minor changes to dp (added an LIS problem), added DP on DAG and some definitions

content/5_Gold/DP.md

@@ -76,7 +76,7 @@ The following USACO problems don't fall into any of the categories below. Arrang
       * Reconsider the state.
  * USACO Gold
    * [Fruit Feast](http://www.usaco.org/index.php?page=viewproblem2&cpid=574)
-     * straightforward
+     * `dp[fullness] = whether you can achieve this fullness` 
    * [Talent Show](http://www.usaco.org/index.php?page=viewproblem2&cpid=839)
      * binary search + knapsack on weight
  * CF
@@ -113,6 +113,8 @@ The following USACO problems don't fall into any of the categories below. Arrang
 (add?)
 
  * [LIS in Quadratic Time](https://leetcode.com/problems/longest-increasing-subsequence/)
-    * Try to improve to $O(N\log N)$. 
+    * Try to improve to $O(N\log N)$: [Solution](https://cp-algorithms.com/sequences/longest_increasing_subsequence.html). 
+ * [Cowjog](http://www.usaco.org/index.php?page=viewproblem2&cpid=489)
+    * Not so easy to see, but direct application of LIS.
  * [Sort It Out (USACO Platinum)](http://www.usaco.org/index.php?page=viewproblem2&cpid=865)
     * Challenging!

content/5_Gold/TopoSort.md

@@ -1,19 +1,15 @@
 ---
 id: toposort
 title: "Topological Sort"
-author: Benjamin Qi
+author: Benjamin Qi, Michael Cao
 prerequisites: 
  - 
      - Gold - Breadth First Search
 ---
 
-A [topological sort](https://en.wikipedia.org/wiki/Topological_sorting) of a directed graph is a linear ordering of its vertices such that for every directed edge $u\to v$ from vertex $u$ to vertex $v$, $u$ comes before $v$ in the ordering. 
+To review, a **directed** graph consists of edges that can only be traversed in one direction. Additionally, a **acyclic** graph defines a graph which does not contain cycles, meaning you are unable to traverse across one or more edges and return to the node you started on. Putting these definitions together, a **directed acyclic** graph, sometimes abbreviated as DAG, is a graph which has edges which can only be traversed in one direction and does not contain cycles. 
 
-## Example Problems
-
- - [CSES Course Schedule](https://cses.fi/problemset/task/1679)
- - [CSES Longest Flight Route](https://cses.fi/problemset/task/1680)
- - [CSES Game Routes](https://cses.fi/problemset/task/1681)
+A [topological sort](https://en.wikipedia.org/wiki/Topological_sorting) of a directed acyclic graph is a linear ordering of its vertices such that for every directed edge $u\to v$ from vertex $u$ to vertex $v$, $u$ comes before $v$ in the ordering. 
 
 ## Tutorial
 
@@ -23,12 +19,28 @@ A [topological sort](https://en.wikipedia.org/wiki/Topological_sorting) of a dir
    - DFS
  - [CSAcademy](https://csacademy.com/lesson/topological_sorting)
    - both BFS, DFS
+   
+## Dynamic Programming
+One useful property of directed acyclic graphs is, as the name suggests, that there exists no cycles. If we consider each node in the graph as a state, we can perform dynamic programming on the graph if we process the states in an order that guarantees for every edge, $u\to v$ that $u$ is processed before $v$. Fortunately, this is the exact definition of a topological sort!
+
+Let's consider a classical problem (see Longest Flight Route) where we must find the longest path in a Directed Acyclic Graph. Let `dp[curr] = longest path ending at the node curr`. Then, if we process states in topological order, the transition is relatively straightforward: `dp[curr] = max of all dp[prev] where prev represents a node with an edge going into the current node` (word better?). To reiterate, since the states a processed in topological order, we can guarantee that all possible `dp[prev]` are computed before we compute `dp[curr]`.
+
+However, not all problems clearly give you directed acyclic graphs (see [Cave Paintings](http://usaco.org/index.php?page=viewproblem2&cpid=996)). An important step in many problems is to reduce the statement into a directed acyclic graph. See the editorial of the linked problem for more information.
+
+## Example Problems
+
+ - [CSES Course Schedule](https://cses.fi/problemset/task/1679)
+ - [CSES Longest Flight Route](https://cses.fi/problemset/task/1680)
+ - [CSES Game Routes](https://cses.fi/problemset/task/1681)
+
 
 ## Problems
 
  - USACO Gold
    - [Timeline](http://www.usaco.org/index.php?page=viewproblem2&cpid=1017)
+     - Dynamic Programming on DAG.
    - [Milking Order](http://www.usaco.org/index.php?page=viewproblem2&cpid=838)
+     - Binary search and check if a valid topological sort exists. Consider [Khan's Algorithm](https://en.wikipedia.org/wiki/Topological_sorting#Kahn's_algorithm) for topological sorting.
  - Other
    - [Minimal Labels](http://codeforces.com/contest/825/problem/E) [](53)
-   - [Quantum](https://open.kattis.com/contests/acpc17open/problems/quantumsuperposition) [](84)
+   - [Quantum](https://open.kattis.com/contests/acpc17open/problems/quantumsuperposition) [](84)