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Lesson 4 of 8

Cancel equal values with XOR

Use occurrence promises to find one or two odd-frequency values.

Understand the idea

XOR is associative and commutative, x XOR x = 0, and x XOR 0 = x. XORing every item therefore cancels values appearing an even number of times. It identifies a single odd-frequency value only when the problem promises that there is exactly one.

Trace a small example

For [4, 7, 4, 2, 2], cancellation leaves 7. If exactly two values a and b have odd counts, the total is a XOR b, which is nonzero. Choose one set bit of that total to split the input into two groups; a and b fall into different groups, and equal values cancel within each.

Conditions and cost

One scan takes O(n) time and O(1) additional state; the two-value method uses another scan. The XOR of arbitrary data is not a general duplicate detector and does not prove the promised occurrence pattern.

Pause & predict

What does XOR of all values tell you without any occurrence promise?

Choose an answer
Try it yourself

Turn the idea into code

Find the two odd-frequency values in [3, 5, 3, 8, 5, 12].

Give me a hint

Cancel equal pairs, then split on a set bit of 8 XOR 12.

Show the explained solution

The total XOR is 4. Bit 2 separates 8 from 12, so the two group XORs recover 8 and 12.

Your notes

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Mark complete when you can explain it yourself.

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