Merge pull request #256 from cpinitiative/content/commuter-pass-sol

content: add commuter pass solution notes

Author: thecodingwizard

solutions/JOI 18-commuter-pass.mdx

@@ -1,13 +1,22 @@
 ---
 id: joi-18-commuter-pass
 title: JOI 2018 Commuter Pass
-author: Andi Qu
+author: Andi Qu, Nathan Wang
 ---
 
-<IncompleteSection />
+1. Use Dijkstra's to calculate distances from $s$, $u$, and $v$.
+2. Consider a subset of directed edges where each directed edge is part of a shortest path from $s$ to $t$. Note that this subset is a DAG, and can be found by keeping track of the parents of every node when running Dijkstra's.
+3. Note that an optimal path will be $u \rightarrow x \rightarrow y \rightarrow v$, where $x \rightarrow y$ is a path on the DAG. Note that any path on the DAG is a valid commuter pass option, and the only node with in-degree 0 is $u$, and the only node with out-degree 0 is $t$ in the DAG.
+4. Define `dpU(i)` = minimum distance from $u$ to any node $x$ such that $x$ is in a path from $s$ to $i$ in the DAG.
+5. Define `dpV(i)` = minimum distance from $v$ to any node $x$ such that $x$ is in a path from $s$ to $i$ in the DAG.
+6. The DP transition for `dpU(i)` is `dpU(i) = min(distU[i], dpU(parent_i))` for each parent of `i` (see `dijkstra2` function implementation for more details). `dpV`'s transition is defined similarly.
+7. Our answer is `dpU(v) + dpV(v)`. Since edges are bidirectional, an optimal path can also be $v \rightarrow x \rightarrow y \rightarrow u$ where $x \rightarrow y$ is a path on the DAG. We can solve this by rerunning steps 2-7 while switching $u$ and $v$. In Andi's code, steps 2-7 are implemented in the `dijkstra2` function.
+8. Alternatively, the answer could just be from $u$ directly to $v$ without using the commuter pass.
 
 ## Code
 
+Andi's implementation:
+
 ```cpp
 #include <bits/stdc++.h>
 typedef long long ll;
@@ -90,4 +99,6 @@ int main() {
     cout << ans << '\n';
     return 0;
 }
-```
+```
+
+Alternatively, [Nathan's implementation](https://github.com/thecodingwizard/competitive-programming/blob/master/JOI/JOI%2018-commuterpass.cpp). Note that the DP definitions in Nathan's implementation are slightly different than above.