[SPARK-37711][PS] Reduce pandas describe job count from O(N) to O(1)

[devin-petersohn]

I generated some benchmarks for new implementation and compared against the old implementation. The performance numbers are show below.

• Row counts: 1,000 and 10,000
• Column counts: 2, 5, 10, 20, 40, 100
• Data distribution: Random uniform distribution over 10 distinct values per column
• Total tests: 11 configurations

Rows	Columns	Old Time	New Time	Speedup	Time Saved	Improvement	Jobs (Old→New)	Jobs Saved
1,000	1	0.125s	0.188s	0.66x	-0.063s	-50.6%	2 → 3	-1
1,000	2	0.226s	0.233s	0.97x	-0.007s	-2.9%	4 → 3	1
1,000	5	0.501s	0.225s	2.23x	0.276s	55.1%	10 → 3	7
1,000	10	0.861s	0.351s	2.46x	0.511s	59.3%	20 → 3	17
1,000	20	1.539s	0.418s	3.68x	1.120s	72.8%	40 → 3	37
1,000	40	3.176s	0.514s	6.18x	2.662s	83.8%	80 → 3	77
1,000	100	7.483s	0.586s	12.77x	6.897s	92.2%	200 → 3	197
10,000	1	0.073s	0.111s	0.66x	-0.038s	-51.9%	2 → 3	-1
10,000	5	0.362s	0.194s	1.87x	0.168s	46.5%	10 → 3	7
10,000	10	1.446s	0.257s	5.61x	1.188s	82.2%	20 → 3	17
10,000	20	1.424s	0.382s	3.72x	1.041s	73.1%	40 → 3	37
10,000	40	3.171s	0.521s	6.09x	2.650s	83.6%	80 → 3	77
10,000	100	10.953s	1.163s	9.41x	9.789s	89.4%	200 → 3	197

Aggregate Statistics:

• Average speedup: 4.33x
• Average improvement: 48.7%
• Average jobs saved: 54.2 per operation
• Maximum speedup: 12.77x (100 columns)
• Regression case: 0.66x for N=1 (new approach is 50% slower)

What changes were proposed in this pull request?

Fixes describe for string-only dataframes to have a fixed number of jobs rather than one job per column

Why are the changes needed?

Performance

Does this PR introduce any user-facing change?

No

How was this patch tested?

CI

Was this patch authored or co-authored using generative AI tooling?

Co-authored-by: Claude Sonnet 4.5

[holdenk]

I like this :) Quick first look, haven't had a chance for a proper review yet but love to see these old TODOs getting fixed :)

[holdenk]

Reason for dropping the coment?

[devin-petersohn]

I can add the comments back. I removed them because I felt the comments didn't add extra context over the code (code is straightforward IMO).

[holdenk]

Note for the future: This seems like a good follow up issue, I think we could do something smarter here long term. I've been thinking about some kind of bounded collection types for aggregations and this might fit. (although tbf describe isn't used all that often, but would love to put these together if we can). They do still end up being large but on the executors and the final driver part is a bit smaller.

[HyukjinKwon]

cc @gaogaotiantian FYI

[gaogaotiantian]

The improvement for multiple columns is great. A few questions:

1. Is it worth it to have a fast path for single column? Just fallback the original implementation?
2. If I understand correctly, we stack data right? What does it mean for memory usage when the df is large?