Invariants are a powerful tool in programming. They’re conditions that always remain true at specific points in an algorithm, no matter what. Establishing invariants early in the problem-solving process helps you write correct, efficient, and easy-to-understand code.
In this blog, we’ll cover how to quickly establish invariants, explain their benefits, and show how they can give you a coding edge.
What Is an Invariant?
An invariant is a rule that stays true:
• Before the algorithm starts.
• At every step of the algorithm.
• When the algorithm finishes.
The three rules of invariants—initialization, maintenance, and termination—are critical because they provide a systematic way to ensure an algorithm is correct, efficient, and aligned with its goal. These rules outline when and how the invariant is established and maintained, giving structure to the algorithm and validating its logic.
How To Spot Invariants (Step-By-Step)
Let’s say you have an array of numbers, and you want to find the largest number in the array. For example:
If the array is [3, 7, 2, 9, 5], the largest number is 9.
1. Understand the Problem
Before identifying an invariant, fully understand the problem you’re solving. Break it into smaller components:
• What are the inputs and outputs?
• What does the algorithm aim to achieve?
Input: An array of numbers, e.g., [3, 7, 2, 9, 5].
Output: The largest number in the array, e.g., 9.2. Identify the State
Invariants are tied to the state of the algorithm, which includes:
• Variables
• Data structures (arrays, lists, etc.)
• Loop counters
The invariant is a rule about the relationship between the variables in the state.
In this algorithm, the invariant describes how max relates to the portion of the array that has been processed.
State Variables
• max: Holds the largest number seen so far.
• i: The current position in the array.
int max = 0;
int i = 0;3. Identify the Property That Must Stay True
Now, ask yourself:
• What must be true at the start, during, and at the end of the algorithm?
The invariant for this problem is:
“At any point during the algorithm, the variable max contains the largest value seen so far.”
4. Confirm the Invariant in Steps
Step 1: Initialization
At the very beginning:
• max is set to the first element of the array.
• At this point, we’ve only seen one number, so it’s correct to say max is the largest number seen so far.
Step 2: Maintenance
As the algorithm iterates through the array:
• For each number, we compare it to max.
• If the current number is larger, we update max.
• This ensures max always reflects the largest number seen so far.
Step 3: Termination
When the loop ends:
• The algorithm has processed every number in the array.
• Since max was updated whenever a larger number was found, it now holds the largest number in the entire array.
Why Is This the Invariant?
This invariant works because:
1. It’s true at the start (only one number has been checked, so it’s the largest by default).
2. It stays true at every step (we update max whenever a larger number is found).
3. It guarantees the correct result at the end (all numbers have been checked, so max must be the largest).
Key Tips for Identifying an Invariant
1. Focus on What Must Always Be True: Look for the property that ensures the algorithm is progressing correctly.
• In this case, max always holds the largest number seen so far.
2. Think About Initialization: How do you start the algorithm so the invariant holds from the beginning?
• Start with max = arr[0].
3. Check Maintenance: How does each step of the algorithm ensure the invariant stays true?
• Compare and update max when a larger number is found.
4. Verify at Termination: Does the invariant guarantee the final result?
• Yes, max contains the largest number after all elements are checked.
public class LargestElement {
public static int findLargest(int[] arr) {
// Check if the array is empty to avoid errors
if (arr == null || arr.length == 0) {
throw new IllegalArgumentException("Array cannot be null or empty");
}
// Initialize the largest element to the first element
int max = arr[0];
// Loop through the array starting from the second element
for (int i = 1; i < arr.length; i++) {
// Update max if the current element is larger
if (arr[i] > max) {
max = arr[i];
}
}
// Return the largest element
return max;
}
}Conclusion
To identify the invariant, think about the key property that guides the algorithm’s logic. For finding the largest number in an array, the invariant is simple but powerful: “max is always the largest number seen so far.” By keeping this invariant true, the algorithm guarantees the correct result every time.
