# Minimum distance to visit all the nodes of an undirected weighted tree - GeeksforGeeks (2023)

Given a weighted tree withNnodes starting from1 to N. The distance between any two nodes is given by the edge weight. Node1is the source, the task is to visit all the nodes of the tree with the minimum distance traveled.

Examples:

Input:
u[] = {1, 1, 2, 2, 1}
v[] = {2, 3, 5, 6, 4}
w[] = {1, 4, 2, 50, 5}
Output:73

Input:
u[] = {1, 2}
v[] = {2, 3}
w[] = {3, 1}
Output:4

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Approach:Let’s suppose there arenleafl1,l2,l3, ……,lnand the cost of the path from root to each leaf isc1,c2,c3, ……,cn.
To travel froml1tol2some of the edges will be visited twice ( till theLCAofl1andl2all the edges will be visited twice ), forl2tol3and some of the edges will be visited ( till theLCAofl2andl3all the edges will be visited twice ) twice and similarly every edge of the tree will be visited twice ( observation ).

To minimize the cost of travelling, the maximum cost path from the root to some leaf should be avoided.
Hence the cost = (c1+c2+c3+ …… +cn) –max(c1,c2,c3, ……,cn)
min cost=(2 * sum of edge weight)max(c1,c2,c3, ……,cn)
DFScan be used with some modification to find the largest distance.

Below is the implementation of the above approach:

## C++

// C++ implementation of above approach

#include

using namespace std;

class Edge{

public:

// from - The source of an edge

// to - destination of an edge

// wt - distance between two nodes

int from;

int to;

long wt;

Edge(int a,int b,long w)

{

from = a;

to = b;

wt = w;

}

};

// Method to add an edge between two nodes

void add_edge(vector> &adj_lis,

int to,int from,long wt)

{

adj_lis[from].push_back(Edge(from, to, wt));

adj_lis[to].push_back(Edge(to, from, wt));

}

// DFS method to find distance

// between node 1 to other nodes

void dfs(vector> &adj_lis,

long val[],int v,int par,

long sum,bool visited[])

{

val[v] = sum;

visited[v] =true;

for(Edge e : adj_lis[v])

{

if (!visited[e.to])

dfs(adj_lis, val, e.to,

v, sum + e.wt, visited);

}

}

// Driver code

int main()

{

// Number of nodes

// V - Total number of

// nodes in a tree

int v = 6;

// adj_lis - It is used to

// make the adjacency list of a tree

vector> adj_lis(v);

// val - This array stores the

// distance from node 1 to node 'n'

long val[v];

bool visited[v];

int sum = 0;

// Edge from a node to another

// node with some weight

int from[] = { 2, 3, 5, 6, 4 };

int to[] = { 1, 1, 2, 2, 1 };

int wt[] = { 1, 4, 2, 50, 5 };

for(int i = 0; i < v - 1; i++)

{

sum += 2 * wt[i];

add_edge(adj_lis, to[i] - 1,

from[i] - 1, wt[i]);

}

dfs(adj_lis, val, 0, -1, 0, visited);

long large = INT_MIN;

// Loop to find largest

// distance in a val.

int size =sizeof(val) /sizeof(long);

for(int i = 1; i < size; i++)

if (val[i] > large)

large = val[i];

(Video) All Nodes Distance K In A Binary Tree - Performing Bidirectional Search On A Tree Using A Hashtable

cout << (sum - large);

}

// This code is contributed by sanjeev2552

## Java

// Java implementation of the approach

import java.util.LinkedList;

import java.util.Scanner;

class Graph {

class Edge {

// from - The source of an edge

// to - destination of an edge

// wt - distance between two nodes

int from;

int to;

long wt;

Edge(int a,int b,long w)

{

from = a;

to = b;

wt = w;

}

}

// adj_lis - It is used to

// make the adjacency list of a tree

// V - Total number of nodes in a tree

// val - This array stores the

// distance from node 1 to node 'n'

static LinkedList[] adj_lis;

static int V;

static long val[];

Graph(int v)

{

this.V = v;

adj_lis =new LinkedList[V];

for (int i =0; i < V; i++)

adj_lis[i] =new LinkedList<>();

}

// Method to add an edge between two nodes

void add_edge(int to,int from,long wt)

