator will compute the resulting magnitude for various common complexity functions. Try inputting n = 10, then n = 100, and finally n = 1000 to see how drastically the gap widens between efficient functions (like O(n log n)) and inefficient ones (like O(n²)).