225 lines
7.8 KiB
Markdown
225 lines
7.8 KiB
Markdown
---
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id: dfs
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title: Depth First Search
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author: Siyong Huang
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---
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- Introduction to Graphs
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- Depth First Search (DFS)
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- Flood Fill
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- Graph Two-Coloring
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- Cycle Detection
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<!-- END DESCRIPTION -->
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## Introduction to Graphs
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- Recommended
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- CPH 11
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- [CSAcademy Graph Intro](https://csacademy.com/lesson/introduction_to_graphs)
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- [CSAcademy Graph Representations](https://csacademy.com/lesson/graph_representation)
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- Usually, adjacency lists are used.
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- Additional
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- [Topcoder Graphs Pt 1](https://www.topcoder.com/community/data-science/data-science-tutorials/introduction-to-graphs-and-their-data-structures-section-1/)
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- [Topcoder Graphs Pt 2](https://www.topcoder.com/community/data-science/data-science-tutorials/introduction-to-graphs-and-their-data-structures-section-2/)
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## Depth First Search (DFS)
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*Depth First Search*, more commonly DFS, is a fundamental graph algorithm that traverses an entire connected component. The rest of this document describes various applications of DFS. Of course, it is one possible way to implement flood fill. *Breadth first search* (BFS) is **not** required for silver.
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- [CSES Building Roads](https://cses.fi/problemset/task/1666)
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### Tutorial
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- Recommended:
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- CPH 12.1
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- [CSAcademy DFS](https://csacademy.com/lesson/depth_first_search/)
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- Additional:
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- [cp-algo DFS](https://cp-algorithms.com/graph/depth-first-search.html)
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- hard to parse if this is your first time learning about DFS
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- [CPC.7](https://github.com/SuprDewd/T-414-AFLV/tree/master/07_graphs_1)
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### Problems
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- CF
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- [PolandBall & Forest](http://codeforces.com/problemset/problem/755/C) [](56)
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- [Bear & Friendship](http://codeforces.com/problemset/problem/771/A)
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- [Journey](http://codeforces.com/contest/839/problem/C) [](54)
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- DFS on Tree
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- [Wizard's Tour](http://codeforces.com/contest/860/problem/D) [](59)
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- USACO
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- [Mootube, Silver (Easy)](http://usaco.org/index.php?page=viewproblem2&cpid=788)
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- [Closing the Barn, Silver (Easy)](http://usaco.org/index.php?page=viewproblem2&cpid=644)
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- [Moocast, Silver (Easy)](http://usaco.org/index.php?page=viewproblem2&cpid=668)
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- [Pails (Normal)](http://usaco.org/index.php?page=viewproblem2&cpid=620)
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- [Milk Visits (Normal)](http://www.usaco.org/index.php?page=viewproblem2&cpid=968)
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- Other
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- [POI Hotels](https://szkopul.edu.pl/problemset/problem/gDw3iFkeVm7ZA3j_16-XR7jI/site/?key=statement) [](61)
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- [Kattis Birthday Party (Easy)](https://open.kattis.com/problems/birthday)
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- DFS with each edge removed
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## Flood Fill
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[Flood Fill](https://en.wikipedia.org/wiki/Flood_fill) refers to finding the number of connected components in a graph, usually when the graph is a grid.
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- [CSES Counting Rooms](https://cses.fi/problemset/task/1192)
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- [CSES Labyrinth](https://cses.fi/problemset/task/1193)
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### Tutorial
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- Recommended:
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- Ch 10 of https://www.overleaf.com/project/5e73f65cde1d010001224d8a
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### Problems
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- [Ice Perimeter (Easy)](http://usaco.org/index.php?page=viewproblem2&cpid=895)
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- [Switching on the Lights (Normal)](http://www.usaco.org/index.php?page=viewproblem2&cpid=570)
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- [Build Gates (Normal)](http://www.usaco.org/index.php?page=viewproblem2&cpid=596)
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- [Why Did the Cow Cross the Road III, Silver (Normal)](http://usaco.org/index.php?page=viewproblem2&cpid=716)
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- [Multiplayer Moo (Hard)](http://usaco.org/index.php?page=viewproblem2&cpid=836)
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## Graph Two-Coloring
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*Graph two-colorings* is assigning a boolean value to each node of the graph, dictated by the edge configuration
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The most common example of a two-colored graph is a *bipartite graph*, in which each edge connects two nodes of opposite colors.
