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Funny question: why are some LL algs short? (ie ZBLL algs with <12 HTM)

NukesAway

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More clearly defined as: within a given 1LLL alg set (one that end with a solved state and deal with permutation and orientation of LL pieces simultaneously), what do the shortest algs (arbitrarily defined as below the 33th percentile in movecount; need a better definition) have in common?
What can we learn from these common traits for algset development and selection?
Take ZBLL as an example
I'm not a maths major and only know basic group theory (call it cube theory) so please put answers in simple terms.
I can't check the forums very often and I'm not in a hurry for an answer so please take your time
Sorry for the vagueness
 
At 10 moves, there are already a few algs with little discernible structure, and I believe most of the ≥11-movers just look like nonsense.

That said, there are common patterns to a lot of short last layer algs.

Some are (3-cycle) commutators:
r U R' U' r' F R F' = [L, F R' F']
R U R' U' R' F R F' = [Lw, F R' F']
These ones are generally 3-cycles, although sometimes the AUF causes it to not be a 3-cycle (e.g. AUF-less Niklas L' U R U' L U R'). If the commutator involves the corners, it must be adj CP.

Some are sexy-move or sledgehammer setups:
F R U R' U' F' = [F : R U R' U']
R U R' U R U2 R' = [R U : R' U R U']
R U2 R2 F R F' R U2 R' = [R U2 R' : R' F R F']
r U2 R2 F R F' R U2 r' = [r U2 R' : R' F R F']
These ones are always 2c'2c'3e, although with AUF it could become something else. These are always either solved CP or diag CP. Quite a few diag ZBLL algs falls in this category.

Some are setups into U/U2/U':
R U' R2 D' r U2 r' D R2 U R' = [R U' R2 D' r : U2]
F' R D2 R' F U2 F' R D2 R' F = [F' R D2 R' F : U2]
R' F2 R2 U2 R' F2 R U2 R2 F2 R = [R' F2 R2 U2 R' : F2]
F R U' R' U2 R U R' F' = [F R U' R' : U2] (not optimal! optimal for this 1LLL is F R U' R' F' L' U L)
The setups to half turns are always 2c2c2e2e, so like the previous category, they're always solved CP or diag CP.

Some involve taking out F2L pairs and reinserting them.
R U R' U R U2 R' (remove FR pair and reinsert) (note: this is the second time Sune is showing up here; these categories are not disjoint!)
R U2 R2 U' R2 U' R2 U2 R (remove FR pair, remove BR pair, insert FR pair, insert BR pair)
If you want to cancel moves between the pairs, the only option is between R U* R' and R' U* R triggers, which leads to many 2-gen algs being in this category, which in turn means many of the short algs here have solved CP. Some of the ones that don't are 3-cycle commutators in disguise, e.g. Niklas.

Some are cancellations of more basic algs:
R' F' U' F U' R U R' U R = (R' F' U' F U R) (R' U2 R U R' U R)
R U R' U R' U' R2 U' R2 U2 R = (R U R' U R U2 R') (R U2 R2 U' R2 U' R2 U2 R)
If the building blocks are all solved CP or diag CP, then the alg combo also has to be one of the two.

Some are cyclic shifts of other short, "low-piece-count" algs like J perm (2c2e):
R U' L U L' R' U2 L U L'
r U2 R2 F R F' R U2 r' = [r U2 : R2 F R F' R U2 r' U r U2] U'
These ones have some AUF that will solve most pieces.

Edge permutation is kind of a mess, but the Z perm kind of EP (opposite edge swap) is relatively rare among short ZBLLs (only three of the 10-movers have opposite edge swap iirc). On the other hand, opposite edge swap for some reason is fairly common among dot OLLs (necessarily at least 11 moves):
R U2 R2 F R F' U2 R' F R F' (has other 11-move solutions)
[r' D R2 U R' : U2]
[R' U2 r U' r' : U2]
L R F R2 U2 R2 U2 R2 U2 F' L' R' = [L R F : R2 U2 R2 U2 R2 U2]

---

There kind of are common patterns, but the patterns aren't strong enough that you can cleanly categorise them as e.g. "solved CP cases are much better" or the like.
 
At 10 moves, there are already a few algs with little discernible structure, and I believe most of the ≥11-movers just look like nonsense.

That said, there are common patterns to a lot of short last layer algs.

Some are (3-cycle) commutators:
r U R' U' r' F R F' = [L, F R' F']
R U R' U' R' F R F' = [Lw, F R' F']
These ones are generally 3-cycles, although sometimes the AUF causes it to not be a 3-cycle (e.g. AUF-less Niklas L' U R U' L U R'). If the commutator involves the corners, it must be adj CP.

Some are sexy-move or sledgehammer setups:
F R U R' U' F' = [F : R U R' U']
R U R' U R U2 R' = [R U : R' U R U']
R U2 R2 F R F' R U2 R' = [R U2 R' : R' F R F']
r U2 R2 F R F' R U2 r' = [r U2 R' : R' F R F']
These ones are always 2c'2c'3e, although with AUF it could become something else. These are always either solved CP or diag CP. Quite a few diag ZBLL algs falls in this category.

Some are setups into U/U2/U':
R U' R2 D' r U2 r' D R2 U R' = [R U' R2 D' r : U2]
F' R D2 R' F U2 F' R D2 R' F = [F' R D2 R' F : U2]
R' F2 R2 U2 R' F2 R U2 R2 F2 R = [R' F2 R2 U2 R' : F2]
F R U' R' U2 R U R' F' = [F R U' R' : U2] (not optimal! optimal for this 1LLL is F R U' R' F' L' U L)
The setups to half turns are always 2c2c2e2e, so like the previous category, they're always solved CP or diag CP.

