Binary Search Overflow: The Bug That Fails Hidden Tests

Originally published on LeetCopilot Blog

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Mid-point overflow and off-by-one loops are the two fastest ways to fail hidden tests. Here's the safe formula.

Binary search sounds simple until boundary bugs derail your run. For beginners, mid-point overflow and off-by-one loops are the two fastest ways to fail hidden tests. This article gives you a repeatable binary search recipe for arrays so your pointers converge without surprises.

TL;DR

• Overflow happens when mid = (l + r) / 2 with large indices; use the safe formula instead.
• Interviews care because boundary bugs reveal shaky fundamentals even if your high-level idea is right.
• Core steps: choose an invariant, use the safe mid formula, define a termination rule, and dry run on tiny arrays.
• Beginners often mix while l < r vs. while l <= r and accidentally skip the last element.
• You'll learn a visual pointer diagram, a safe code pattern, and common pitfalls to avoid.

Beginner-Friendly Explanation: Why Overflow Happens

JavaScript and Python won't overflow easily, but C++ and Java can when l and r are large. Even without overflow, a sloppy mid formula can break invariants. Use this mental model:

• Invariant first: Decide what holds true after each iteration (e.g., target is always in [l, r]).
• Safe mid: Compute mid = l + (r - l) // 2 to avoid addition overflow.
• Shrink correctly: Update l or r so the invariant stays true.

For a deeper intuition about search spaces, see Binary Search on Answer in LeetCode for Beginners.

Step-by-Step Learning Guidance: A Reliable Loop Template

1) Pick the invariant and loop condition

• Use while l <= r when you check nums[mid] and can return immediately.
• Use while l < r when you want to keep at least one candidate open (common in lower/upper bound problems).

2) Apply the safe mid formula

mid = l + (r - l) // 2 prevents overflow and respects integer division.

3) Update boundaries with intent

• If nums[mid] < target, move l = mid + 1.
• If nums[mid] > target, move r = mid - 1.
• Return when equal, or tailor to lower/upper bound rules.

4) Prove termination

Draw two steps of the loop on a 5-element array. Ensure the window shrinks every time. This habit complements the checklist in Questions to Ask Before Coding in an Interview.

Visualizable Example: Lower Bound Search

Imagine finding the first index where nums[i] >= target in a sorted array.

• Invariant: answer is always in [l, r].
• Loop: while l < r keeps at least one candidate alive.
• Mid: mid = l + (r - l) // 2.
• Update: if nums[mid] < target, set l = mid + 1; else set r = mid.

Pointer Diagram (conceptual)

• Start: l = 0, r = n-1.
• After each iteration, either the left half is discarded or the right half is narrowed, but the invariant holds.

Code Example: Overflow-Safe Template

def lower_bound(nums, target):
    l, r = 0, len(nums) - 1
    while l < r:
        mid = l + (r - l) // 2
        if nums[mid] < target:
            l = mid + 1
        else:
            r = mid
    return l if nums and nums[l] >= target else -1

This template avoids overflow, keeps the invariant, and returns the first index meeting the condition.

Practical Preparation Strategies

• Dry-run tiny arrays: Practice on length-1, length-2 arrays to confirm termination.
• Shift languages: Rehearse the same template in your interview language to catch integer division rules.
• Time complexity callout: Mention the O(log n) shrink each iteration to show confidence, similar to how Why Does My Code Work on Small Test Cases but Fail on Large Ones? frames complexity.
• Use tooling sparingly: A hinting assistant like LeetCopilot can replay pointer movement on your code, reinforcing the invariant without giving away answers.

Common Mistakes to Avoid

Mistake 1: Using mid = (l + r) / 2

In Java/C++, this can overflow. Even in safe languages, it can create floating midpoints if you forget integer division.

Mistake 2: Mixing loop conditions

while l <= r with r = mid can trap you in an infinite loop when l == r. Align updates with the loop condition.

Mistake 3: Ignoring duplicates

For lower/upper bound searches, update rules must bias to the side you need; otherwise you return the wrong duplicate index.

Mistake 4: Forgetting empty arrays

Check if not nums: return -1 before starting to avoid index errors.

FAQ

• How do I know which loop condition to use? If you return inside the loop, <= is fine; if you return after the loop, prefer < and bias updates appropriately.
• Is overflow really an issue in interviews? Yes for C++/Java; using the safe formula shows attention to detail in all languages.
• What should I practice first? Start with exact-target search, then add lower and upper bound variations.
• How do I debug pointer drift? Print l, mid, r for two iterations and ensure the window strictly shrinks.

Conclusion

Binary search reliability comes from respecting invariants, shrinking windows, and using an overflow-safe mid. By rehearsing a single clear template and testing it on tiny arrays, you avoid the classic off-by-one traps and present confident solutions in interviews.

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