advent_of_code/2019/day18_many-worlds_interpretation/challenge.txt

As you approach Neptune, a planetary security system detects you and activates a giant [tractor beam](https://en.wikipedia.org/wiki/Tractor_beam) on [Triton](https://en.wikipedia.org/wiki/Triton_(moon))! You have no choice but to land.

A scan of the local area reveals only one interesting feature: a massive underground vault. You generate a map of the tunnels (your puzzle input). The tunnels are too narrow to move diagonally.

Only one *entrance* (marked `@`) is present among the *open passages* (marked `.`) and *stone walls* (`#`), but you also detect an assortment of *keys* (shown as lowercase letters) and *doors* (shown as uppercase letters). Keys of a given letter open the door of the same letter: `a` opens `A`, `b` opens `B`, and so on. You aren't sure which key you need to disable the tractor beam, so you'll need to *collect all of them*.

For example, suppose you have the following map:

```
#########
#b.A.@.a#
#########

```

Starting from the entrance (`@`), you can only access a large door (`A`) and a key (`a`). Moving toward the door doesn't help you, but you can move `2` steps to collect the key, unlocking `A` in the process:

```
#########
#b.....@#
#########

```

Then, you can move `6` steps to collect the only other key, `b`:

```
#########
#@......#
#########

```

So, collecting every key took a total of `*8*` steps.

Here is a larger example:

```
########################
#f.D.E.e.C.b.A.@.a.B.c.#
######################.#
#d.....................#
########################

```

The only reasonable move is to take key `a` and unlock door `A`:

```
########################
#f.D.E.e.C.b.....@.B.c.#
######################.#
#d.....................#
########################

```

Then, do the same with key `b`:

```
########################
#f.D.E.e.C.@.........c.#
######################.#
#d.....................#
########################

```

...and the same with key `c`:

```
########################
#f.D.E.e.............@.#
######################.#
#d.....................#
########################

```

Now, you have a choice between keys `d` and `e`. While key `e` is closer, collecting it now would be slower in the long run than collecting key `d` first, so that's the best choice:

```
########################
#f...E.e...............#
######################.#
#@.....................#
########################

```

Finally, collect key `e` to unlock door `E`, then collect key `f`, taking a grand total of `*86*` steps.

Here are a few more examples:

* ```
  ########################
  #...............b.C.D.f#
  #.######################
  #.....@.a.B.c.d.A.e.F.g#
  ########################

  ```

  Shortest path is `132` steps: `b`, `a`, `c`, `d`, `f`, `e`, `g`

* ```
  #################
  #i.G..c...e..H.p#
  ########.########
  #j.A..b...f..D.o#
  ########@########
  #k.E..a...g..B.n#
  ########.########
  #l.F..d...h..C.m#
  #################

  ```

  Shortest paths are `136` steps;  
  one is: `a`, `f`, `b`, `j`, `g`, `n`, `h`, `d`, `l`, `o`, `e`, `p`, `c`, `i`, `k`, `m`

* ```
  ########################
  #@..............ac.GI.b#
  ###d#e#f################
  ###A#B#C################
  ###g#h#i################
  ########################

  ```

  Shortest paths are `81` steps; one is: `a`, `c`, `f`, `i`, `d`, `g`, `b`, `e`, `h`

*How many steps is the shortest path that collects all of the keys?*

Your puzzle answer was `5182`.

\--- Part Two ---
----------

You arrive at the vault only to discover that there is not one vault, but *four* - each with its own entrance.

On your map, find the area in the middle that looks like this:

```
...
.@.
...

```

Update your map to instead use the correct data:

```
@#@
###
@#@

```

This change will split your map into four separate sections, each with its own entrance:

```
#######       #######
#a.#Cd#       #a.#Cd#
##...##       ##@#@##
##.@.##  -->  #######
##...##       ##@#@##
#cB#Ab#       #cB#Ab#
#######       #######

```

Because some of the keys are for doors in other vaults, it would take much too long to collect all of the keys by yourself. Instead, you deploy four remote-controlled robots. Each starts at one of the entrances (`@`).

Your goal is still to *collect all of the keys in the fewest steps*, but now, each robot has its own position and can move independently. You can only remotely control a single robot at a time. Collecting a key instantly unlocks any corresponding doors, regardless of the vault in which the key or door is found.

For example, in the map above, the top-left robot first collects key `a`, unlocking door `A` in the bottom-right vault:

```
#######
#@.#Cd#
##.#@##
#######
##@#@##
#cB#.b#
#######

```

Then, the bottom-right robot collects key `b`, unlocking door `B` in the bottom-left vault:

```
#######
#@.#Cd#
##.#@##
#######
##@#.##
#c.#.@#
#######

```

Then, the bottom-left robot collects key `c`:

```
#######
#@.#.d#
##.#@##
#######
##.#.##
#@.#.@#
#######

```

Finally, the top-right robot collects key `d`:

```
#######
#@.#.@#
##.#.##
#######
##.#.##
#@.#.@#
#######

```

In this example, it only took `*8*` steps to collect all of the keys.

Sometimes, multiple robots might have keys available, or a robot might have to wait for multiple keys to be collected:

```
###############
#d.ABC.#.....a#
######@#@######
###############
######@#@######
#b.....#.....c#
###############

```

First, the top-right, bottom-left, and bottom-right robots take turns collecting keys `a`, `b`, and `c`, a total of `6 + 6 + 6 = 18` steps. Then, the top-left robot can access key `d`, spending another `6` steps; collecting all of the keys here takes a minimum of `*24*` steps.

Here's a more complex example:

```
#############
#DcBa.#.GhKl#
#.###@#@#I###
#e#d#####j#k#
###C#@#@###J#
#fEbA.#.FgHi#
#############

```

* Top-left robot collects key `a`.
* Bottom-left robot collects key `b`.
* Top-left robot collects key `c`.
* Bottom-left robot collects key `d`.
* Top-left robot collects key `e`.
* Bottom-left robot collects key `f`.
* Bottom-right robot collects key `g`.
* Top-right robot collects key `h`.
* Bottom-right robot collects key `i`.
* Top-right robot collects key `j`.
* Bottom-right robot collects key `k`.
* Top-right robot collects key `l`.

In the above example, the fewest steps to collect all of the keys is `*32*`.

Here's an example with more choices:

```
#############
#g#f.D#..h#l#
#F###e#E###.#
#dCba@#@BcIJ#
#############
#nK.L@#@G...#
#M###N#H###.#
#o#m..#i#jk.#
#############

```

One solution with the fewest steps is:

* Top-left robot collects key `e`.
* Top-right robot collects key `h`.
* Bottom-right robot collects key `i`.
* Top-left robot collects key `a`.
* Top-left robot collects key `b`.
* Top-right robot collects key `c`.
* Top-left robot collects key `d`.
* Top-left robot collects key `f`.
* Top-left robot collects key `g`.
* Bottom-right robot collects key `k`.
* Bottom-right robot collects key `j`.
* Top-right robot collects key `l`.
* Bottom-left robot collects key `n`.
* Bottom-left robot collects key `m`.
* Bottom-left robot collects key `o`.

This example requires at least `*72*` steps to collect all keys.

After updating your map and using the remote-controlled robots, *what is the fewest steps necessary to collect all of the keys?*

Your puzzle answer was `2154`.

Both parts of this puzzle are complete! They provide two gold stars: \*\*

At this point, you should [return to your Advent calendar](/2019) and try another puzzle.

If you still want to see it, you can [get your puzzle input](18/input).