Dado un diccionario, un método para realizar una búsqueda en el diccionario y un tablero M x N donde cada celda tiene un carácter. Encuentra todas las palabras posibles que pueden estar formadas por una secuencia de caracteres adyacentes. Tenga en cuenta que podemos movernos a cualquiera de los 8 caracteres adyacentes, pero una palabra no debe tener varias instancias de la misma celda.
Ejemplo:
Input: dictionary[] = {"GEEKS", "FOR", "QUIZ", "GO"};
boggle[][] = {{'G', 'I', 'Z'},
{'U', 'E', 'K'},
{'Q', 'S', 'E'}};
Output: Following words of the dictionary are present
GEEKS
QUIZ
Explanation:
Input: dictionary[] = {"GEEKS", "ABCFIHGDE"};
boggle[][] = {{'A', 'B', 'C'},
{'D', 'E', 'F'},
{'G', 'H', 'I'}};
Output: Following words of the dictionary are present
ABCFIHGDE
Explanation:
Hemos discutido una solución basada en Graph DFS en la publicación a continuación.
Boggle (Encuentra todas las palabras posibles en un tablero de personajes) | Conjunto 1
Aquí discutimos una solución basada en Trie que es mejor que la solución basada en DFS.
Diccionario de diccionario dado[] = {“GEEKS”, “PARA”, “PRUEBA”, “GO”}
1. Cree un trie vacío e inserte todas las palabras del diccionario dado en el trie
After insertion, Trie looks like(leaf nodes are in RED)
root
/
G F Q
/ | | |
O E O U
| | |
E R I
| |
K Z
|
S
2. Después de eso, hemos elegido solo aquellos caracteres en boggle[][] que son hijos de la raíz de Trie
Let para arriba, elegimos ‘G’ boggle[0][0], ‘Q’ boggle[2][0] ( ambos están presentes en boggle matrix)
3. busque una palabra en un trie que comience con el carácter que elegimos en el paso 2
1) Create bool visited boolean matrix (Visited[M][N] = false )
2) Call SearchWord() for every cell (i, j) which has one of the
first characters of dictionary words. In above example,
we have 'G' and 'Q' as first characters.
SearchWord(Trie *root, i, j, visited[][N])
if root->leaf == true
print word
if we have seen this element first time then make it visited.
visited[i][j] = true
do
traverse all child of current root
k goes (0 to 26 ) [there are only 26 Alphabet]
add current char and search for next character
find next character which is adjacent to boggle[i][j]
they are 8 adjacent cells of boggle[i][j] (i+1, j+1),
(i+1, j) (i-1, j) and so on.
make it unvisited visited[i][j] = false
A continuación se muestra la implementación de la idea anterior:
C++
// C++ program for Boggle game
#include <bits/stdc++.h>
using namespace std;
// Converts key current character into index
// use only 'A' through 'Z'
#define char_int(c) ((int)c - (int)'A')
// Alphabet size
#define SIZE (26)
#define M 3
#define N 3
// trie Node
struct TrieNode {
TrieNode* Child[SIZE];
// isLeaf is true if the node represents
// end of a word
bool leaf;
};
// Returns new trie node (initialized to NULLs)
TrieNode* getNode()
{
TrieNode* newNode = new TrieNode;
newNode->leaf = false;
for (int i = 0; i < SIZE; i++)
newNode->Child[i] = NULL;
return newNode;
}
// If not present, inserts a key into the trie
// If the key is a prefix of trie node, just
// marks leaf node
void insert(TrieNode* root, char* Key)
{
int n = strlen(Key);
TrieNode* pChild = root;
for (int i = 0; i < n; i++) {
int index = char_int(Key[i]);
if (pChild->Child[index] == NULL)
pChild->Child[index] = getNode();
pChild = pChild->Child[index];
}
// make last node as leaf node
pChild->leaf = true;
}
// function to check that current location
// (i and j) is in matrix range
bool isSafe(int i, int j, bool visited[M][N])
{
return (i >= 0 && i < M && j >= 0 && j < N && !visited[i][j]);
}
// A recursive function to print all words present on boggle
void searchWord(TrieNode* root, char boggle[M][N], int i,
int j, bool visited[][N], string str)
