Dynamic Programming (DP) and Directed Acyclic Graphs (DAG)

Pre-Requisite:
What is Directed Graph? | Directed Graph meaning
Dynamic Programming (DP) Tutorial with Problems

Every Dynamic Programming problem can be represented as a Directed Acyclic Graph(DAG). The nodes of the DAG represent the subproblems and the edges represents the transitions between the subproblems.

Relationship Between Dynamic Programming (DP) and Directed Acyclic Graphs (DAG):

1. Subproblems and DAG Nodes:

• Dynamic Programming involves breaking down a problem into similar subproblems and these subproblems are solved independently and their result is used to solve the original problem.
• Nodes of Directed Acyclic Graph represents the subproblems. We can also say these nodes represent a state.

2. Transitions and DAG Edges:

• Like the Nodes represents subproblems in the DAG, similarly the edges represent transition.
• The transition from one subproblem to another is represented by an edge.

3. Cycles and Redundancy:

• The reason it’s a directed acyclic graph is that if there are cycles in the directed graph then some states would lead back to themselves after a series of transitions leading to redundancy.

Let’s take an example to understand the above points:

Problem: Finding Longest Path in Directed Acyclic Graphs (DAG):

The dp state and the transition in this case will be:

Let dp[v] denote the length of the longest path ending at the node v. Initialize all the positions in dp array as 1. Clearly,

For every edge from node u to node v, dp[v] = max (dp[u]+1)

At the end check for the maximum value in dp[] array, which will be the longest path in the DAG.

DP on Directed Graphs

What happens when the graph is not Directed Acyclic Graphs (DAG)?

If the graph contains a cycle, then some states would lead back to themselves after a series of transitions. This would lead to infinite calculations. Thus, answer will not exist if the graph is cyclic in this case. As shown in below image, we potentially keep traversing the cycle indefinitely, making it impossible to find the longest path.

DP on Directed Graphs

How to Solve Dynamic Programming (DP) Problems on Directed Graphs:

Let us consider few problems which will tell how to define the states and make the transition:

Count the number of different ways to reach a node:

Given a Directed Acyclic Graph consisting of N nodes and M edges. The task is to determine the number of different ways to reach Node N from Node 1.

Solution: The problem can be solved in following steps:

1. Define the States: As mentioned above each Nodes of Directed Acyclic Graph represents a state. Let dp[v] be the number of ways to reach Node N from the vertex v.

2. Make the Transitions: Like the Nodes represents states in the DAG, similarly the edges represent transition. Thus dp[v] can be calculated by:

dp[v] = Σdp[u], for every edge v->u

3. Topological Consideration: We process the nodes topologically i.e., if edge v->u exists then dp[u] should be computed before dp[v].

Hence ,dp[1] will be our final answer.

Below is the implementation of above approach:

C++

#include <bits/stdc++.h>
using namespace std;

// Recursive function to perform depth-first search
// and calculate the number of ways to reach Node N from
// vertex curr
int dfs(int n, vector<vector<int> >& adj, vector<int>& dp,
        vector<int>& vis, int curr)
{
    // If the node has already been visited, return its
    // stored value
    if (vis[curr] == 1)
        return dp[curr];

    // Mark the current node as visited
    vis[curr] = 1;

    // Traverse all adjacent nodes and recursively calculate
    // the number of ways
    for (auto x : adj[curr]) {
        if (!vis[x]) {
            dfs(n, adj, dp, vis, x);
        }

        // Update the number of ways to reach Node N from
        // the current node
        dp[curr] += dp[x];
    }

    // Return the calculated value for the current node
    return dp[curr];
}

// Function to calculate the number of ways to reach Node N
// from Node 1
int noOfWays(int n, vector<vector<int> >& edges)
{
    // Adjacency list to represent the directed acyclic
    // graph
    vector<vector<int> > adj(n + 1);

    // Populate the adjacency list using the provided edges
    for (vector<int> i : edges) {
        adj[i[0]].push_back(i[1]);
    }

    // Initialize dp array to store the number of ways to
    // reach Node N from each vertex
    vector<int> dp(n + 1, 0);

    // Initialize visited array to keep track of visited
    // nodes during DFS
    vector<int> vis(n + 1, 0);

    // Set the base case for Node N, i.e., there is one way
    // to reach N from N
    dp[n] = 1;

    // Mark Node N as visited
    vis[n] = 1;

    // Call the recursive DFS function to calculate the
    // number of ways
    int res = dfs(n, adj, dp, vis, 1);