{

adj_lis[from].add(

new Edge(from, to, wt));

adj_lis[to].add(

new Edge(to, from, wt));

}

// DFS method to find distance

// between node 1 to other nodes

void dfs(int v,

int par,

long sum,

boolean[] visited)

{

val[v] = sum;

visited[v] =true;

for (Edge e : adj_lis[v]) {

if (!visited[e.to])

dfs(e.to,

v,

sum + e.wt,

visited);

}

}

// Driver code

public static void main(String a[])

{

// Number of nodes

int v =6;

Graph obj =new Graph(v);

val =new long[v];

boolean[] visited

=new boolean[v];

int sum =0;

// Edge from a node to another

// node with some weight

int from[] = {2,3,5,6,4 };

int to[] = {1,1,2,2,1 };

int wt[] = {1,4,2,50,5 };

for (int i =0; i < v -1; i++) {

sum +=2 * wt[i];

obj.add_edge(to[i] -1,

from[i] -1,

wt[i]);

(Video) Problem of The Day: 20/03/2023 | Shortest XY Distance in Grid | Jay Dalsaniya

}

obj.dfs(0, -1,0, visited);

long large = Integer.MIN_VALUE;

// Loop to find largest

// distance in a val.

for (int i =1; i < val.length;

i++)

if (val[i] > large)

large = val[i];

System.out.println(sum - large);

}

}

## C#

// C# implementation of above approach

using System;

using System.Collections.Generic;

class Graph

{

public class Edge

{

// from - The source of an edge

// to - destination of an edge

// wt - distance between two nodes

public int from;

public int to;

public long wt;

public Edge(int a,int b,long w)

{

from = a;

to = b;

wt = w;

}

}

// adj_lis - It is used to

// make the adjacency list of a tree

// V - Total number of nodes in a tree

// val - This array stores the

// distance from node 1 to node 'n'

public static List[] adj_lis;

public static int V;

public static long []val;

public Graph(int v)

{

V = v;

adj_lis =new List[V];

for (int i = 0; i < V; i++)

adj_lis[i] =new List();

}

// Method to add an edge between two nodes

void add_edge(int to,int from,long wt)

{

adj_lis[from].Add(

new Edge(from, to, wt));

adj_lis[to].Add(

new Edge(to,from, wt));

}

// DFS method to find distance

// between node 1 to other nodes

void dfs(int v,

int par,

long sum,

bool[] visited)

{

val[v] = sum;

visited[v] =true;

foreach (Edge ein adj_lis[v])

{

if (!visited[e.to])

dfs(e.to, v,

sum + e.wt, visited);

}

}

// Driver code

public static void Main(String []a)

{

// Number of nodes

int v = 6;

Graph obj =new Graph(v);

val =new long[v];

bool []visited =new bool[v];

int sum = 0;

// Edge from a node to another

(Video) Number of Connected Components in an Undirected Graph - Union Find - Leetcode 323 - Python

// node with some weight

int []from = { 2, 3, 5, 6, 4 };

int []to = { 1, 1, 2, 2, 1 };

int []wt = { 1, 4, 2, 50, 5 };

for (int i = 0; i < v - 1; i++)

{

sum += 2 * wt[i];

obj.add_edge(to[i] - 1,

from[i] - 1, wt[i]);

}

obj.dfs(0, -1, 0, visited);

long large =int.MinValue;

// Loop to find largest

// distance in a val.

for (int i = 1; i < val.Length;

i++)

if (val[i] > large)

large = val[i];

Console.WriteLine(sum - large);

}

}

// This code is contributed by Princi Singh

## Javascript

// Javascript implementation of above approach

class Edge

{

// from - The source of an edge

// to - destination of an edge

// wt - distance between two nodes

constructor(a, b, w)

{

this.from = a;

this.to = b;

this.wt = w;

}

}

// adj_lis - It is used to

// make the adjacency list of a tree

// V - Total number of nodes in a tree

// val - This array stores the

// distance from node 1 to node 'n'

was adj_lis = [];

was V =0;

was val =[];

function Graph(v)

{

V = v;

adj_lis = Array.from(Array(V), ()=>Array());

}

// Method to add an edge between two nodes

function add_edge(to, from, wt)

{

adj_lis[from].push(

new Edge(from, to, wt));

adj_lis[to].push(

new Edge(to, from, wt));