{
// if we found word in trie / dictionary
if (root->leaf == true)
cout << str << endl;
// If both I and j in range and we visited
// that element of matrix first time
if (isSafe(i, j, visited)) {
// make it visited
visited[i][j] = true;
// traverse all childs of current root
for (int K = 0; K < SIZE; K++) {
if (root->Child[K] != NULL) {
// current character
char ch = (char)K + (char)'A';
// Recursively search reaming character of word
// in trie for all 8 adjacent cells of boggle[i][j]
if (isSafe(i + 1, j + 1, visited)
&& boggle[i + 1][j + 1] == ch)
searchWord(root->Child[K], boggle,
i + 1, j + 1, visited, str + ch);
if (isSafe(i, j + 1, visited)
&& boggle[i][j + 1] == ch)
searchWord(root->Child[K], boggle,
i, j + 1, visited, str + ch);
if (isSafe(i - 1, j + 1, visited)
&& boggle[i - 1][j + 1] == ch)
searchWord(root->Child[K], boggle,
i - 1, j + 1, visited, str + ch);
if (isSafe(i + 1, j, visited)
&& boggle[i + 1][j] == ch)
searchWord(root->Child[K], boggle,
i + 1, j, visited, str + ch);
if (isSafe(i + 1, j - 1, visited)
&& boggle[i + 1][j - 1] == ch)
searchWord(root->Child[K], boggle,
i + 1, j - 1, visited, str + ch);
if (isSafe(i, j - 1, visited)
&& boggle[i][j - 1] == ch)
searchWord(root->Child[K], boggle,
i, j - 1, visited, str + ch);
if (isSafe(i - 1, j - 1, visited)
&& boggle[i - 1][j - 1] == ch)
searchWord(root->Child[K], boggle,
i - 1, j - 1, visited, str + ch);
if (isSafe(i - 1, j, visited)
&& boggle[i - 1][j] == ch)
searchWord(root->Child[K], boggle,
i - 1, j, visited, str + ch);
}
}
// make current element unvisited
visited[i][j] = false;
}
}
// Prints all words present in dictionary.
void findWords(char boggle[M][N], TrieNode* root)
{
// Mark all characters as not visited
bool visited[M][N];
memset(visited, false, sizeof(visited));
TrieNode* pChild = root;
string str = "";
// traverse all matrix elements
for (int i = 0; i < M; i++) {
for (int j = 0; j < N; j++) {
// we start searching for word in dictionary
// if we found a character which is child
// of Trie root
if (pChild->Child[char_int(boggle[i][j])]) {
str = str + boggle[i][j];
searchWord(pChild->Child[char_int(boggle[i][j])],
boggle, i, j, visited, str);
str = "";
}
}
}
}
// Driver program to test above function
int main()
{
// Let the given dictionary be following
char* dictionary[] = { "GEEKS", "FOR", "QUIZ", "GEE" };
// root Node of trie
TrieNode* root = getNode();
// insert all words of dictionary into trie
int n = sizeof(dictionary) / sizeof(dictionary[0]);
for (int i = 0; i < n; i++)
insert(root, dictionary[i]);
char boggle[M][N] = { { 'G', 'I', 'Z' },
{ 'U', 'E', 'K' },
{ 'Q', 'S', 'E' } };
findWords(boggle, root);
return 0;
}
Java
// Java program for Boggle game
public class Boggle {
// Alphabet size
static final int SIZE = 26;
static final int M = 3;
static final int N = 3;
// trie Node
static class TrieNode {
TrieNode[] Child = new TrieNode[SIZE];
// isLeaf is true if the node represents
// end of a word
boolean leaf;
// constructor
public TrieNode()
{
leaf = false;
for (int i = 0; i < SIZE; i++)
Child[i] = null;
}
}
// If not present, inserts a key into the trie
// If the key is a prefix of trie node, just
// marks leaf node
static void insert(TrieNode root, String Key)
{
int n = Key.length();
TrieNode pChild = root;
for (int i = 0; i < n; i++) {
int index = Key.charAt(i) - 'A';
if (pChild.Child[index] == null)
pChild.Child[index] = new TrieNode();
pChild = pChild.Child[index];
}
// make last node as leaf node
pChild.leaf = true;
}
// function to check that current location
// (i and j) is in matrix range
static boolean isSafe(int i, int j, boolean visited[][])
{
return (i >= 0 && i < M && j >= 0
&& j < N && !visited[i][j]);
}
