86 lines
2.5 KiB
Text
86 lines
2.5 KiB
Text
---
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id: func-graphs
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title: Functional Graphs
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author: Siyong Huang
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prerequisites:
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- dfs
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description: "Properties of a functional graphs, directed graphs in which every vertex has exactly one outgoing edge."
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frequency: 1
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---
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import { Problem } from "../models";
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export const metadata = {
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problems: {
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sample: [
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new Problem("CF", "Div 2 B - Badge", "contest/1020/problem/B", "Very Easy", false, ["Func Graph"], "Try to solve the problem in $O(N)$!"),
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],
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general: [
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new Problem("Silver", "The Bovine Shuffle", "764", "Normal", false, ["Func Graph"], "Try to solve the problem in $O(N)$!"),
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new Problem("CSES", "Planets Cycles", "1751", "Normal", false, ["Func Graph"], ""),
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new Problem("Silver", "Swapity Swapity Swap", "1014", "Normal", false, ["Permutation"], ""),
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new Problem("POI", "Mafia", "https://szkopul.edu.pl/problemset/problem/w3YAoAT3ej27YeiaNWjK57_G/site/?key=statement", "Hard", false, ["Func Graph"], ""),
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new Problem("POI", "Spies", "https://szkopul.edu.pl/problemset/problem/r6tMTfvQFPAEfQioYMCQndQe/site/?key=statement", "Hard", false, [], ""),
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new Problem("POI", "Frog", "https://szkopul.edu.pl/problemset/problem/qDH9CkBHZKHY4vbKRBlXPrA7/site/?key=statement", "Hard", false, [], ""),
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],
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}
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};
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We'll consider graphs like the one presented in this problem:
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<problems-list problems={metadata.problems.sample} />
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< br/>
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Aka **successor graph**.
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<resources>
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<resource source="CPH" title="16.3, 16.4 - Successor Graphs"></resource>
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</resources>
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## Implementation
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The following sample code counts the number of cycles in such a graph. The "stack" contains nodes that can reach the current node. If the current node points to a node `v` on the stack (`on_stack[v]` is true), then we know that a cycle has been created. However, if the current node points to a node `v` that has been previously visited but is not on the stack, then we know that the current chain of nodes points into a cycle that has already been considered.
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<LanguageSection>
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<CPPSection>
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```cpp
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//UNTESTED
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//Each node points to next_node[node]
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bool visited[MAXN], on_stack[MAXN];
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int number_of_cycles = 0, next_node[MAXN];
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void dfs(int n)
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{
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visited[n] = on_stack[n] = true;
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int u = next_node[n];
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if(on_stack[u])
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number_of_cycles++;
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else if(!visited[u])
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dfs(u);
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on_stack[n] = false;
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}
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int main()
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{
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//read input, etc
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for(int i = 1;i <= N;i++)
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if(!visited[i])
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dfs(i);
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}
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```
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</CPPSection>
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</LanguageSection>
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(floyd's algo?)
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<IncompleteSection />
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## Problems
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<problems-list problems={metadata.problems.general} /> |