Three-Pass Reading

Fast solvers do not read a problem once. They read it three times, asking a different question each pass.

Pass 1 — What is the shape?

Skip the story. Find: what is given (n, m, arrays, graph), what is asked (count? construct? maximize? minimum possible x?), and the constraint table. Write down the shape in one line: n ≤ 2·10^5, q ≤ 2·10^5, range add, range sum is a shape; "orcs attack villages" is not.

Pass 2 — What do the examples say?

Recompute every sample by hand. Two reasons: it verifies you understood the operation, and samples hide edge cases (a sample where the answer is 0 is telling you 0 is possible). If your mental model cannot reproduce the samples, you are not ready to think about algorithms.

Pass 3 — What fits the budget?

Convert constraints to a complexity budget (next lesson covers the table). Then ask: which family has that complexity and matches the shape? If n ≤ 20, almost anything works — the setter is testing correct brute force. If n ≤ 2·10^5 with queries interleaved, only O((n+q) log n)-ish survives.

The Algorithm Selection Guide (keep this)

| If you see | Consider (in order) | |---|---| | n ≤ 20 | brute force / bitmask enumeration | | n ≤ 500, pairs | O(n^2) DP or Floyd-Warshall | | n ≤ 5000, sequences | O(n^2) DP, divide and conquer | | n ≤ 2·10^5, offline | sorting + two pointers, sweep line | | range add / range sum | Fenwick (BIT) or lazy segment tree | | static range min | sparse table | | connectivity, edges only added | DSU | | shortest path, non-negative weights | Dijkstra | | shortest path, unit weights | BFS | | unweighted, 0/1 weights | 0-1 BFS | | all-pairs small n | Floyd-Warshall | | spanning tree weight | Kruskal / Prim | | strongly connected components | Kosaraju / Tarjan | | k-th element online | balanced BST / Fenwick descent / merge sort tree | | count pairs i<j with condition | sort + Fenwick (inversion-style) | | pattern occurrences in text | KMP / Z-function | | count numbers ≤ N with digit property | digit DP | | assignment / pairing | bipartite matching, flow | | game, piles, last move wins | Nim / Grundy (XOR) | | geometry, minimize distance | sweep + closest pair, binary search | | answer is monotone in x | binary search on answer |

The table is a shortlist generator, not an answer key. Drills in this module train the reflex; later modules train the verification (does the family actually fit the operation semantics?).