    // Return the result, which represents the number of
    // ways to reach Node N from Node 1
    return res;
}

int main()
{
    // Example inputs
    int N = 5;
    vector<vector<int> > edges = {
        { 1, 2 }, { 1, 3 }, { 2, 4 }, { 3, 4 }, { 4, 5 }
    };

    // Function call
    int result = noOfWays(N, edges);

    // Output the result
    cout << "No. of ways to reach Node N from Node 1: "
         << result << endl;

    return 0;
}

Java

import java.util.ArrayList;
import java.util.List;

public class WaysToReachNodeN {

    // Recursive function to perform depth-first search
    // and calculate the number of ways to reach Node N from
    // vertex curr
    private static int dfs(int n, List<List<Integer>> adj, int[] dp,
                           int[] vis, int curr) {
        // If the node has already been visited, return its
        // stored value
        if (vis[curr] == 1)
            return dp[curr];

        // Mark the current node as visited
        vis[curr] = 1;

        // Traverse all adjacent nodes and recursively calculate
        // the number of ways
        for (int x : adj.get(curr)) {
            if (vis[x] == 0) {
                dfs(n, adj, dp, vis, x);
            }

            // Update the number of ways to reach Node N from
            // the current node
            dp[curr] += dp[x];
        }

        // Return the calculated value for the current node
        return dp[curr];
    }

    // Function to calculate the number of ways to reach Node N
    // from Node 1
    private static int noOfWays(int n, List<List<Integer>> edges) {
        // Adjacency list to represent the directed acyclic
        // graph
        List<List<Integer>> adj = new ArrayList<>(n + 1);

        // Populate the adjacency list using the provided edges
        for (int i = 0; i <= n; i++) {
            adj.add(new ArrayList<>());
        }

        for (List<Integer> edge : edges) {
            adj.get(edge.get(0)).add(edge.get(1));
        }

        // Initialize dp array to store the number of ways to
        // reach Node N from each vertex
        int[] dp = new int[n + 1];

        // Initialize visited array to keep track of visited
        // nodes during DFS
        int[] vis = new int[n + 1];

        // Set the base case for Node N, i.e., there is one way
        // to reach N from N
        dp[n] = 1;

        // Mark Node N as visited
        vis[n] = 1;

        // Call the recursive DFS function to calculate the
        // number of ways
        int res = dfs(n, adj, dp, vis, 1);

        // Return the result, which represents the number of
        // ways to reach Node N from Node 1
        return res;
    }

    public static void main(String[] args) {
        // Example inputs
        int N = 5;
        List<List<Integer>> edges = List.of(
                List.of(1, 2), List.of(1, 3), List.of(2, 4), List.of(3, 4), List.of(4, 5)
        );

        // Function call
        int result = noOfWays(N, edges);

        // Output the result
        System.out.println("No. of ways to reach Node N from Node 1: " + result);
    }
}

C#

using System;
using System.Collections.Generic;

class Program
{
    // Recursive function to perform depth-first search
    // and calculate the number of ways to reach Node N from
    // vertex curr
    static int DFS(int n, List<List<int>> adj, List<int> dp,
                   List<int> vis, int curr)
    {
        // If the node has already been visited, return its
        // stored value
        if (vis[curr] == 1)
            return dp[curr];

        // Mark the current node as visited
        vis[curr] = 1;

        // Traverse all adjacent nodes and recursively calculate
        // the number of ways
        foreach (int x in adj[curr])
        {
            if (vis[x] == 0)
                DFS(n, adj, dp, vis, x);

            // Update the number of ways to reach Node N from
            // the current node
            dp[curr] += dp[x];
        }

        // Return the calculated value for the current node
        return dp[curr];
    }

    // Function to calculate the number of ways to reach Node N
    // from Node 1
    static int NoOfWays(int n, List<List<int>> edges)
    {
        // Adjacency list to represent the directed acyclic
        // graph
        List<List<int>> adj = new List<List<int>>(n + 1);

        // Initialize adjacency list
        for (int i = 0; i <= n; i++)
        {
            adj.Add(new List<int>());
        }

        // Populate the adjacency list using the provided edges
        foreach (List<int> i in edges)
        {
            adj[i[0]].Add(i[1]);
        }

        // Initialize dp array to store the number of ways to
        // reach Node N from each vertex
        List<int> dp = new List<int>(new int[n + 1]);

        // Initialize visited array to keep track of visited
        // nodes during DFS
        List<int> vis = new List<int>(new int[n + 1]);

