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Certainly! Let's analyze your code and optimize it.
**Original code:**
### Optimization opportunities.
1. **lru_cache**: The cache helps, but there's still some overhead in calling the function and the join/map/str conversions.
2. **Integer to string conversion**: `" ".join(map(str, ...))` is already efficient, but there is a slightly more performant method using a generator expression or f-strings in some Python versions.
3. **String concatenation**: No improvement recommended over join.
4. **Range**: Already memory-efficient.
#### The real bottleneck.
- The biggest cost here is converting numbers to strings and joining them. `map(str, ...)` is already faster than a list comprehension.
#### Optional: Using Precomputed Cache for Small Numbers (up to 1000)
- Since the function is only cached for up to 1001 unique values, we could **precompute** all results up front for numbers 0..1000 using a tuple or list and use **direct lookup**, which will be much faster for repeated calls, at the cost of a small amount of memory but no dynamic LRU lookup cost.
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**Optimized code:**
**Key Improvements:**
- The first 1001 values (same as your cache size) incur zero runtime cost and no LRU lookup overhead.
- For numbers >1000, the code works just as before.
- You keep exactly the same function signature and results; faster runtime for all practical (cached) use cases.
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If you have constraints on memory (though for 1001 joined strings it's negligible), or your cache size can be dynamically changed, let me know for an alternative solution!
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