usaco-guide/content/4_Silver/DFS.mdx

---
id: dfs
title: Depth First Search (DFS)
author: Siyong Huang
prerequisites: 
 - Bronze - Introduction to Graphs
description: A way to traverse a graph using recursion.
frequency: 4
---

import { Problem } from "../models";

export const metadata = {
  problems: {
    sample: [
      new Problem("CSES", "Building Roads", "1666", "Intro|Very Easy", false, ["DFS"]),
    ],
    tree: [
      new Problem("CSES", "Subordinates", "1674", "Very Easy", false, ["Tree", "DFS"]),
      new Problem("Silver", "Mootube", "788", "Easy", false, ["Tree", "DFS"]),
      new Problem("CF", "Journey", "contest/839/problem/C", "Easy", false, ["Tree", "DFS"]),
      new Problem("CF", "PolandBall & Forest", "problemset/problem/755/C", "Easy", false, ["Tree", "DFS"]),
      new Problem("CSES", "Tree Diameter", "1131", "Normal", false, ["Tree", "DFS"]),
      new Problem("CSES", "Tree Distances I", "1132", "Normal", false, ["Tree", "DFS"]),
      new Problem("CSES", "Tree Distances II", "1133", "Normal", false, ["Tree", "DFS"]),
      new Problem("CF", "Wizard's Tour", "contest/860/problem/D", "Normal", false, ["Tree", "DFS"]),
      new Problem("POI", "Hotels", "https://szkopul.edu.pl/problemset/problem/gDw3iFkeVm7ZA3j_16-XR7jI/site/?key=statement", "Normal", false, ["Tree", "DFS"]),
      new Problem("HE", "Birthday Gifts", "https://www.hackerearth.com/practice/math/combinatorics/inclusion-exclusion/practice-problems/algorithm/mancunian-and-birthday-gifts-d44faa15/description/", "Normal", false, ["Tree", "PIE"], ""),
      new Problem("CSA", "Tree Construction", "contest/860/problem/D", "Hard", false, ["Tree", "DFS"], "several cases"),
    ],
    general: [
      new Problem("CF", "Bear & Friendship", "problemset/problem/771/A", "Easy", false, ["DFS"]),
      new Problem("Silver", "Closing the Farm", "644", "Easy", false, ["DFS"]),
      new Problem("Silver", "Moocast", "668", "Easy", false, ["DFS"]),
      new Problem("Silver", "Fence Planning", "944", "Easy", false, ["DFS"]),
      new Problem("Kattis", "Birthday Party", "birthday", "Easy", false, ["DFS"], "DFS with each edge removed"),
      new Problem("Silver", "Milk Visits", "968", "Normal", false, ["DFS"]),
      new Problem("Silver", "Milk Pails", "620", "Normal", false, ["DFS"]),
      new Problem("Silver", "Wormhole Sort", "992", "Normal", false, ["DFS", "Binary Search"]),
      new Problem("Silver", "Moo Particle", "1040", "Normal", false, ["Sorting"]),
    ],
    bipsample: [
      new Problem("CSES", "Building Teams", "1668", "Easy", false, ["Bipartite"]),
    ],    
    bip: [
      new Problem("CF", "Bipartiteness", "contest/862/problem/B", "Easy", false, ["Bipartite"]),
      new Problem("Silver", "The Great Revegetation", "920", "Easy", false, ["Bipartite"]),
    ],
  }
};


<problems-list problems={metadata.problems.sample} />

## Tutorial

<resources>
  <resource source="CPH" title="12.1 - DFS, 14 - Tree algorithms" starred></resource>
  <resource source="CSA" title="Depth First Search" url="depth_first_search" starred>interactive</resource>
  <resource source="IUSACO" title="10.4 - Graph Traversal Algorithms"></resource>
  <resource source="PAPS" title="12.2 - Depth-First Search">with an example problem</resource>
  <resource source="CPC" title="7 - Graphs 1" url="07_graphs_1">fast-paced</resource>
  <resource source="cp-algo" title="Depth First Search" url="graph/depth-first-search.html">hard to parse for a beginner</resource>
  <resource source="TC" title="Graphs Section 2" url="introduction-to-graphs-and-their-data-structures-section-2"></resource>
</resources>

### Problems

 - Trees

<problems-list problems={metadata.problems.tree} />

 - General

<problems-list problems={metadata.problems.general} />

## Graph Two-Coloring

*Graph two-coloring* refers to 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.

<problems-list problems={metadata.problems.bipsample} />

### Tutorial

<resources>
  <resource source="IUSACO" title="10.7 - Bipartite Graphs" starred></resource>
  <resource source="CPH" title="12.3 - Graph Traversal: Applications"></resource>
  <resource source="cp-algo" title="Bipartite Check" url="graph/bipartite-check.html">Uses BFS, but DFS accomplishes the same task.</resource>
</resources>
 
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);
    }
}
```

### Problems

<problems-list problems={metadata.problems.bip} />