}

// DFS method to find distance

// between node 1 to other nodes

function dfs(v, par, sum, visited)

{

val[v] = sum;

visited[v] =true;

for(was e of adj_lis[v])

{

if (!visited[e.to])

dfs(e.to, v, sum + e.wt, visited);

}

}

// Driver code

// Number of nodes

was v = 6;

Graph(v);

val =new Array(v).fill(0);

was visited = Array(v).fill(false);

was sum = 0;

// Edge from a node to another

// node with some weight

was from = [2, 3, 5, 6, 4];

was to = [1, 1, 2, 2, 1];

was wt = [1, 4, 2, 50, 5];

(Video) Shortest Path Visiting All Nodes | Leetcode 847 | Live coding session 🔥🔥 | BFS + Bit Manipulation

for(was i = 0; i < v - 1; i++)

{

sum += 2 * wt[i];

add_edge(to[i] - 1,

from[i] - 1, wt[i]);

}

dfs(0, -1, 0, visited);

was large = -100000;

// Loop to find largest

// distance in a val.

for (was i = 1; i < val.length;i++)

if (val[i] > large)

large = val[i];

document.write(sum - large);

## Python3

# Python 3 implementation of above approach

class Edge:

# src - The source of an edge

# to - destination of an edge

# wt - distance between two nodes

def __init__(self, a, b, w):

self.src= a

self.to= b

self.wt= w

# Method to add an edge between two nodes

def add_edge(adj_lis, to, src, wt):

adj_lis[src].append(Edge(src, to, wt))

adj_lis[to].append(Edge(to, src, wt))

# DFS method to find distance

# between node 1 to other nodes

def dfs(adj_lis, val, v, par,sum, visited):

val[v]= sum

visited[v]= True

for ein adj_lis[v]:

if (not visited[e.to]):

dfs(adj_lis, val, e.to, v,sum + e.wt, visited)

# Driver code

if __name__== '__main__':

# Number of nodes

# V - Total number of

# nodes in a tree

v= 6

# adj_lis - It is used to

# make the adjacency list of a tree

adj_lis=[[]for _in range(v)]

# val - This array stores the

# distance src node 1 to node 'n'

val=[-1]*v

visited=[False]*v

sum = 0

# Edge src a node to another

# node with some weight

src= [2,3,5,6,4]

to= [1,1,2,2,1 ]

wt= [1,4,2,50,5 ]

for iin range(v- 1):

sum += 2 * wt[i]

add_edge(adj_lis, to[i]- 1,

src[i]- 1, wt[i])

dfs(adj_lis, val,0,-1,0, visited)

large= -1e9

# Loop to find largest

# distance in a val.

size= len(val)

for iin range(1,size):

if (val[i] > large):

large= val[i]

print(sum - large)

# This code is contributed by Amartya Ghosh

Output:

73

Time Complexity:O(N).
Auxiliary Space: O(N).

My Personal Notesarrow_drop_up

Last Updated :30 Nov, 2021

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(Video) Implementing Dijkstra Algorithm || GeeksforGeeks || Problem of the Day || Must Watch

## FAQs

### Which algorithm to find the minimum distance from any node to all other nodes in the graph? ›

Dijkstra's Algorithm finds the shortest path between a given node (which is called the "source node") and all other nodes in a graph. This algorithm uses the weights of the edges to find the path that minimizes the total distance (weight) between the source node and all other nodes.

How do you find the shortest path in an undirected weighted graph? ›

One common way to find the shortest path in a weighted graph is using Dijkstra's Algorithm. Dijkstra's algorithm finds the shortest path between two vertices in a graph. It can also be used to generate a Shortest Path Tree - which will be the shortest path to all vertices in the graph (from a given source vertex).

What is the algorithm to find the shortest path visiting all nodes? ›

Recall that the Floyd-Warshall algorithm calculates the shortest path between all pairs of nodes inside a graph. Finally, the shortest path visiting all nodes in a graph will have the minimum cost among all possible paths.

How do you calculate the distance between nodes in a tree? ›

It could be calculated by finding the LCA(Least Common Ancestor) of the two given nodes and then summing - (the distance between LCA and node1) + (the distance between LCA and node2).