// A recursive function to print
// all words present on boggle
static void searchWord(TrieNode root, char boggle[][], int i,
int j, boolean visited[][], String str)
{
// if we found word in trie / dictionary
if (root.leaf == true)
System.out.println(str);
// If both I and j in range and we visited
// that element of matrix first time
if (isSafe(i, j, visited)) {
// make it visited
visited[i][j] = true;
// traverse all child of current root
for (int K = 0; K < SIZE; K++) {
if (root.Child[K] != null) {
// current character
char ch = (char)(K + 'A');
// Recursively search reaming character of word
// in trie for all 8 adjacent cells of
// boggle[i][j]
if (isSafe(i + 1, j + 1, visited)
&& boggle[i + 1][j + 1] == ch)
searchWord(root.Child[K], boggle,
i + 1, j + 1,
visited, str + ch);
if (isSafe(i, j + 1, visited)
&& boggle[i][j + 1] == ch)
searchWord(root.Child[K], boggle,
i, j + 1,
visited, str + ch);
if (isSafe(i - 1, j + 1, visited)
&& boggle[i - 1][j + 1] == ch)
searchWord(root.Child[K], boggle,
i - 1, j + 1,
visited, str + ch);
if (isSafe(i + 1, j, visited)
&& boggle[i + 1][j] == ch)
searchWord(root.Child[K], boggle,
i + 1, j,
visited, str + ch);
if (isSafe(i + 1, j - 1, visited)
&& boggle[i + 1][j - 1] == ch)
searchWord(root.Child[K], boggle,
i + 1, j - 1,
visited, str + ch);
if (isSafe(i, j - 1, visited)
&& boggle[i][j - 1] == ch)
searchWord(root.Child[K], boggle,
i, j - 1,
visited, str + ch);
if (isSafe(i - 1, j - 1, visited)
&& boggle[i - 1][j - 1] == ch)
searchWord(root.Child[K], boggle,
i - 1, j - 1,
visited, str + ch);
if (isSafe(i - 1, j, visited)
&& boggle[i - 1][j] == ch)
searchWord(root.Child[K], boggle,
i - 1, j,
visited, str + ch);
}
}
// make current element unvisited
visited[i][j] = false;
}
}
// Prints all words present in dictionary.
static void findWords(char boggle[][], TrieNode root)
{
// Mark all characters as not visited
boolean[][] visited = new boolean[M][N];
TrieNode pChild = root;
String str = "";
// traverse all matrix elements
for (int i = 0; i < M; i++) {
for (int j = 0; j < N; j++) {
// we start searching for word in dictionary
// if we found a character which is child
// of Trie root
if (pChild.Child[(boggle[i][j]) - 'A'] != null) {
str = str + boggle[i][j];
searchWord(pChild.Child[(boggle[i][j]) - 'A'],
boggle, i, j, visited, str);
str = "";
}
}
}
}
// Driver program to test above function
public static void main(String args[])
{
// Let the given dictionary be following
String dictionary[] = { "GEEKS", "FOR", "QUIZ", "GEE" };
// root Node of trie
TrieNode root = new TrieNode();
// insert all words of dictionary into trie
int n = dictionary.length;
for (int i = 0; i < n; i++)
insert(root, dictionary[i]);
char boggle[][] = { { 'G', 'I', 'Z' },
{ 'U', 'E', 'K' },
{ 'Q', 'S', 'E' } };
findWords(boggle, root);
}
}
// This code is contributed by Sumit Ghosh
Python3
# Python program for Boggle game
class TrieNode:
# Trie node class
def __init__(self):
self.children = [None] * 26
# isEndOfWord is True if node represent the end of the word
self.isEndOfWord = False
M = 3
N = 3
class Boogle:
# Trie data structure class
def __init__(self):
self.root = self.getNode()
def getNode(self):
# Returns new trie node (initialized to NULLs)
return TrieNode()
def _charToIndex(self, ch):
# private helper function
# Converts key current character into index
# use only 'A' through 'Z' and upper case
return ord(ch) - ord('A')
def insert(self, key):
# If not present, inserts key into trie
# If the key is prefix of trie node,
# just marks leaf node
pCrawl = self.root
length = len(key)
for level in range(length):
index = self._charToIndex(key[level])
# if current character is not present
if not pCrawl.children[index]:
