Counting sort counts how often each key occurs and uses cumulative counts to place values without comparisons.
How to reason about it
Start by identifying the state the method maintains and the operation that changes it. Then trace one small input by hand. This separates the invariant that makes the method correct from implementation details such as arrays, pointers, counters, or protocol messages.
Worked example
For [3,1,2,1], the count array records two 1s, one 2, and one 3 before output is assembled.
Write down the state before and after each step. For a larger input, repeat the same transition rather than guessing from the final output. That habit makes edge cases visible and gives you a concrete basis for testing.
Trade-offs and common mistakes
Its memory depends on the value range, so it is wasteful when keys are extremely sparse.'
Choose this technique when its assumptions match the problem. State those assumptions explicitly, test an empty or boundary input, and compare the cost of the operation you perform most often with the memory or setup cost it introduces.
Takeaway
Counting Sort for Bounded Integer Ranges becomes easier to use once its invariant, input assumptions, and failure cases are clear. A small worked example is a useful first test; measurements and proofs should follow when the system grows.