给定一个二叉树，任务是打印该树的所有互质路径。
A path of a binary tree is said to be a co-prime path if all the nodes of this path are co-prime to each other.
例子：
Input:
1
/ \
12 11
/ / \
3 4 13
\ /
15 3
Output:
1 11 4 15
1 11 13 3
Explanation:
{1, 11, 4, 15} and {1, 11, 13, 3}
are coprimes because their GCD is 1.
Input:
5
/ \
21 77
/ \ \
61 16 16
\ /
10 3
/
23
Output:
5 21 61
5 77 16 3
Explanation:
{5, 21, 61} and {5, 77, 16, 3}
are coprimes because their GCD is 1.
方法：想法是遍历树并检查该路径中的所有元素是否互质。因此，一种有效的解决方案是通过使用Eratosthenes筛子生成整数的所有质数因子。使用哈希，存储每个元素的计数，它是数组中任何数字的素数。如果该元素与其他元素不包含任何公共质数因子，则它总是与其他元素形成互质对。
下面是上述方法的实现：
C++
// C++ program for printing Co-prime
// paths of binary Tree
#include
using namespace std;
// A Tree node
struct Node {
int key;
struct Node *left, *right;
};
// Utility function to create
// a new node
Node* newNode(int key)
{
Node* temp = new Node;
temp->key = key;
temp->left = temp->right = NULL;
return (temp);
}
int N = 1000000;
// Vector to store all the
// prime numbers
vector prime;
// Function to store all the
// prime numbers in an array
void SieveOfEratosthenes()
{
// Create a boolean array "prime[0..N]"
// and initialize all the entries in it
// as true. A value in prime[i]
// will finally be false if
// i is Not a prime, else true.
bool check[N + 1];
memset(check, true, sizeof(check));
for (int p = 2; p * p <= N; p++) {
// If prime[p] is not changed,
// then it is a prime
if (check[p] == true) {
prime.push_back(p);
// Update all multiples of p
// greater than or equal to
// the square of it
// numbers which are multiples of p
// and are less than p^2
// are already marked.
for (int i = p * p; i <= N; i += p)
check[i] = false;
}
}
}
// Function to check whether Path
// is Co-prime or not
bool isPathCo_Prime(vector& path)
{
int max = 0;
// Iterating through the array
// to find the maximum element
// in the array
for (auto x : path) {
if (max < x)
max = x;
}
for (int i = 0;
i * prime[i] <= max / 2;
i++) {
int ct = 0;
// Incrementing the variable
// if any of the value has
// a factor
for (auto x : path) {
if (x % prime[i] == 0)
ct++;
}
// If not co-prime
if (ct > 1) {
return false;
}
}
return true;
}
// Function to print a Co-Prime path
void printCo_PrimePaths(vector& path)
{
for (auto x : path) {
cout << x << " ";
}
cout << endl;
}
// Function to find co-prime paths of
// binary tree
void findCo_PrimePaths(struct Node* root,
vector& path)
{
// Base case
if (root == NULL)
return;
// Store the value in path vector
path.push_back(root->key);
// Recursively call for left sub tree
findCo_PrimePaths(root->left, path);
// Recursively call for right sub tree
findCo_PrimePaths(root->right, path);
// Condition to check, if leaf node
if (root->left == NULL
&& root->right == NULL) {
// Condition to check,
// if path co-prime or not
if (isPathCo_Prime(path)) {
// Print co-prime path
printCo_PrimePaths(path);
}
}
// Remove the last element
// from the path vector
path.pop_back();
}
// Function to find Co-Prime paths
// In a given binary tree
void printCo_PrimePaths(struct Node* node)
{
// To save all prime numbers
SieveOfEratosthenes();
vector path;
// Function call
findCo_PrimePaths(node, path);
}
// Driver Code
int main()
{
/* 10
/ \
48 3
/ \
11 37
/ \ / \
7 29 42 19
/
7
*/
// Create Binary Tree as shown
Node* root = newNode(10);
root->left = newNode(48);
root->right = newNode(3);
root->right->left = newNode(11);
root->right->right = newNode(37);
root->right->left->left = newNode(7);
root->right->left->right = newNode(29);
root->right->right->left = newNode(42);
root->right->right->right = newNode(19);
root->right->right->right->left = newNode(7);
// Print Co-Prime Paths
printCo_PrimePaths(root);
return 0;
} Java
// Java program for printing Co-prime
// paths of binary Tre
import java.util.*;
class GFG{
// A Tree node
static class Node {
int key;
Node left, right;
};
// Utility function to create
// a new node
static Node newNode(int key)
{
Node temp = new Node();
temp.key = key;
temp.left = temp.right = null;
return (temp);
}
static int N = 1000000;
// Vector to store all the
// prime numbers
static Vector prime = new Vector();
// Function to store all the
// prime numbers in an array
static void SieveOfEratosthenes()
{
// Create a boolean array "prime[0..N]"
// and initialize all the entries in it
// as true. A value in prime[i]
// will finally be false if
// i is Not a prime, else true.
