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Lesson 8 of 8

Visit submasks and larger bitsets

Choose between a machine-word mask, a fixed bitset, and an exponential search.

Understand the idea

To enumerate submasks of mask, start at mask and repeatedly use (sub − 1) & mask. Process zero once and stop before updating it, or unsigned wraparound restarts the cycle. A fixed std::bitset<N> stores more than one machine word and supports whole-set operations.

Trace a small example

The submasks of 1010 are 1010, 1000, 0010, and 0000. For bounded nonnegative subset sums, start with reachable[0] = 1 and update reachable |= reachable << value once per item. The highest represented sum must cover every query you intend to answer.

Conditions and cost

There are 2^popcount(mask) submasks, and all-mask/all-submask enumeration totals 3^n states. Bitset operations cost work across their storage words. Optional extensions in the PDF cover per-bit pair contributions, copied-mask backtracking, and subset-total recurrences.

Pause & predict

What happens if the submask loop updates zero before stopping?

Choose an answer
Try it yourself

Turn the idea into code

List the reachable subset sums of [2, 3] using a bitset trace.

Give me a hint

Begin with sum 0 reachable, then shift by each value.

Show the explained solution

Start {0}; after 2 obtain {0, 2}; after 3 obtain {0, 2, 3, 5}. Each update combines sums without and with the new item.

Your notes

Ready for the next idea?

Mark complete when you can explain it yourself.

From a problem to a programIdentify the input, output, and constraints, then translate a small plan into C++.Variables, types, and remembered valuesChoose a representation, initialize it, and understand what assignment really changes.Arithmetic without surprisesPredict division and remainder, widen before multiplication, and format decimal answers.Conditions and logical decisionsBuild exclusive categories, combine comparisons, and protect operations with short-circuit logic.Optional: other ways to chooseRecognize switch for exact choices and the conditional operator for choosing a value.For loops: make repetition preciseTrace initialization, condition, body, and update—and avoid an extra iteration.While, do-while, and stopping rulesChoose a loop from its stopping condition and read the exact input format.Break, continue, and early answersDistinguish leaving a loop from skipping one iteration, without losing required input.Five patterns worth understandingUse sums, counts, best-so-far candidates, products, and flags to process a stream.Nested loops: rows and columnsTranslate a shape into two loop roles and count the total work.Take a number apart, one digit at a timeUse remainder and division to process decimal digits, including zero.Multiple cases and a debugging mindsetCombine the patterns, isolate each case, and test the boundaries that can break your idea.Arrays: give every value a placeUnderstand indices, initialization, capacity, active length, and the first safe array loop.Scan, count, search, and compareDistinguish values from positions and preserve the exact tie rule a problem asks for.Change a sequence without losing itUpdate, copy, compare, reverse, and inspect neighbours using the correct bounds.Matrices: think in two coordinatesTraverse tables, total rows and columns, inspect diagonals, and protect grid neighbours.Characters become stringsSeparate characters from numbers, then safely inspect and edit managed text.Read the text the problem actually givesChoose words or full lines, preserve empty cases, and reset only the state that needs it.Scan text: count, filter, transformBuild a result from matching characters without losing leading zeros or skipping shifted text.Join, compare, and find textLearn content comparison, lexicographical order, substring counts, and the not-found sentinel.