In computer science and mathematics, analyzing the growth rate of a function is critical for understanding algorithm efficiency. This Function Growth Rate Comparison Calculator allows you to visualize and compare how different time complexities behave as the input size ($n$) increases.
Why Compare Growth Rates?
When designing software or solving mathematical problems, the "speed" of an algorithm is often defined by how the number of operations grows relative to the input. This is commonly expressed using Big O notation.
Scalability: An algorithm with $O(n)$ complexity scales linearly, meaning if you double the data, the time doubles. However, an $O(n^2)$ algorithm would take four times as long.
Performance Bottlenecks: For small inputs, the difference between $n$ and $n^2$ might be negligible. However, as inputs reach thousands or millions, the gap becomes enormous, potentially freezing systems.
Common Complexity Classes
This calculator compares the following standard growth rates:
Constant $O(1)$: The execution time remains the same regardless of input size (e.g., accessing an array index).
Logarithmic $O(\log n)$: Growth slows as $n$ increases. Common in binary search algorithms.
Linear $O(n)$: Growth is directly proportional to input. Typical for single loops over an array.
Linearithmic $O(n \log n)$: Slightly faster than linear, standard for efficient sorting algorithms like Merge Sort.
Quadratic $O(n^2)$: Growth accelerates quickly. Common in nested loops (e.g., Bubble Sort).
Exponential $O(2^n)$: Growth explodes rapidly. Often seen in brute-force solutions to complex problems like the Traveling Salesman.
How to Use This Calculator
Enter an Input Size (n) to represent the magnitude of data (e.g., number of items in a database). Select two different function types to compare. The calculator will compute the theoretical operation count for both and project how they diverge as $n$ increases, helping you make informed decisions about algorithm selection.
function calculateGrowth() {
var nInput = document.getElementById('inputN');
var funcASelect = document.getElementById('funcA');
var funcBSelect = document.getElementById('funcB');
var n = parseFloat(nInput.value);
var typeA = funcASelect.value;
var typeB = funcBSelect.value;
// Validation
if (isNaN(n) || n 1e12 || (num 0)) {
return num.toExponential(4);
}
return num.toLocaleString(undefined, {maximumFractionDigits: 2});
}
// Helper to calculate value based on type
function getVal(type, x) {
if (x === 0 && type === 'log') return 0; // handle log(0) edge case
switch (type) {
case '1': return 1;
case 'log': return Math.log2(x);
case 'n': return x;
case 'nlog': return x * Math.log2(x);
case 'n2': return Math.pow(x, 2);
case 'n3': return Math.pow(x, 3);
case '2n': return Math.pow(2, x);
default: return 0;
}
}
// Helper to get nice text label
function getLabel(type) {
switch (type) {
case '1': return 'O(1)';
case 'log': return 'O(log n)';
case 'n': return 'O(n)';
case 'nlog': return 'O(n log n)';
case 'n2': return 'O(n²)';
case 'n3': return 'O(n³)';
case '2n': return 'O(2ⁿ)';
default: return ";
}
}
// Calculate main results
var valA = getVal(typeA, n);
var valB = getVal(typeB, n);
var diff = valB – valA;
// Display results
document.getElementById('resValA').innerText = formatNum(valA);
document.getElementById('resValB').innerText = formatNum(valB);
document.getElementById('resDiff').innerText = formatNum(diff);
// Comparison Text Logic
var compText = document.getElementById('comparisonText');
var labelA = getLabel(typeA);
var labelB = getLabel(typeB);
if (valA < valB) {
var ratio = (valA === 0) ? "infinite" : (valB / valA).toFixed(1);
compText.innerHTML = "At n=" + n + ", Function A " + labelA + " is significantly more efficient.Function B is " + ratio + "x larger than Function A.";
} else if (valA > valB) {
var ratio = (valB === 0) ? "infinite" : (valA / valB).toFixed(1);
compText.innerHTML = "At n=" + n + ", Function B " + labelB + " is significantly more efficient.Function A is " + ratio + "x larger than Function B.";
} else {
compText.innerHTML = "At n=" + n + ", both functions have the same value.";
}
// Generate Projection Table
var tbody = document.getElementById('projectionBody');
tbody.innerHTML = ""; // Clear previous
var multipliers = [1, 2, 5, 10]; // Multipliers of original n
for (var i = 0; i 1000 && (typeA === '2n' || typeB === '2n')) {
// skip massive numbers for projection if exponential
if(i > 0) continue;
}
var rowValA = getVal(typeA, currentN);
var rowValB = getVal(typeB, currentN);
var tr = document.createElement('tr');
var tdN = document.createElement('td');
tdN.innerText = currentN.toLocaleString();
var tdA = document.createElement('td');
tdA.innerText = formatNum(rowValA);
var tdB = document.createElement('td');
tdB.innerText = formatNum(rowValB);
tr.appendChild(tdN);
tr.appendChild(tdA);
tr.appendChild(tdB);
tbody.appendChild(tr);
}
// Show result area
document.getElementById('result-area').style.display = 'block';
}