        // Set the base case for Node N, i.e., there is one way
        // to reach N from N
        dp[n] = 1;

        // Mark Node N as visited
        vis[n] = 1;

        // Call the recursive DFS function to calculate the
        // number of ways
        int res = DFS(n, adj, dp, vis, 1);

        // Return the result, which represents the number of
        // ways to reach Node N from Node 1
        return res;
    }

    static void Main()
    {
        // Example inputs
        int N = 5;
        List<List<int>> edges = new List<List<int>>
        {
            new List<int> {1, 2},
            new List<int> {1, 3},
            new List<int> {2, 4},
            new List<int> {3, 4},
            new List<int> {4, 5}
        };

        // Function call
        int result = NoOfWays(N, edges);

        // Output the result
        Console.WriteLine($"No. of ways to reach Node N from Node 1: {result}");
    }
}

Javascript

// Function to perform depth-first search
// and calculate the number of ways to reach Node N from vertex curr
function dfs(n, adj, dp, vis, curr) {
    // If the node has already been visited, return its stored value
    if (vis[curr] === 1)
        return dp[curr];

    // Mark the current node as visited
    vis[curr] = 1;

    // Traverse all adjacent nodes and recursively calculate the number of ways
    for (let x of adj[curr]) {
        if (vis[x] === 0) {
            dfs(n, adj, dp, vis, x);
        }

        // Update the number of ways to reach Node N from the current node
        dp[curr] += dp[x];
    }

    // Return the calculated value for the current node
    return dp[curr];
}

// Function to calculate the number of ways to reach Node N from Node 1
function noOfWays(n, edges) {
    // Adjacency list to represent the directed acyclic graph
    let adj = new Array(n + 1).fill().map(() => []);

    // Populate the adjacency list using the provided edges
    for (let edge of edges) {
        adj[edge[0]].push(edge[1]);
    }

    // Initialize dp array to store the number of ways to reach Node N from each vertex
    let dp = new Array(n + 1).fill(0);

    // Initialize visited array to keep track of visited nodes during DFS
    let vis = new Array(n + 1).fill(0);

    // Set the base case for Node N, i.e., there is one way to reach N from N
    dp[n] = 1;

    // Mark Node N as visited
    vis[n] = 1;

    // Call the recursive DFS function to calculate the number of ways
    let res = dfs(n, adj, dp, vis, 1);

    // Return the result, which represents the number of ways to reach Node N from Node 1
    return res;
}

// Example inputs
let N = 5;
let edges = [
    [1, 2], [1, 3], [2, 4], [3, 4], [4, 5]
];

// Function call
let result = noOfWays(N, edges);

// Output the result
console.log("No. of ways to reach Node N from Node 1:", result);

Python3

from collections import defaultdict

def dfs(n, adj, dp, vis, curr):
    # If the node has already been visited, return its stored value
    if vis[curr] == 1:
        return dp[curr]

    # Mark the current node as visited
    vis[curr] = 1

    # Traverse all adjacent nodes and recursively calculate the number of ways
    for x in adj[curr]:
        if vis[x] == 0:
            dfs(n, adj, dp, vis, x)

        # Update the number of ways to reach Node N from the current node
        dp[curr] += dp[x]

    # Return the calculated value for the current node
    return dp[curr]

def no_of_ways(n, edges):
    # Adjacency list to represent the directed acyclic graph
    adj = defaultdict(list)

    # Populate the adjacency list using the provided edges
    for edge in edges:
        adj[edge[0]].append(edge[1])

    # Initialize dp array to store the number of ways to reach Node N from each vertex
    dp = [0] * (n + 1)

    # Initialize visited array to keep track of visited nodes during DFS
    vis = [0] * (n + 1)

    # Set the base case for Node N, i.e., there is one way to reach N from N
    dp[n] = 1

    # Mark Node N as visited
    vis[n] = 1

    # Call the recursive DFS function to calculate the number of ways
    res = dfs(n, adj, dp, vis, 1)

    # Return the result, which represents the number of ways to reach Node N from Node 1
    return res

# Example inputs
N = 5
edges = [
    (1, 2), (1, 3), (2, 4), (3, 4), (4, 5)
]

# Function call
result = no_of_ways(N, edges)

# Output the result
print("No. of ways to reach Node N from Node 1:", result)

Output

No. of ways to reach Node N from Node 1: 2

Time Complexity: O(N), Where ‘N’ is the number of nodes in the given graph.
Auxiliary Space: O(N)

Find the largest number of nodes colored with the most occurring color on any path of DAG:

Given a Directed Acyclic Graph consisting of N nodes and M edges. Each node is assigned a lowercase alphabet representing the color of that node (‘a’ to ‘z’). The task is to determine the largest value of any path. The value of a path is defined as the number of nodes which are colored with the most occurring color in the path.

