• i. Divide and Conquer: Split into independent halves, recurse, and handle what crosses the split.
• ii. CDQ Divide and Conquer: CDQ (after Chen Danqi) divides the sequence of operations in half, recurses on both halves, and lets the left half’s updates act on the right half’s queries.
• iii. Offline Techniques: If all queries are known in advance, you can reorder them, reverse time, batch them, or split time into a tree.
• iv. Parallel Binary Search: $q$ queries each ask “after how many of the $m$ operations does my condition first become true?”, where the condition is monotone in time.