---
slug: /silver/dfs
title: "DFS"
author: Siyong Huang
order: 7
---
- Prerequisites
- Depth First Search (DFS)
- Flood Fill
- Graph Two-Coloring
- Cycle Detection
## Overview
- Prerequisites
- Depth First Search (DFS)
- Flood Fill
- Graph Two-Coloring
- Cycle Detection
## Prerequisites
- [CSAcademy Graph Intro](https://csacademy.com/lesson/introduction_to_graphs)
- [CSAcademy Graph Representations](https://csacademy.com/lesson/graph_representation)
- Note: DFS is most commonly implemented with adjacency lists.
- CPH 11
## Depth First Search (DFS)
*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.
- [CSES Building Roads](https://cses.fi/problemset/task/1666)
### Tutorial
- Recommended:
- CPH 12.1
- [CSAcademy DFS](https://csacademy.com/lesson/depth_first_search/)
- Additional:
- [cp-algo DFS](https://cp-algorithms.com/graph/depth-first-search.html)
- hard to parse if this is your first time learning about DFS
- [CPC.7](https://github.com/SuprDewd/T-414-AFLV/tree/master/07_graphs_1)
### Problems
- [Mootube, Silver (Easy)](http://usaco.org/index.php?page=viewproblem2&cpid=788)
- [Closing the Barn, Silver (Easy)](http://usaco.org/index.php?page=viewproblem2&cpid=644)
- [Moocast, Silver (Easy)](http://usaco.org/index.php?page=viewproblem2&cpid=668)
- [Pails (Normal)](http://usaco.org/index.php?page=viewproblem2&cpid=620)
- [Milk Visits (Normal)](http://www.usaco.org/index.php?page=viewproblem2&cpid=968)
## Flood Fill
[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.
- [CSES Counting Rooms](https://cses.fi/problemset/task/1192)
- [CSES Labyrinth](https://cses.fi/problemset/task/1193)
### Tutorial
- Recommended:
- Ch 10 of https://www.overleaf.com/project/5e73f65cde1d010001224d8a
### Problems
- [Ice Perimeter (Easy)](http://usaco.org/index.php?page=viewproblem2&cpid=895)
- [Switching on the Lights (Moderate)](http://www.usaco.org/index.php?page=viewproblem2&cpid=570)
- [Build Gates (Moderate)](http://www.usaco.org/index.php?page=viewproblem2&cpid=596)
- [Why Did the Cow Cross the Road III, Silver (Moderate)](http://usaco.org/index.php?page=viewproblem2&cpid=716)
- [Multiplayer Moo (Hard)](http://usaco.org/index.php?page=viewproblem2&cpid=836)
## Graph Two-Coloring
*Graph two-colorings* is assigning a boolean value to each node of the graph, dictated by the edge configuration
The most common example of a two-colored graph is a *bipartite graph*, in which each edge connects two nodes of opposite colors.
- [CSES Building Teams](https://cses.fi/problemset/task/1668)
### Tutorial
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.
```cpp
//UNTESTED
bool is_bipartite = true;
void dfs(int node)
{
visited[node] = true;
for(int u:adj_list[node])
if(visited[u])
{
if(color[u] == color[node])
is_bipartite = false;
}
else
{
color[u] = !color[node];
dfs(u);
}
}
```
- Additional:
- [Bipartite Graphs: cp-alg bipartite check](https://cp-algorithms.com/graph/bipartite-check.html)
- Note: CP-Algorithms uses BFS, but DFS accomplishes the same task
### Problems
- [The Great Revegetation (Normal)](http://usaco.org/index.php?page=viewproblem2&cpid=920)
## Cycle Detection
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.
A related topic is **strongly connected components**, a platinum level concept.
### Functional Graphs
Links:
- CPH 16.3: successor paths
- CPH 16.4: cycle detection in successor graph
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.**
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.
```cpp
//UNTESTED
//Each node points to next_node[node]
bool visited[MAXN], on_stack[MAXN];
int number_of_cycles = 0, next_node[MAXN];
void dfs(int n)
{
visited[n] = on_stack[n] = true;
int u = next_node[n];
if(on_stack[u])
number_of_cycles++;
else if(!visited[u])
dfs(u);
on_stack[n] = false;
}
int main()
{
//read input, etc
for(int i = 1;i <= N;i++)
if(!visited[i])
dfs(i);
}
```
The same general idea is implemented below to find any cycle in a directed graph (if one exists).
```cpp
//UNTESTED
bool visited[MAXN], on_stack[MAXN];
vector adj[MAXN];
vector cycle;
bool dfs(int n)
{
visited[n] = on_stack[n] = true;
for(int u:adj[n])
{
if(on_stack[u])
return cycle.push_back(v), cycle.push_back(u), on_stack[n] = on_stack[u] = false, true;
else if(!visited[u])
{
if(dfs(u))
if(on_stack[n])
return cycle.push_back(n), on_stack[n] = false, true;
else
return false;
if(!cycle.empty())
return false;
}
}
on_stack[n] = false;
return false;
}
int main()
{
//take input, etc
for(int i = 1;cycle.empty() && i <= N;i++)
dfs(i);
if(cycle.empty())
printf("No cycle found!\n");
else
{
reverse(cycle.begin(), cycle.end());
printf("Cycle of length %u found!\n", cycle.size());
for(int n : cycle) printf("%d ", n);
printf("\n");
}
}
```
### Problems
- [Codeforces 1020B. Badge (Very Easy)](https://codeforces.com/contest/1020/problem/B)
- Try to solve the problem in O(N)!
- [The Bovine Shuffle (Normal)](http://usaco.org/index.php?page=viewproblem2&cpid=764)
- [Swapity Swapity Swap (Very Hard)](http://www.usaco.org/index.php?page=viewproblem2&cpid=1014)
- [CSES Round Trip (undirected)](https://cses.fi/problemset/task/1669)
- [CSES Round Trip II (directed)](https://cses.fi/problemset/task/1678)