Solution: The problem can be solved in following steps:

1. Define the States: We can observe that there are total 26 colors possible. So, we can make a 2D dp[][] array of size N*26. Let dp[v][i] represent the maximum count of vertices having color i of any path starting from vertex v.

2. Make the Transitions: Again, like the Nodes represent states in the DAG, similarly the edges represent transitions. Thus dp[v][i] can be calculated by:

dp[v][i] = max(dp[u][i]+col, dp[v][i]), for each edge v->u and each color i from 0 to 25. The value of col will be 1 if color of vertex v is equal to i otherwise it will be 0.

3. Topological Consideration: Again, we process the nodes topologically i.e., if edge v->u exists then the node u should be processed before node v.

Hence, our final answer will be maximum of dp[v][i] for each vertex from 1 to N and color from 0 to 25.

Below is the implementation of above approach:

C++

#include <bits/stdc++.h>
using namespace std;

// Recursive function to perform depth-first search
// and calculate the maximum count of vertices having a specific color
void dfs(int n, vector<vector<int>>& adj, vector<vector<int>>& dp,
         vector<int>& vis, int curr, string& s) {
    // If the node has already been visited, return
    if (vis[curr] == 1)
        return;

    // Mark the current node as visited
    vis[curr] = 1;

    // Initialize the color count for the current node
    dp[curr][s[curr - 1] - 97] = 1;

    // Traverse all adjacent nodes and recursively calculate
    // the maximum count of vertices having each color
    for (auto x : adj[curr]) {
        if (!vis[x]) {
            dfs(n, adj, dp, vis, x, s);
        }

        // Update the count of vertices having each color
        for (int j = 0; j < 26; j++) {
            int col = 0;
            if (s[curr - 1] - 97 == j)
                col = 1;
            dp[curr][j] = max(dp[curr][j], dp[x][j] + col);
        }
    }
}

// Function to calculate the maximum count of vertices
// having the most occurring color in any path
int maxcolorpath(int n, vector<vector<int>>& edges, string s) {
    // Adjacency list to represent the directed acyclic graph
    vector<vector<int>> adj(n + 1);

    // Populate the adjacency list using the provided edges
    for (vector<int> i : edges) {
        adj[i[0]].push_back(i[1]);
    }

    // Initialize dp array to store the maximum count of vertices
    // having each color for each vertex
    vector<vector<int>> dp(n + 1, vector<int>(26, 0));

    // Initialize visited array to keep track of visited nodes during DFS
    vector<int> vis(n + 1);

    // Call the recursive DFS function to calculate the maximum count
    dfs(n, adj, dp, vis, 1, s);

    // Find the maximum count of vertices having the most occurring color
    int res = 0;
    for (int i = 1; i <= n; i++) {
        for (int j = 0; j < 26; j++) {
            res = max(res, dp[i][j]);
        }
    }
    return res;
}

int main() {
    // Example inputs
    int N = 5;
    vector<vector<int>> edges = {
        {1, 2}, {1, 3}, {2, 4}, {3, 4}, {4, 5}
    };
    string color = "abaca";

    // Function call
    int result = maxcolorpath(N, edges, color);

    // Output the result
    cout << "Maximum count of vertices having the most occurring color: "
         << result << endl;

    return 0;
}

Java

import java.util.ArrayList;
import java.util.Arrays;

public class MaxColorPath {

    // Recursive function to perform depth-first search
    // and calculate the maximum count of vertices having a specific color
    static void dfs(int n, ArrayList<ArrayList<Integer>> adj, int[][] dp,
                    int[] vis, int curr, String s) {
        // If the node has already been visited, return
        if (vis[curr] == 1)
            return;

        // Mark the current node as visited
        vis[curr] = 1;

        // Initialize the color count for the current node
        dp[curr][s.charAt(curr - 1) - 'a'] = 1;

        // Traverse all adjacent nodes and recursively calculate
        // the maximum count of vertices having each color
        for (int x : adj.get(curr)) {
            if (vis[x] == 0) {
                dfs(n, adj, dp, vis, x, s);
            }

            // Update the count of vertices having each color
            for (int j = 0; j < 26; j++) {
                int col = (s.charAt(curr - 1) - 'a' == j) ? 1 : 0;
                dp[curr][j] = Math.max(dp[curr][j], dp[x][j] + col);
            }
        }
    }