Which method can be used to find the shortest distance between every node and starting node of the graph Mcq? ›

Dijkstra's algorithm is an algorithm used for finding the shortest paths between nodes or vertices in a graph. This algorithm is basically used to find the shortest path from a starting node to a target node in a weighted graph.

What is the shortest path problem in a weighted undirected graph? ›

In graph theory, the shortest path problem is the problem of finding a path between two vertices (or nodes) in a graph such that the sum of the weights of its constituent edges is minimized.

What is the shortest path algorithm for weighted graph? ›

One algorithm for finding the shortest path from a starting node to a target node in a weighted graph is Dijkstra's algorithm. The algorithm creates a tree of shortest paths from the starting vertex, the source, to all other points in the graph.

Which is the best algorithm for shortest path in weighted graph? ›

Many graph use cases rely on finding the shortest path between nodes. When the weight of a path is of no concern, the simplest and best algorithms are Breadth-First Search and Depth-First Search, both of which have a time complexity of O(V + E), where V is the number of vertices and E is the number of edges.

What is the shortest path between every pair of nodes? ›

The all pair shortest path algorithm is also known as Floyd-Warshall algorithm is used to find all pair shortest path problem from a given weighted graph. As a result of this algorithm, it will generate a matrix, which will represent the minimum distance from any node to all other nodes in the graph.

Which path visits all nodes exactly once? ›

First, an Eulerian path is one where every relationship is visited exactly once. Second, a Hamiltonian path is one where every node is visited exactly once. A path can be both Eulerian and Hamiltonian, and if you start and finish at the same node it's considered a cycle or tour.

### Which algorithm is used to calculate the shortest path between the nodes when there are no negative cost edges in the graph? ›

Dijkstra's algorithm is one of the SSSP (Single Source Shortest Path) algorithms. Therefore, it calculates the shortest path from a source node to all the nodes inside the graph. Although it's known that Dijkstra's algorithm works with weighted graphs, it works with non-negative weights for the edges.

What is the shortest distance from one node to other node? ›

The distance between two nodes is defined as the total number of edges in the shortest path from one node to other. For example, consider the binary tree. The distance between node 7 and node 6 is 3. This problem is a standard application of the lowest common ancestor of given nodes.

What is the distance between nodes and nodes? ›

The distance between two adjacent nodes ortwo adjacent antinodes is equal to half of the wavelength.

What is distance between all nodes in graph? ›

The distance between two nodes is the length of the shortest path between them.

How do you find the shortest distance? ›

The Distance Formula. The shortest distance between two points is a straight line. This distance can be calculated by using the distance formula. The distance between two points (x1, y1) and (x2, y2) can be defined as d=√(x2−x1)2+(y2−y1)2.

Which algorithm is used to find shortest distances in a graph? ›

Bellman-Ford algorithm is used to find all pairs shortest distances in a graph and it is dynamic programming technique.

What is the shortest path distance? ›

A shortest path, or geodesic path, between two nodes in a graph is a path with the minimum number of edges. If the graph is weighted, it is a path with the minimum sum of edge weights. The length of a geodesic path is called geodesic distance or shortest distance.

What is the minimum distance between a node? ›

Hence, the required distance is λ/4.

Which method should be applied to find the shortest distance? ›

We wish to find its shortest distance from the line L : y = mx + c. Let B(bx,by) be the point on line L such that PB ⊥ L. It can be shown, using the Pythagoras theorem, that the perpendicular distance d = l(PB) (see the Figure) is the shortest distance between point P and line L.

What are the different methods for finding shortest path spanning tree? ›

1) Dijkstra's Shortest Path First (SPF) Algorithm. 2) Bellman-Ford Shortest Path Algorithm. We have different techniques to find the spanning tree of a network or graphs are 1) Kruskal's algorithm. 2) Jarnik prim algorithm.

### What is shortest path in undirected tree? ›

In mathematics and computer science, a shortest-path tree rooted at a vertex v of a connected, undirected graph G is a spanning tree T of G, such that the path distance from root v to any other vertex u in T is the shortest path distance from v to u in G.

What is the longest path in weighted undirected graph? ›

A longest path between two given vertices s and t in a weighted graph G is the same thing as a shortest path in a graph −G derived from G by changing every weight to its negation. Therefore, if shortest paths can be found in −G, then longest paths can also be found in G.

