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Lesson 7 of 8

Enumerate subsets and optimize a choice

Give every position one take-or-skip bit and evaluate each mask exactly once.

Understand the idea

For n positions, masks from 0 through 2^n − 1 describe every index subset. Bit i indicates whether position i is selected. To test a target sum or minimize a partition difference, scan selected positions and evaluate that candidate.

Trace a small example

For [−2, 3, 5], mask 101 selects −2 and 5, giving sum 3. Mask 010 also gives 3 but selects a different subset. For partitioning, if total is T and one selected side sums to S, the difference between the two sides is |T − 2S|.

Conditions and cost

A straightforward scan costs O(n × 2^n), so use small n and a shift width that contains 2^n. Include the empty subset if allowed. Negative values and zero need no special enumeration trick, but they invalidate some sum-based pruning rules.

Pause & predict

How many index subsets does an array of four values have, including the empty subset?

Choose an answer
Try it yourself

Turn the idea into code

Find the minimum partition difference for [1, 2, 4].

Give me a hint

Try subset sums near half the total, 3.5.

Show the explained solution

Total is 7. Sums 3 and 4 are closest to half; both give difference 1. The partition {1, 2} and {4} achieves it.

Your notes

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