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- [CSES Building Teams](https://cses.fi/problemset/task/1668)
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### Tutorial
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The idea is that we can arbitrarily label a node and then run DFS. Every time we visit a new (unvisited) node, we set its color based on the edge rule. When we visit a previously visited node, check to see whether its color matches the edge rule. For example, an implementation of coloring a bipartite graph is shown below.
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```cpp
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//UNTESTED
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bool is_bipartite = true;
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void dfs(int node)
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{
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visited[node] = true;
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for(int u:adj_list[node])
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if(visited[u])
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{
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if(color[u] == color[node])
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is_bipartite = false;
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}
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else
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{
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color[u] = !color[node];
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dfs(u);
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}
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}
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```
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- Additional:
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- [Bipartite Graphs: cp-alg bipartite check](https://cp-algorithms.com/graph/bipartite-check.html)
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- Note: CP-Algorithms uses BFS, but DFS accomplishes the same task
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### Problems
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- [CF Bipartiteness](http://codeforces.com/contest/862/problem/B) [](49)
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- [The Great Revegetation (Normal)](http://usaco.org/index.php?page=viewproblem2&cpid=920)
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## Cycle Detection
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A *cycle* is a non-empty path of distinct edges that start and end at the same node. *Cycle detection* determines properties of cycles in a directed or undirected graph, such as whether each node of the graph is part of a cycle or just checking whether a cycle exists.
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A related topic is **strongly connected components**, a platinum level concept.
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### Functional Graphs
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Links:
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- CPH 16.3: successor paths
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- CPH 16.4: cycle detection in successor graph
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In silver-level directed cycle problems, it is generally the case that each node has exactly one edge going out of it. This is known as a **successor graph** or a **functional graph.**
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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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```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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The same general idea is implemented below to find any cycle in a directed graph (if one exists).
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```cpp
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//UNTESTED
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bool visited[MAXN], on_stack[MAXN];
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vector<int> adj[MAXN];
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vector<int> cycle;
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bool dfs(int n)
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{
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visited[n] = on_stack[n] = true;
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for(int u:adj[n])
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{
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if(on_stack[u])
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return cycle.push_back(v), cycle.push_back(u), on_stack[n] = on_stack[u] = false, true;
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else if(!visited[u])
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{
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if(dfs(u))
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if(on_stack[n])
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return cycle.push_back(n), on_stack[n] = false, true;
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else
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return false;
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if(!cycle.empty())
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return false;
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}
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}
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on_stack[n] = false;
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return false;
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}
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int main()
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{
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//take input, etc
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for(int i = 1;cycle.empty() && i <= N;i++)
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dfs(i);
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if(cycle.empty())
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printf("No cycle found!\n");
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else
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{
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reverse(cycle.begin(), cycle.end());
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printf("Cycle of length %u found!\n", cycle.size());
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for(int n : cycle) printf("%d ", n);
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printf("\n");
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}
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}
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```
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### Problems
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- [Codeforces 1020B. Badge (Very Easy)](https://codeforces.com/contest/1020/problem/B)
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- Try to solve the problem in O(N)!
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- [The Bovine Shuffle (Normal)](http://usaco.org/index.php?page=viewproblem2&cpid=764)
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- [Swapity Swapity Swap (Very Hard)](http://www.usaco.org/index.php?page=viewproblem2&cpid=1014)
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- [CSES Round Trip (undirected cycle)](https://cses.fi/problemset/task/1669)
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- [CSES Round Trip II (directed cycle)](https://cses.fi/problemset/task/1678)
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- POI
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- [Mafia](https://szkopul.edu.pl/problemset/problem/w3YAoAT3ej27YeiaNWjK57_G/site/?key=statement)
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- [Spies](https://szkopul.edu.pl/problemset/problem/r6tMTfvQFPAEfQioYMCQndQe/site/?key=statement)
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- [Frog](https://szkopul.edu.pl/problemset/problem/qDH9CkBHZKHY4vbKRBlXPrA7/site/?key=statement)
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