    // Function to calculate the maximum count of vertices
    // having the most occurring color in any path
    static int maxColorPath(int n, int[][] edges, String s) {
        // Adjacency list to represent the directed acyclic graph
        ArrayList<ArrayList<Integer>> adj = new ArrayList<>(n + 1);

        for (int i = 0; i <= n; i++) {
            adj.add(new ArrayList<>());
        }

        // Populate the adjacency list using the provided edges
        for (int[] edge : edges) {
            adj.get(edge[0]).add(edge[1]);
        }

        // Initialize dp array to store the maximum count of vertices
        // having each color for each vertex
        int[][] dp = new int[n + 1][26];

        // Initialize visited array to keep track of visited nodes during DFS
        int[] vis = new int[n + 1];

        // Call the recursive DFS function to calculate the maximum count
        dfs(n, adj, dp, vis, 1, s);

        // Find the maximum count of vertices having the most occurring color
        int res = 0;
        for (int i = 1; i <= n; i++) {
            for (int j = 0; j < 26; j++) {
                res = Math.max(res, dp[i][j]);
            }
        }
        return res;
    }

    public static void main(String[] args) {
        // Example inputs
        int N = 5;
        int[][] edges = {
                {1, 2}, {1, 3}, {2, 4}, {3, 4}, {4, 5}
        };
        String color = "abaca";

        // Function call
        int result = maxColorPath(N, edges, color);

        // Output the result
        System.out.println("Maximum count of vertices having the most occurring color: " + result);
    }
}

C#

using System;
using System.Collections.Generic;
using System.Linq;

class Program
{
    // Recursive function to perform depth-first search
    // and calculate the maximum count of vertices having a specific color
    static void Dfs(int n, List<int>[] adj, int[][] dp, int[] vis, int curr, string s)
    {
        // If the node has already been visited, return
        if (vis[curr] == 1)
            return;

        // Mark the current node as visited
        vis[curr] = 1;

        // Initialize the color count for the current node
        dp[curr][s[curr - 1] - 97] = 1;

        // Traverse all adjacent nodes and recursively calculate
        // the maximum count of vertices having each color
        foreach (var x in adj[curr])
        {
            if (vis[x] == 0)
            {
                Dfs(n, adj, dp, vis, x, s);
            }

            // Update the count of vertices having each color
            for (int j = 0; j < 26; j++)
            {
                int col = 0;
                if (s[curr - 1] - 97 == j)
                    col = 1;
                dp[curr][j] = Math.Max(dp[curr][j], dp[x][j] + col);
            }
        }
    }

    // Function to calculate the maximum count of vertices
    // having the most occurring color in any path
    static int MaxColorPath(int n, int[][] edges, string s)
    {
        // Adjacency list to represent the directed acyclic graph
        var adj = new List<int>[n + 1];
        for (int i = 0; i < adj.Length; i++)
        {
            adj[i] = new List<int>();
        }

        // Populate the adjacency list using the provided edges
        foreach (var i in edges)
        {
            adj[i[0]].Add(i[1]);
        }

        // Initialize dp array to store the maximum count of vertices
        // having each color for each vertex
        var dp = new int[n + 1][];
        for (int i = 0; i < dp.Length; i++)
        {
            dp[i] = new int[26];
        }

        // Initialize visited array to keep track of visited nodes during DFS
        var vis = new int[n + 1];

        // Call the recursive DFS function to calculate the maximum count
        Dfs(n, adj, dp, vis, 1, s);

        // Find the maximum count of vertices having the most occurring color
        int res = 0;
        for (int i = 1; i <= n; i++)
        {
            for (int j = 0; j < 26; j++)
            {
                res = Math.Max(res, dp[i][j]);
            }
        }
        return res;
    }

    static void Main()
    {
        int N = 5;
        int[][] edges = {
            new int[] {1, 2}, new int[] {1, 3}, new int[] {2, 4}, new int[] {3, 4}, new int[] {4, 5}
        };
        string color = "abaca";

        // Function call
        int result = MaxColorPath(N, edges, color);

        // Output the result
        Console.WriteLine("Maximum count of vertices having the most occurring color: "
             + result);
    }
}

Javascript

// Recursive function to perform depth-first search
// and calculate the maximum count of vertices having a specific color
function dfs(n, adj, dp, vis, curr, s) {
    // If the node has already been visited, return
    if (vis[curr] === 1)
        return;