pCrawl.children[index] = self.getNode()
pCrawl = pCrawl.children[index]
# print('h', self.root.children)
# mark last node as leaf
pCrawl.isEndOfWord = True
def is_Safe(i, j, vis):
return 0 <= i < M and 0 <= j < N and not vis[i][j]
def search_word(root, boggle, i, j, vis, string):
if root.isEndOfWord:
print(string)
if is_Safe(i, j, vis):
vis[i][j] = True
for K in range(26):
if root.children[K] is not None:
ch = chr(K+ord('A'))
# Recursively search reaming character of word
# in trie for all 8 adjacent cells of boggle[i][j]
if is_Safe(i + 1, j + 1, vis) and boggle[i + 1][j + 1] == ch:
search_word(root.children[K], boggle,
i + 1, j + 1, vis, string + ch)
if is_Safe(i, j + 1, vis) and boggle[i][j + 1] == ch:
search_word(root.children[K], boggle,
i, j + 1, vis, string + ch)
if is_Safe(i - 1, j + 1, vis) and boggle[i - 1][j + 1] == ch:
search_word(root.children[K], boggle,
i - 1, j + 1, vis, string + ch)
if is_Safe(i + 1, j, vis) and boggle[i + 1][j] == ch:
search_word(root.children[K], boggle,
i + 1, j, vis, string + ch)
if is_Safe(i + 1, j - 1, vis) and boggle[i + 1][j - 1] == ch:
search_word(root.children[K], boggle,
i + 1, j - 1, vis, string + ch)
if is_Safe(i, j - 1, vis) and boggle[i][j - 1] == ch:
search_word(root.children[K], boggle,
i, j - 1, vis, string + ch)
if is_Safe(i - 1, j - 1, vis) and boggle[i - 1][j - 1] == ch:
search_word(root.children[K], boggle,
i - 1, j - 1, vis, string + ch)
if is_Safe(i - 1, j, vis) and boggle[i - 1][j] == ch:
search_word(root.children[K], boggle,
i - 1, j, vis, string + ch)
vis[i][j] = False
def char_int(ch):
# private helper function
# Converts key current character into index
# use only 'A' through 'Z' and upper case
return ord(ch) - ord('A')
def findWords(boggle, root):
# Mark all characters as not visited
visited = [[False for i in range(N)] for i in range(M)]
pChild = root
string = ""
# traverse all matrix elements
for i in range(M):
for j in range(N):
# we start searching for word in dictionary
# if we found a character which is child
# of Trie root
if pChild.children[char_int(boggle[i][j])]:
# print('h')
string = string + boggle[i][j]
search_word(pChild.children[char_int(boggle[i][j])],
boggle, i, j, visited, string)
string = ""
dictionary = ["GEEKS", "FOR", "QUIZ", "GEE"]
# root Node of trie
t = Boogle()
# insert all words of dictionary into trie
n = len(dictionary)
for i in range(n):
t.insert(dictionary[i])
root = t.root
boggle = [['G', 'I', 'Z'],
['U', 'E', 'K'],
['Q', 'S', 'E']]
# print(root.children)
findWords(boggle, root)
# This code is contributed by Yashwant Kumar
C#
// C# program for Boggle game
using System;
public class Boggle {
// Alphabet size
static readonly int SIZE = 26;
static readonly int M = 3;
static readonly int N = 3;
// trie Node
public class TrieNode {
public TrieNode[] Child = new TrieNode[SIZE];
// isLeaf is true if the node represents
// end of a word
public bool leaf;
// constructor
public TrieNode()
{
leaf = false;
for (int i = 0; i < SIZE; i++)
Child[i] = null;
}
}
// If not present, inserts a key into the trie
// If the key is a prefix of trie node, just
// marks leaf node
static void insert(TrieNode root, String Key)
{
int n = Key.Length;
TrieNode pChild = root;
for (int i = 0; i < n; i++) {
int index = Key[i] - 'A';
if (pChild.Child[index] == null)
pChild.Child[index] = new TrieNode();
pChild = pChild.Child[index];
}
// make last node as leaf node
pChild.leaf = true;
}
// function to check that current location
// (i and j) is in matrix range
static bool isSafe(int i, int j, bool[, ] visited)
{
return (i >= 0 && i < M && j >= 0 && j < N && !visited[i, j]);
}
// A recursive function to print all words present on boggle
static void searchWord(TrieNode root, char[, ] boggle, int i,
