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| id | title | author | prerequisites | description | ||
|---|---|---|---|---|---|---|
| toposort | Topological Sort | Benjamin Qi, Michael Cao |
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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.
Topological Sort
A topological sort 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
- CPH 16.1, 16.2
- DFS
- cp-algorithms
- DFS
- CSAcademy
- both BFS, DFS
Consider Khan's Algorithm for topological sorting.
Problems
- Milking Order
- Binary search and check if a valid topological sort exists.
- CSES Course Schedule II
- Tricky!
- equivalent to Minimal Labels
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 the classical problem CSES Longest Flight Route, where we must find the longest path in a DAG.
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 (ex. Plat - Cave Paintings). 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.
(Ben - this last paragraph doesn't seem very helpful.)
Problems
- CSES Game Routes
- counting paths on DAG
- Quantum
- enumerating paths on DAG
- USACO Gold - Timeline
- not explicitly given, but graph is a DAG