66 lines
3.4 KiB
Markdown
66 lines
3.4 KiB
Markdown
---
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id: toposort
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title: "Topological Sort"
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author: Benjamin Qi, Michael Cao
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prerequisites:
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- Gold - Breadth First Search
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- Gold - Introduction to Dynamic Programming
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description: "?"
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---
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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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## Topological Sort
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- [CSES Course Schedule](https://cses.fi/problemset/task/1679)
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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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- CPH 16.1, 16.2
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- DFS
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- [cp-algorithms](https://cp-algorithms.com/graph/topological-sort.html)
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- DFS
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- [CSAcademy](https://csacademy.com/lesson/topological_sorting)
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- both BFS, DFS
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Consider [Khan's Algorithm](https://en.wikipedia.org/wiki/Topological_sorting#Kahn's_algorithm) for topological sorting.
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### Problems
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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.
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- [CSES Course Schedule II](https://cses.fi/problemset/task/1757)
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- Tricky!
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## Dynamic Programming
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- [PAPS 9.1](https://www.csc.kth.se/~jsannemo/slask/main.pdf)
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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 the classical problem [CSES Longest Flight Route](https://cses.fi/problemset/task/1680), where we must find the longest path in a DAG.
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<spoiler title="Solution">
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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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</spoiler>
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However, not all problems clearly give you directed acyclic graphs (ex. [Plat - 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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(Ben - this last paragraph doesn't seem very helpful.)
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## Problems
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- [CSES Game Routes](https://cses.fi/problemset/task/1681)
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- counting paths on DAG
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- [Quantum](https://open.kattis.com/contests/acpc17open/problems/quantumsuperposition)
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- enumerating paths on DAG
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- [USACO Gold - Timeline](http://www.usaco.org/index.php?page=viewproblem2&cpid=1017)
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- not explicitly given, but graph is a DAG
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## Problems
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- Other
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- [Minimal Labels](http://codeforces.com/contest/825/problem/E) [](53)
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