binary search boolean array framework

Topics

• binary search
• algorithmic paradigm

Most binary search problems can be reframed as finding the first true in a conceptual boolean array [F, F, ..., F, T, T, ..., T]. Define a function check(mid) that returns true if mid is “good enough” (meets the condition) and false otherwise.

1. Set up the search space.
   • Typically, lo = 0 and hi = n - 1 for an array of size n
   • Many problems require search on answer space: (e.g., lo = 0, hi = 10^9)
2. Binary Search Loop.
   • While lo <= hi, compute mid = lo + (hi - lo) / 2 (tackles overflow)
   • If check(mid) == true, record mid as a candidate (e.g., ans = mid) and move hi left to search for an earlier true.
   • Otherwise, move lo right to find a possible true.
3. Interpret ans.
   • If ans was updated, it’s the first index where check was true.
   • If it was never updated, no valid solution exists under your check condition.

Why This Works:

Once check(mid) flips to true, everything at or beyond mid remains true. Once it’s false, everything before it remains false. You’re leveraging this monotonic property to discard half the search space each step.

int lo = 0;
int hi = n - 1;
int ans = n;  // or some 'not found' default, often n means 'beyond last index'
 
while (lo <= hi) {
    int mid = lo + (hi - lo) / 2; // safer than (lo + hi) to avoid overflow
 
    if (check(mid)) {
        // 'mid' is a valid index (we found a '1'), try finding an earlier valid index.
        ans = mid;     // record this index
        hi = mid - 1;  // search in [lo .. mid-1]
    } else {
        // 'mid' is not valid (we found a '0'), so search in [mid+1 .. hi]
        lo = mid + 1;
    }
}
cout << ans << "\n";  // 'ans' is the first index for which check(mid) was true

Variations

Sometimes, you want the last false instead of the first true. Similar approach, but you might have:

if (check(mid)) {
    // 1 found, skip left portion because we want the last false
    hi = mid - 1;
} else {
    ans = mid; // store the last false
    lo = mid + 1;
}

In some problems (like searching for a floating-point answer), you do repeated halving with a loop that ends when (hi - lo) < some_epsilon. You still rely on a monotonic feasibility check, but the loop termination condition changes.

Example

If we are asked to find index of min element in a rotated sorted array, our check function would look like: bool check(int mid) { return (v[mid] <= v[n-1]); }

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