    // Mark the current node as visited
    vis[curr] = 1;

    // Initialize the color count for the current node
    dp[curr][s.charCodeAt(curr - 1) - 97] = 1;

    // Traverse all adjacent nodes and recursively calculate
    // the maximum count of vertices having each color
    for (let x of adj[curr]) {
        if (!vis[x]) {
            dfs(n, adj, dp, vis, x, s);
        }

        // Update the count of vertices having each color
        for (let j = 0; j < 26; j++) {
            let col = 0;
            if (s.charCodeAt(curr - 1) - 97 === j)
                col = 1;
            dp[curr][j] = Math.max(dp[curr][j], dp[x][j] + col);
        }
    }
}

// Function to calculate the maximum count of vertices
// having the most occurring color in any path
function maxcolorpath(n, edges, s) {
    // Adjacency list to represent the directed acyclic graph
    let adj = new Array(n + 1).fill(null).map(() => []);

    // Populate the adjacency list using the provided edges
    for (let i of edges) {
        adj[i[0]].push(i[1]);
    }

    // Initialize dp array to store the maximum count of vertices
    // having each color for each vertex
    let dp = new Array(n + 1).fill(null).map(() => new Array(26).fill(0));

    // Initialize visited array to keep track of visited nodes during DFS
    let vis = new Array(n + 1).fill(0);

    // Call the recursive DFS function to calculate the maximum count
    dfs(n, adj, dp, vis, 1, s);

    // Find the maximum count of vertices having the most occurring color
    let res = 0;
    for (let i = 1; i <= n; i++) {
        for (let j = 0; j < 26; j++) {
            res = Math.max(res, dp[i][j]);
        }
    }
    return res;
}

// Example inputs
let N = 5;
let edges = [
    [1, 2], [1, 3], [2, 4], [3, 4], [4, 5]
];
let color = "abaca";

// Function call
let result = maxcolorpath(N, edges, color);

// Output the result
console.log("Maximum count of vertices having the most occurring color: " + result);
//this code is contributed by Adarsh.

Python3

# Recursive function to perform depth-first search
# and calculate the maximum count of vertices having a specific color
def dfs(n, adj, dp, vis, curr, s):
    # If the node has already been visited, return
    if vis[curr] == 1:
        return

    # Mark the current node as visited
    vis[curr] = 1

    # Initialize the color count for the current node
    dp[curr][ord(s[curr - 1]) - 97] = 1

    # Traverse all adjacent nodes and recursively calculate
    # the maximum count of vertices having each color
    for x in adj[curr]:
        if not vis[x]:
            dfs(n, adj, dp, vis, x, s)

        # Update the count of vertices having each color
        for j in range(26):
            col = 0
            if ord(s[curr - 1]) - 97 == j:
                col = 1
            dp[curr][j] = max(dp[curr][j], dp[x][j] + col)

# Function to calculate the maximum count of vertices
# having the most occurring color in any path
def maxcolorpath(n, edges, s):
    # Adjacency list to represent the directed acyclic graph
    adj = [[] for _ in range(n + 1)]

    # Populate the adjacency list using the provided edges
    for i in edges:
        adj[i[0]].append(i[1])

    # Initialize dp array to store the maximum count of vertices
    # having each color for each vertex
    dp = [[0] * 26 for _ in range(n + 1)]

    # Initialize visited array to keep track of visited nodes during DFS
    vis = [0] * (n + 1)

    # Call the recursive DFS function to calculate the maximum count
    dfs(n, adj, dp, vis, 1, s)

    # Find the maximum count of vertices having the most occurring color
    res = 0
    for i in range(1, n + 1):
        for j in range(26):
            res = max(res, dp[i][j])
    return res

# Example inputs
N = 5
edges = [
    [1, 2], [1, 3], [2, 4], [3, 4], [4, 5]
]
color = "abaca"

# Function call
result = maxcolorpath(N, edges, color)

# Output the result
print("Maximum count of vertices having the most occurring color:", result)
# This code is contributed by prachi

Output

Maximum count of vertices having the most occurring color: 3

Time Complexity: O(N), Where ‘N’ is the number of nodes in the given graph.
Auxiliary Space: O(N)

Practice Problems Dynamic Programming (DP) on Directed Graphs for Competitive Programming:

Longest path in a directed Acyclic graph
Longest Path with Maximum Letter Frequency
Minimize the Maximum value
Maximum Edges In the Path
Minimum path need to reverse to reach each node in a tree from each vertex
Two-Color Sorting