What is minimum path cover in undirected graph? ›

Given an undirected graph G=(V,E), a path cover is a set of disjoint paths such that every vertex v∈V belongs to exactly one path. The minimum path cover problem consists of finding a path cover of G having the least number of paths.

Which algorithm is best for shortest path with negative weights? ›

Bellman-Ford Algorithm. Suppose that we are given a weighted directed graph with vertices and edges, and some specified vertex . You want to find the length of shortest paths from vertex to every other vertex. Unlike the Dijkstra algorithm, this algorithm can also be applied to graphs containing negative weight edges .

Which algorithm can be used to find the minimum spanning tree in an undirected weighted graph? ›

Kruskal's greedy algorithm finds a minimum spanning tree for a weighted, undirected graph.

Which of these algorithms solves the weighted single source shortest path problem? ›

The Bellman-Ford algorithm also solves the Single-Source Shortest Paths problem, but in contrast to Dijkstra's algorithm, edge weights may be negative.

Which algorithm is best for finding the shortest distance between two points in an unweighted graph? ›

Dijkstra's Algorithm. This algorithm might be the most famous one for finding the shortest path. Its advantage over a DFS, BFS, and bidirectional search is that you can use it in all graphs with positive edge weights.

Is a shortest path between two vertices in a weighted graph unique if the weights of edges are distinct? ›

(b) T F [3 points] If all edges in a graph have distinct weights, then the shortest path between two vertices is unique. Solution: FALSE. Even if no two edges have the same weight, there could be two paths with the same weight.

Which data structure to be used to implement shortest path algorithm on a weighted graph so it runs in linear time? ›

The correct option is A Heap.

What is the difference between single source shortest path and all pair shortest path? ›

The single-source shortest-path problem requires that we find the shortest path from a single vertex to all other vertices in a graph. The all-pairs shortest-path problem requires that we find the shortest path between all pairs of vertices in a graph.

### What approach is being followed in all pair shortest path algorithm? ›

What approach is being followed in Floyd Warshall Algorithm? Explanation: Floyd Warshall Algorithm follows dynamic programming approach because the all pair shortest paths are computed in bottom up manner.

What is the longest path from all nodes in A tree? ›

The diameter of a binary tree is the length of the longest path between any two nodes in a tree. This path may or may not pass through the root.

How many path exists between any pair of nodes in tree network? ›

A tree of an undirected network is a connected subgraph that has no cycles. Thus, a connected tree with t nodes has exactly t-I edges and there exists a single path between any two nodes on the tree.

What path in graph visits every edge exactly once? ›

An Euler path , in a graph or multigraph, is a walk through the graph which uses every edge exactly once. An Euler circuit is an Euler path which starts and stops at the same vertex. Our goal is to find a quick way to check whether a graph (or multigraph) has an Euler path or circuit.

What connects every node to every other node? ›

Mesh network topology

A mesh topology is another nonhierarchical structure where each network node directly connects to all others.

Which algorithm is used to find the minimal distance from one node to all other nodes in A weighted graph? ›

Dijkstra's Algorithm finds the shortest path between a given node (which is called the "source node") and all other nodes in a graph. This algorithm uses the weights of the edges to find the path that minimizes the total distance (weight) between the source node and all other nodes.

Which algorithm is shortest path visiting all nodes? ›

Shortest Path Visiting All Nodes in C++

We can start and stop at any node, we can revisit nodes multiple times, and we can reuse edges. So, if the input is like [[1],[0,2,4],[1,3,4],[2],[1,2]], then the output will be 4. Now here one possible path is [0,1,4,2,3].

Which simpler algorithm solves the problem of shortest paths from one node to all other nodes if all edges are equally long? ›

Dijkstra's algorithm is a simple and efficient way to find the shortest path from a root node to all other nodes in a graph.

What is the shortest path in A weighted graph? ›

For a weighted graph G(V,E,w) a shortest (weighted) path from vertex u to vertex v is a path from u to v with minimum weight. There might be multiple paths with equal weight, and if so they are all shortest weighted paths from u to v. We use σG(u, v) to indicate the weight of a shortest path from u to v.