int j, bool[, ] visited, String str)
{
// if we found word in trie / dictionary
if (root.leaf == true)
Console.WriteLine(str);
// If both I and j in range and we visited
// that element of matrix first time
if (isSafe(i, j, visited)) {
// make it visited
visited[i, j] = true;
// traverse all child of current root
for (int K = 0; K < SIZE; K++) {
if (root.Child[K] != null) {
// current character
char ch = (char)(K + 'A');
// Recursively search reaming character of word
// in trie for all 8 adjacent cells of
// boggle[i, j]
if (isSafe(i + 1, j + 1, visited) && boggle[i + 1, j + 1] == ch)
searchWord(root.Child[K], boggle, i + 1, j + 1,
visited, str + ch);
if (isSafe(i, j + 1, visited) && boggle[i, j + 1] == ch)
searchWord(root.Child[K], boggle, i, j + 1,
visited, str + ch);
if (isSafe(i - 1, j + 1, visited) && boggle[i - 1, j + 1] == ch)
searchWord(root.Child[K], boggle, i - 1, j + 1,
visited, str + ch);
if (isSafe(i + 1, j, visited) && boggle[i + 1, j] == ch)
searchWord(root.Child[K], boggle, i + 1, j,
visited, str + ch);
if (isSafe(i + 1, j - 1, visited) && boggle[i + 1, j - 1] == ch)
searchWord(root.Child[K], boggle, i + 1, j - 1,
visited, str + ch);
if (isSafe(i, j - 1, visited) && boggle[i, j - 1] == ch)
searchWord(root.Child[K], boggle, i, j - 1,
visited, str + ch);
if (isSafe(i - 1, j - 1, visited) && boggle[i - 1, j - 1] == ch)
searchWord(root.Child[K], boggle, i - 1, j - 1,
visited, str + ch);
if (isSafe(i - 1, j, visited) && boggle[i - 1, j] == ch)
searchWord(root.Child[K], boggle, i - 1, j,
visited, str + ch);
}
}
// make current element unvisited
visited[i, j] = false;
}
}
// Prints all words present in dictionary.
static void findWords(char[, ] boggle, TrieNode root)
{
// Mark all characters as not visited
bool[, ] visited = new bool[M, N];
TrieNode pChild = root;
String str = "";
// traverse all matrix elements
for (int i = 0; i < M; i++) {
for (int j = 0; j < N; j++) {
// we start searching for word in dictionary
// if we found a character which is child
// of Trie root
if (pChild.Child[(boggle[i, j]) - 'A'] != null) {
str = str + boggle[i, j];
searchWord(pChild.Child[(boggle[i, j]) - 'A'],
boggle, i, j, visited, str);
str = "";
}
}
}
}
// Driver program to test above function
public static void Main(String[] args)
{
// Let the given dictionary be following
String[] dictionary = { "GEEKS", "FOR", "QUIZ", "GEE" };
// root Node of trie
TrieNode root = new TrieNode();
// insert all words of dictionary into trie
int n = dictionary.Length;
for (int i = 0; i < n; i++)
insert(root, dictionary[i]);
char[, ] boggle = { { 'G', 'I', 'Z' },
{ 'U', 'E', 'K' },
{ 'Q', 'S', 'E' } };
findWords(boggle, root);
}
}
// This code has been contributed by 29AjayKumar
Javascript
<script>
// Javascript program for Boggle game
// Alphabet size
let SIZE = 26;
let M = 3;
let N = 3;
// trie Node
class TrieNode
{
constructor()
{
this.leaf=false;
this.Child = new Array(SIZE);
for (let i = 0; i < SIZE; i++)
this.Child[i]=null;
}
}
// If not present, inserts a key into the trie
// If the key is a prefix of trie node, just
// marks leaf node
function insert(root,Key)
{
let n = Key.length;
let pChild = root;
for (let i = 0; i < n; i++) {
let index = Key[i].charCodeAt(0) - 'A'.charCodeAt(0);
if (pChild.Child[index] == null)
pChild.Child[index] = new TrieNode();
pChild = pChild.Child[index];
}
// make last node as leaf node
pChild.leaf = true;
}
// function to check that current location
// (i and j) is in matrix range
function isSafe(i,j,visited)
{
return (i >= 0 && i < M && j >= 0
&& j < N && !visited[i][j]);
}
// A recursive function to print
// all words present on boggle
function searchWord(root,boggle,i,j,visited,str)
