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Here is a faster version of your program. The current code converts each integer to a string individually using map and then joins them. This can be improved by.
- Using an explicit list comprehension with preallocation is not much faster here.
- However, since all numbers are from 0 up to (number-1), Python's internal `" ".join()` with map is already very efficient.
- For small fixed max size (as in lru_cache(maxsize=1001)), you can precompute the strings up to that number and fetch them directly, reducing both time and memory for repetitive access.
But keeping the same signature and caching behavior, here's a further optimized version using a single global list to store precomputed results, which is faster than lru_cache with repeated access patterns (and fully thread-safe for read-only access).
This version will be significantly faster for numbers from 0 to 1000, which matches the intent of the original lru_cache(maxsize=1001), and is just as correct for larger (non-cached) values. The memory overhead is similar to lru_cache at its maximum, but with much lower per-call computational cost for the most common cases.
**Comments are unchanged unless relevant to the modification.**
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