minor changes to dp (added an LIS problem), added DP on DAG and some definitions
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@ -76,7 +76,7 @@ The following USACO problems don't fall into any of the categories below. Arrang
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* Reconsider the state.
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* USACO Gold
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* [Fruit Feast](http://www.usaco.org/index.php?page=viewproblem2&cpid=574)
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* straightforward
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* `dp[fullness] = whether you can achieve this fullness`
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* [Talent Show](http://www.usaco.org/index.php?page=viewproblem2&cpid=839)
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* binary search + knapsack on weight
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* CF
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@ -113,6 +113,8 @@ The following USACO problems don't fall into any of the categories below. Arrang
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(add?)
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* [LIS in Quadratic Time](https://leetcode.com/problems/longest-increasing-subsequence/)
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* Try to improve to $O(N\log N)$.
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* Try to improve to $O(N\log N)$: [Solution](https://cp-algorithms.com/sequences/longest_increasing_subsequence.html).
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* [Cowjog](http://www.usaco.org/index.php?page=viewproblem2&cpid=489)
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* Not so easy to see, but direct application of LIS.
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* [Sort It Out (USACO Platinum)](http://www.usaco.org/index.php?page=viewproblem2&cpid=865)
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* Challenging!
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@ -1,19 +1,15 @@
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---
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id: toposort
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title: "Topological Sort"
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author: Benjamin Qi
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author: Benjamin Qi, Michael Cao
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prerequisites:
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-
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- Gold - Breadth First Search
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---
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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.
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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.
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## Example Problems
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- [CSES Course Schedule](https://cses.fi/problemset/task/1679)
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- [CSES Longest Flight Route](https://cses.fi/problemset/task/1680)
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- [CSES Game Routes](https://cses.fi/problemset/task/1681)
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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.
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## Tutorial
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@ -24,11 +20,27 @@ A [topological sort](https://en.wikipedia.org/wiki/Topological_sorting) of a dir
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- [CSAcademy](https://csacademy.com/lesson/topological_sorting)
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- both BFS, DFS
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## Dynamic Programming
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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!
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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]`.
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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.
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## Example Problems
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- [CSES Course Schedule](https://cses.fi/problemset/task/1679)
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- [CSES Longest Flight Route](https://cses.fi/problemset/task/1680)
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- [CSES Game Routes](https://cses.fi/problemset/task/1681)
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## Problems
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- USACO Gold
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- [Timeline](http://www.usaco.org/index.php?page=viewproblem2&cpid=1017)
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- Dynamic Programming on DAG.
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- [Milking Order](http://www.usaco.org/index.php?page=viewproblem2&cpid=838)
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- 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.
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- Other
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- [Minimal Labels](http://codeforces.com/contest/825/problem/E) [](53)
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- [Quantum](https://open.kattis.com/contests/acpc17open/problems/quantumsuperposition) [](84)
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