boolean []check = new boolean[N + 1];
Arrays.fill(check, true);
for (int p = 2; p * p <= N; p++) {
// If prime[p] is not changed,
// then it is a prime
if (check[p] == true) {
prime.add(p);
// Update all multiples of p
// greater than or equal to
// the square of it
// numbers which are multiples of p
// and are less than p^2
// are already marked.
for (int i = p * p; i <= N; i += p)
check[i] = false;
}
}
}
// Function to check whether Path
// is Co-prime or not
static boolean isPathCo_Prime(Vector path)
{
int max = 0;
// Iterating through the array
// to find the maximum element
// in the array
for (int x : path) {
if (max < x)
max = x;
}
for (int i = 0;
i * prime.get(i) <= max / 2;
i++) {
int ct = 0;
// Incrementing the variable
// if any of the value has
// a factor
for (int x : path) {
if (x % prime.get(i) == 0)
ct++;
}
// If not co-prime
if (ct > 1) {
return false;
}
}
return true;
}
// Function to print a Co-Prime path
static void printCo_PrimePaths(Vector path)
{
for (int x : path) {
System.out.print(x+ " ");
}
System.out.println();
}
// Function to find co-prime paths of
// binary tree
static void findCo_PrimePaths(Node root,
Vector path)
{
// Base case
if (root == null)
return;
// Store the value in path vector
path.add(root.key);
// Recursively call for left sub tree
findCo_PrimePaths(root.left, path);
// Recursively call for right sub tree
findCo_PrimePaths(root.right, path);
// Condition to check, if leaf node
if (root.left == null
&& root.right == null) {
// Condition to check,
// if path co-prime or not
if (isPathCo_Prime(path)) {
// Print co-prime path
printCo_PrimePaths(path);
}
}
// Remove the last element
// from the path vector
path.remove(path.size()-1);
}
// Function to find Co-Prime paths
// In a given binary tree
static void printCo_PrimePaths(Node node)
{
// To save all prime numbers
SieveOfEratosthenes();
Vector path = new Vector();
// Function call
findCo_PrimePaths(node, path);
}
// Driver Code
public static void main(String[] args)
{
/* 10
/ \
48 3
/ \
11 37
/ \ / \
7 29 42 19
/
7
*/
// Create Binary Tree as shown
Node root = newNode(10);
root.left = newNode(48);
root.right = newNode(3);
root.right.left = newNode(11);
root.right.right = newNode(37);
root.right.left.left = newNode(7);
root.right.left.right = newNode(29);
root.right.right.left = newNode(42);
root.right.right.right = newNode(19);
root.right.right.right.left = newNode(7);
// Print Co-Prime Paths
printCo_PrimePaths(root);
}
}
// This code is contributed by 29AjayKumar Python3
# Python3 program for printing Co-prime
# paths of binary Tree
# A Tree node
class Node:
def __init__(self, key):
self.key = key
self.left = None
self.right = None
# Utility function to create
# a new node
def newNode(key):
temp = Node(key)
return temp
N = 1000000
# Vector to store all the
# prime numbers
prime = []
# Function to store all the
# prime numbers in an array
def SieveOfEratosthenes():
# Create a boolean array "prime[0..N]"
# and initialize all the entries in it
# as true. A value in prime[i]
# will finally be false if
# i is Not a prime, else true.
check = [True for i in range(N + 1)]
p = 2
while (p * p <= N):
# If prime[p] is not changed,
# then it is a prime
if (check[p]):
prime.append(p)