How do you find the minimum distance between two nodes on a graph? ›

Algorithm to find the shortest path between two vertices in an undirected graph
1. Input the graph.
2. Input the source and destination nodes.
3. Find the paths between the source and the destination nodes.
4. Find the number of edges in all the paths and return the path having the minimum number of edges.
Jul 8, 2022

### What is the formula for distance between two nodes? ›

Nodes and antinodes are known to generate the stationary waves. In a given stationary wave, the distance between any given two successive nodes or any two successive antinodes is always half of the wavelength. The distance between the two successive nodes is $\dfrac{\lambda }{2}$.

What is the distance between node to node? ›

The distance between two adjacent nodes ortwo adjacent antinodes is equal to half of the wavelength.

What is the shortest distance between two successive nodes? ›

The distance between two successive nodes is λ/2. At nodes air pressure and density both are low. The distance between a node (N) and adjoining antinode (A) is λ/4. ​So, from above explanation, it is clear that distance between two consecutive nodes is λ / 2.

What is the algorithm to find minimum distance between two points? ›

The most common solution for this problem is Dijkstra's algorithm which updates the shortest path between the current node and all of its neighbors. After updating the distance of all of the neighbors it moves to the node with the lowest distance and repeats the process with all unvisited neighbors.

Which algorithm will you use to determine the shortest path between the nodes in a graph where each edge has a positive or negative edge weight with no negative cycles? ›

Johnson's algorithm is a way to find the shortest paths between all pairs of vertices in an edge-weighted directed graph. It allows some of the edge weights to be negative numbers, but no negative-weight cycles may exist.

Which algorithm will you use to determine the shortest path between the nodes in a graph other than Dijkstra? ›

Bellman Ford's Algorithm:

Bellman Ford's algorithm is used to find the shortest paths from the source vertex to all other vertices in a weighted graph. It depends on the following concept: Shortest path contains at most edges, because the shortest path couldn't have a cycle.

Which algorithm will you use to determine the shortest path between the nodes in a graph Mcq? ›

Answer - B) Djikstra's algorithm is used to find the shortest path from a source node to all other nodes in a weighted graph.

What is the formula for finding minimum distance? ›

The shortest distance between two points is a straight line. This distance can be calculated by using the distance formula. The distance between two points [Math Processing Error] ( x 1 , y 1 ) and [Math Processing Error] ( x 2 , y 2 ) can be defined as [Math Processing Error] d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 .

Which algorithm would determine the shortest distance to connect multiple nodes? ›

Dijkstra's algorithm (/ˈdaɪkstrəz/ DYKE-strəz) is an algorithm for finding the shortest paths between nodes in a weighted graph, which may represent, for example, road networks.

What is the shortest path for a weighted graph? ›

For a weighted graph G(V,E,w) a shortest (weighted) path from vertex u to vertex v is a path from u to v with minimum weight. There might be multiple paths with equal weight, and if so they are all shortest weighted paths from u to v. We use σG(u, v) to indicate the weight of a shortest path from u to v.

### What is the fastest algorithm for the shortest path? ›

Dijkstra's algorithm is used for our fastest path algorithm because it can find the shortest path between vertices in the graph. The coordinates on the arena are considered as the vertices in the graph.

Which of the following algorithms can be used to find the shortest path between two nodes in an undirected unweighted graph? ›

We say that BFS is the algorithm to use if we want to find the shortest path in an undirected, unweighted graph.

Which algorithm will you use to determine the shortest path between the nodes in a graph Kruskal? ›

Kruskal's algorithm is the concept that is introduced in the graph theory of discrete mathematics. It is used to discover the shortest path between two points in a connected weighted graph. This algorithm converts a given graph into the forest, considering each node as a separate tree.

Which algorithm will you use to determine the shortest path between the nodes in A graph Prim's algorithm? ›

In the computation aspect, Prim's and Dijkstra's algorithms have three main differences: Dijkstra's algorithm finds the shortest path, but Prim's algorithm finds the MST. Dijkstra's algorithm can work on both directed and undirected graphs, but Prim's algorithm only works on undirected graphs.

What problem does Dijkstra algorithm solve? ›

Dijkstra's algorithm solves the shortest-path problem for any weighted, directed graph with non-negative weights. It can handle graphs consisting of cycles, but negative weights will cause this algorithm to produce incorrect results.

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Author information

Name: Duane Harber

Birthday: 1999-10-17