Some involve taking out F2L pairs and reinserting them.
R U R' U R U2 R' (remove FR pair and reinsert) (note: this is the second time Sune is showing up here; these categories are not disjoint!)
R U2 R2 U' R2 U' R2 U2 R (remove FR pair, remove BR pair, insert FR pair, insert BR pair)
If you want to cancel moves between the pairs, the only option is between R U* R' and R' U* R triggers, which leads to many 2-gen algs being in this category, which in turn means many of the short algs here have solved CP. Some of the ones that don't are 3-cycle commutators in disguise, e.g. Niklas.

Some are cancellations of more basic algs:
R' F' U' F U' R U R' U R = (R' F' U' F U R) (R' U2 R U R' U R)
R U R' U R' U' R2 U' R2 U2 R = (R U R' U R U2 R') (R U2 R2 U' R2 U' R2 U2 R)
If the building blocks are all solved CP or diag CP, then the alg combo also has to be one of the two.

Some are cyclic shifts of other short, "low-piece-count" algs like J perm (2c2e):
R U' L U L' R' U2 L U L'
r U2 R2 F R F' R U2 r' = [r U2 : R2 F R F' R U2 r' U r U2] U'
These ones have some AUF that will solve most pieces.

Edge permutation is kind of a mess, but the Z perm kind of EP (opposite edge swap) is relatively rare among short ZBLLs (only three of the 10-movers have opposite edge swap iirc). On the other hand, opposite edge swap for some reason is fairly common among dot OLLs (necessarily at least 11 moves):
R U2 R2 F R F' U2 R' F R F' (has other 11-move solutions)
[r' D R2 U R' : U2]
[R' U2 r U' r' : U2]
L R F R2 U2 R2 U2 R2 U2 F' L' R' = [L R F : R2 U2 R2 U2 R2 U2]

---

There kind of are common patterns, but the patterns aren't strong enough that you can cleanly categorise them as e.g. "solved CP cases are much better" or the like.
Thanks, great post
Are speed-optimized OR movecount-minimal ZBLLs with diag corner swaps more likely to be structured (like conjugates and combos as you mentioned)? Do they contain more easy-to-memorize OR short algs than other ZBLLs? If so, why?
Correct me if I'm wrong, there are only 76 diag corner swap ZBLLs (excluding X-perm 2GLLs), quite a small subset
 
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Thanks, great post
Are speed-optimized OR movecount-minimal ZBLLs with diag corner swaps more likely to be structured (like conjugates and combos as you mentioned)? Do they contain more easy-to-memorize OR short algs than other ZBLLs? If so, why?
Correct me if I'm wrong, there are only 76 diag corner swap ZBLLs (excluding X-perm 2GLLs), quite a small subset
Idk much about this topic, but there are 84 2glls, so i assume theres about 84 diag corner swap zblls.
might be wrong, idk much about this topic after all.
 
Idk much about this topic, but there are 84 2glls, so i assume theres about 84 diag corner swap zblls.
might be wrong, idk much about this topic after all.
Nah just count them one by one
Should be 71 diag corner ZB+ 84 2GLLs, an decent LL subset for speed I guess
Edit: sorry 84
 
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Do they contain more easy-to-memorize OR short algs than other ZBLLs? If so, why?
Arguably yes, but they also contain some terrible cases which I think balances out the good ones. Like, diag-T and diag-pi used to be considered among the worst sets; don't know if that's still the consensus nowadays though.

Also, diag ZBLL can be put in bijection with 2GLL by doing an N perm, so they have exactly the same number of cases.
 
Arguably yes, but they also contain some terrible cases which I think balances out the good ones. Like, diag-T and diag-pi used to be considered among the worst sets; don't know if that's still the consensus nowadays though.

Also, diag ZBLL can be put in bijection with 2GLL by doing an N perm, so they have exactly the same number of cases.
I recently learnt Diag Pi, and even though I'm not a super fast solver, I think the algs are pretty decent. I know 3 sets of Pi and Diag is probably my best set even ahead of Pi 2GLL. I can currently sub 3 recog+exec+AUF all 3 sets together, and I am most consistent with Diag but that might just be bias for learning it first. I still find the algs to flow really nicely and are just higher quality than they were years ago when people considered Pi to be really bad. I don't know if Diag Pi is a good set outside of Pi algs though, it might still be bad if you consider most of TUL sets.
 
I tried to generate diag cp zblls by sifting through batch solver outputs, but I found it quite hard to find results with the patterns @xyzzy mentioned that were structured, easy to remember and beautiful.
I am curious how people managed to find such algs after browsing speedcube db. Here are some guesses:
  1. Just keep tweaking params of their alg genarators
  2. Throw the basic building blocks into solvers and mine for useful algs
  3. Mess around with comms, cancellations, cyclic shifts and combos by hand
Any more ideas? If not, could anyone please elaborate on how to actually implement the ones above?

Yes I know that these algs weren't specifically created with structuralness in mind but just happened to show up on the same webpage

I want to eventually gather as many Diag CP ZBLLs that are like this as I can (ideally all 84). For speedsolving with minimum pain of course; but I've always dreamed for an algset I can call "elegant" aside for <RrUD> CMLL and BH corners
 
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