# Update all multiples of p
# greater than or equal to
# the square of it numbers
# which are multiples of p
# and are less than p^2
# are already marked.
for i in range(p * p, N + 1, p):
check[i] = False
p += 1
# Function to check whether Path
# is Co-prime or not
def isPathCo_Prime(path):
max = 0
# Iterating through the array
# to find the maximum element
# in the array
for x in path:
if (max < x):
max = x
i = 0
while (i * prime[i] <= max // 2):
ct = 0
# Incrementing the variable
# if any of the value has
# a factor
for x in path:
if (x % prime[i] == 0):
ct += 1
# If not co-prime
if (ct > 1):
return False
i += 1
return True
# Function to print a Co-Prime path
def printCo_PrimePaths(path):
for x in path:
print(x, end = ' ')
print()
# Function to find co-prime paths of
# binary tree
def findCo_PrimePaths(root, path):
# Base case
if (root == None):
return path
# Store the value in path vector
path.append(root.key)
# Recursively call for left sub tree
path = findCo_PrimePaths(root.left, path)
# Recursively call for right sub tree
path = findCo_PrimePaths(root.right, path)
# Condition to check, if leaf node
if (root.left == None and root.right == None):
# Condition to check,
# if path co-prime or not
if (isPathCo_Prime(path)):
# Print co-prime path
printCo_PrimePaths(path)
# Remove the last element
# from the path vector
path.pop()
return path
# Function to find Co-Prime paths
# In a given binary tree
def printCo_PrimePath(node):
# To save all prime numbers
SieveOfEratosthenes()
path = []
# Function call
path = findCo_PrimePaths(node, path)
# Driver code
if __name__=="__main__":
''' 10
/ \
48 3
/ \
11 37
/ \ / \
7 29 42 19
/
7
'''
# Create Binary Tree as shown
root = newNode(10)
root.left = newNode(48)
root.right = newNode(3)
root.right.left = newNode(11)
root.right.right = newNode(37)
root.right.left.left = newNode(7)
root.right.left.right = newNode(29)
root.right.right.left = newNode(42)
root.right.right.right = newNode(19)
root.right.right.right.left = newNode(7)
# Print Co-Prime Paths
printCo_PrimePath(root)
# This code is contributed by rutvik_56C#
// C# program for printing Co-prime
// paths of binary Tre
using System;
using System.Collections.Generic;
class GFG{
// A Tree node
class Node {
public int key;
public Node left, right;
};
// Utility function to create
// a new node
static Node newNode(int key)
{
Node temp = new Node();
temp.key = key;
temp.left = temp.right = null;
return (temp);
}
static int N = 1000000;
// List to store all the
// prime numbers
static List prime = new List();
// Function to store all the
// prime numbers in an array
static void SieveOfEratosthenes()
{
// Create a bool array "prime[0..N]"
// and initialize all the entries in it
// as true. A value in prime[i]
// will finally be false if
// i is Not a prime, else true.
bool []check = new bool[N + 1];
for(int i=0;i<=N;i++)
check[i] = true;
for (int p = 2; p * p <= N; p++) {
// If prime[p] is not changed,
// then it is a prime
if (check[p] == true) {
prime.Add(p);
// Update all multiples of p
// greater than or equal to
// the square of it
// numbers which are multiples of p
// and are less than p^2
// are already marked.
for (int i = p * p; i <= N; i += p)
check[i] = false;
}
}
}
// Function to check whether Path
// is Co-prime or not
static bool isPathCo_Prime(List path)
{
int max = 0;
// Iterating through the array
// to find the maximum element
// in the array
foreach (int x in path) {
if (max < x)
max = x;
}
for (int i = 0;
i * prime[i] <= max / 2;
i++) {
int ct = 0;
// Incrementing the variable
// if any of the value has
// a factor
foreach (int x in path) {
if (x % prime[i] == 0)
ct++;
}
// If not co-prime
if (ct > 1) {
return false;
}
}
return true;
}
// Function to print a Co-Prime path
static void printCo_PrimePaths(List path)
{
foreach (int x in path) {
Console.Write(x+ " ");
}
Console.WriteLine();
}
// Function to find co-prime paths of
// binary tree
static void findCo_PrimePaths(Node root,
List path)
{
// Base case
if (root == null)
return;
// Store the value in path vector
path.Add(root.key);
// Recursively call for left sub tree
findCo_PrimePaths(root.left, path);
// Recursively call for right sub tree
findCo_PrimePaths(root.right, path);
// Condition to check, if leaf node
if (root.left == null
&& root.right == null) {
// Condition to check,
// if path co-prime or not
if (isPathCo_Prime(path)) {
// Print co-prime path
printCo_PrimePaths(path);
}
}
// Remove the last element
// from the path vector
path.RemoveAt(path.Count-1);
}
// Function to find Co-Prime paths
// In a given binary tree
static void printCo_PrimePaths(Node node)
{
// To save all prime numbers
SieveOfEratosthenes();
List path = new List();
// Function call
findCo_PrimePaths(node, path);
}
// Driver Code
public static void Main(String[] args)
{
/* 10
/ \
48 3
/ \
11 37
/ \ / \
7 29 42 19
/
7
*/
// Create Binary Tree as shown
Node root = newNode(10);
root.left = newNode(48);
root.right = newNode(3);
root.right.left = newNode(11);
root.right.right = newNode(37);
root.right.left.left = newNode(7);
root.right.left.right = newNode(29);
root.right.right.left = newNode(42);
root.right.right.right = newNode(19);
root.right.right.right.left = newNode(7);
// Print Co-Prime Paths
printCo_PrimePaths(root);
}
}
// This code is contributed by 29AjayKumar 输出：
10 3 11 7
10 3 11 29
10 3 37 19 7