leetcode/islands/decks/binary-search.md

Binary Search

Halve a monotonic search space each step → O(log n).

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Recognition

• Sorted input + "find / insert position"
• "Minimize the maximum" / "first value where condition flips"
• Answer itself is numeric and checkable → binary search on the answer

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Template

function search(nums, target) {
  let lo = 0;
  let hi = nums.length - 1;
  while (lo <= hi) {
    const mid = lo + ((hi - lo) >> 1);
    if (nums[mid] === target) return mid;
    if (nums[mid] < target) lo = mid + 1;
    else hi = mid - 1;
  }
  return -1; // lo is the insert position
}

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First-true boundary

// smallest index where ok(i) is true; ok is false...false true...true
let lo = 0;
let hi = n; // exclusive; n means "never true"
while (lo < hi) {
  const mid = lo + ((hi - lo) >> 1);
  if (ok(mid)) hi = mid;
  else lo = mid + 1;
}
return lo;

Most "hard" binary searches are this shape in disguise.

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Pitfalls

• Off-by-one: pick lo <= hi or lo < hi and stay consistent
• mid = lo + ((hi - lo) >> 1) avoids overflow and floats
• No progress → infinite loop: every branch must shrink the range

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Recall

• What property must the search space have? (not "sorted" — monotonic)
• Where does lo land when the target is absent?
• When do you search the answer space instead of the array?