{
// if we found word in trie / dictionary
if (root.leaf == true)
document.write(str+" <br>");
// If both I and j in range and we visited
// that element of matrix first time
if (isSafe(i, j, visited)) {
// make it visited
visited[i][j] = true;
// traverse all child of current root
for (let K = 0; K < SIZE; K++) {
if (root.Child[K] != null) {
// current character
let ch = String.fromCharCode(K + 'A'.charCodeAt(0));
// Recursively search reaming character of word
// in trie for all 8 adjacent cells of
// boggle[i][j]
if (isSafe(i + 1, j + 1, visited)
&& boggle[i + 1][j + 1] == ch)
searchWord(root.Child[K], boggle,
i + 1, j + 1,
visited, str + ch);
if (isSafe(i, j + 1, visited)
&& boggle[i][j + 1] == ch)
searchWord(root.Child[K], boggle,
i, j + 1,
visited, str + ch);
if (isSafe(i - 1, j + 1, visited)
&& boggle[i - 1][j + 1] == ch)
searchWord(root.Child[K], boggle,
i - 1, j + 1,
visited, str + ch);
if (isSafe(i + 1, j, visited)
&& boggle[i + 1][j] == ch)
searchWord(root.Child[K], boggle,
i + 1, j,
visited, str + ch);
if (isSafe(i + 1, j - 1, visited)
&& boggle[i + 1][j - 1] == ch)
searchWord(root.Child[K], boggle,
i + 1, j - 1,
visited, str + ch);
if (isSafe(i, j - 1, visited)
&& boggle[i][j - 1] == ch)
searchWord(root.Child[K], boggle,
i, j - 1,
visited, str + ch);
if (isSafe(i - 1, j - 1, visited)
&& boggle[i - 1][j - 1] == ch)
searchWord(root.Child[K], boggle,
i - 1, j - 1,
visited, str + ch);
if (isSafe(i - 1, j, visited)
&& boggle[i - 1][j] == ch)
searchWord(root.Child[K], boggle,
i - 1, j,
visited, str + ch);
}
}
// make current element unvisited
visited[i][j] = false;
}
}
// Prints all words present in dictionary.
function findWords(boggle,root)
{
// Mark all characters as not visited
let visited = new Array(M);
for(let i=0;i<M;i++)
{
visited[i]=new Array(N);
for(let j=0;j<N;j++)
{
visited[i][j]=false;
}
}
let pChild = root;
let str = "";
// traverse all matrix elements
for (let i = 0; i < M; i++) {
for (let j = 0; j < N; j++) {
// we start searching for word in dictionary
// if we found a character which is child
// of Trie root
if (pChild.Child[(boggle[i][j]).charCodeAt(0) - 'A'.charCodeAt(0)] != null) {
str = str + boggle[i][j];
searchWord(pChild.Child[(boggle[i][j]).charCodeAt(0) - 'A'.charCodeAt(0)],
boggle, i, j, visited, str);
str = "";
}
}
}
}
// Driver program to test above function
let dictionary=["GEEKS", "FOR", "QUIZ", "GEE" ];
// root Node of trie
let root = new TrieNode();
// insert all words of dictionary into trie
let n = dictionary.length;
for (let i = 0; i < n; i++)
insert(root, dictionary[i]);
let boggle = [[ 'G', 'I', 'Z' ],
[ 'U', 'E', 'K' ],
[ 'Q', 'S', 'E' ]];
findWords(boggle, root);
// This code is contributed by rag2127
</script>
Producción:
GEE, GEEKS, QUIZ
Análisis de Complejidad:
- Complejidad del tiempo: O(4^(N^2)).
Incluso después de aplicar la prueba, la complejidad del tiempo sigue siendo la misma. Para cada celda hay 4 direcciones y hay N^2 celdas. Entonces la complejidad del tiempo es O(4^(N^2)). - Espacio auxiliar: O(N^2).
La longitud máxima de recursión puede ser N^2, donde N es el lado de la array. Entonces la Complejidad del espacio es O(N^2).
Este artículo es una contribución de Nishant Singh . Si te gusta GeeksforGeeks y te gustaría contribuir, también puedes escribir un artículo usando write.geeksforgeeks.org o enviar tu artículo por correo a review-team@geeksforgeeks.org. Vea su artículo que aparece en la página principal de GeeksforGeeks y ayude a otros Geeks.
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Publicación traducida automáticamente
Artículo escrito por GeeksforGeeks-1 y traducido por Barcelona Geeks. The original can be accessed here. Licence: CCBY-SA