Dada una array arr[] de enteros positivos donde 2 ≤ arr[i] ≤ 10 6 para todos los valores posibles de i . La tarea es verificar si existe al menos un elemento en la array dada que forme un par coprimo con todos los demás elementos de la array. Si no existe tal elemento, imprima No else imprima Sí .
Ejemplos:
Entrada: arr[] = {2, 8, 4, 10, 6, 7}
Salida: Sí
, 7 es coprimo con todos los demás elementos de la arrayEntrada: arr[] = {3, 6, 9, 12}
Salida: No
Enfoque ingenuo: una solución simple es comprobar si el gcd de cada elemento con todos los demás elementos es igual a 1. La complejidad temporal de esta solución es O(n 2 ) .
Enfoque eficiente: una solución eficiente es generar todos los factores primos de los números enteros en la array dada. Usando hash, almacene el conteo de cada elemento que es un factor primo de cualquiera de los números en la array. Si el elemento no contiene ningún factor primo común con otros elementos, siempre forma un par coprimo con otros elementos.
Para generar factores primos, consulte el artículo Factorización prima usando Sieve en O (log n)
A continuación se muestra la implementación del enfoque anterior:
C++
// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;
#define MAXN 1000001
// Stores smallest prime factor for every number
int spf[MAXN];
// Hash to store prime factors count
int hash1[MAXN] = { 0 };
// Function to calculate SPF (Smallest Prime Factor)
// for every number till MAXN
void sieve()
{
spf[1] = 1;
for (int i = 2; i < MAXN; i++)
// Marking smallest prime factor for every
// number to be itself
spf[i] = i;
// Separately marking spf for every even
// number as 2
for (int i = 4; i < MAXN; i += 2)
spf[i] = 2;
// Checking if i is prime
for (int i = 3; i * i < MAXN; i++) {
// Marking SPF for all numbers divisible by i
if (spf[i] == i) {
for (int j = i * i; j < MAXN; j += i)
// Marking spf[j] if it is not
// previously marked
if (spf[j] == j)
spf[j] = i;
}
}
}
// Function to store the prime factors after dividing
// by the smallest prime factor at every step
void getFactorization(int x)
{
int temp;
while (x != 1) {
temp = spf[x];
if (x % temp == 0) {
// Storing the count of
// prime factors in hash
hash1[spf[x]]++;
x = x / spf[x];
}
while (x % temp == 0)
x = x / temp;
}
}
// Function that returns true if there are
// no common prime factors between x
// and other numbers of the array
bool check(int x)
{
int temp;
while (x != 1) {
temp = spf[x];
// Checking whether it common
// prime factor with other numbers
if (x % temp == 0 && hash1[temp] > 1)
return false;
while (x % temp == 0)
x = x / temp;
}
return true;
}
// Function that returns true if there is
// an element in the array which is coprime
// with all the other elements of the array
bool hasValidNum(int arr[], int n)
{
// Using sieve for generating prime factors
sieve();
for (int i = 0; i < n; i++)
getFactorization(arr[i]);
// Checking the common prime factors
// with other numbers
for (int i = 0; i < n; i++)
if (check(arr[i]))
return true;
return false;
}
// Driver code
int main()
{
int arr[] = { 2, 8, 4, 10, 6, 7 };
int n = sizeof(arr) / sizeof(arr[0]);
if (hasValidNum(arr, n))
cout << "Yes";
else
cout << "No";
return 0;
}
Java
// Java implementation of the approach
class GFG
{
static int MAXN = 1000001;
// Stores smallest prime factor for every number
static int[] spf = new int[MAXN];
// Hash to store prime factors count
static int[] hash1 = new int[MAXN];
// Function to calculate SPF (Smallest Prime Factor)
// for every number till MAXN
static void sieve()
{
spf[1] = 1;
for (int i = 2; i < MAXN; i++)
// Marking smallest prime factor for every
// number to be itself
spf[i] = i;
// Separately marking spf for every even
// number as 2
for (int i = 4; i < MAXN; i += 2)
spf[i] = 2;
// Checking if i is prime
for (int i = 3; i * i < MAXN; i++)
{
// Marking SPF for all numbers divisible by i
if (spf[i] == i)
{
for (int j = i * i; j < MAXN; j += i)
// Marking spf[j] if it is not
// previously marked
if (spf[j] == j)
spf[j] = i;
}
}
}
// Function to store the prime factors after dividing
// by the smallest prime factor at every step
static void getFactorization(int x)
{
int temp;
while (x != 1)
{
temp = spf[x];
if (x % temp == 0)
{
// Storing the count of
// prime factors in hash
hash1[spf[x]]++;
x = x / spf[x];
}
while (x % temp == 0)
x = x / temp;
}
}
// Function that returns true if there are
// no common prime factors between x
// and other numbers of the array
static boolean check(int x)
{
int temp;
while (x != 1)
{
temp = spf[x];
// Checking whether it common
// prime factor with other numbers
if (x % temp == 0 && hash1[temp] > 1)
return false;
while (x % temp == 0)
x = x / temp;
}
return true;
}
// Function that returns true if there is
// an element in the array which is coprime
// with all the other elements of the array
static boolean hasValidNum(int []arr, int n)
{
// Using sieve for generating prime factors
sieve();
for (int i = 0; i < n; i++)
getFactorization(arr[i]);
// Checking the common prime factors
// with other numbers
for (int i = 0; i < n; i++)
if (check(arr[i]))
return true;
return false;
}
// Driver code
public static void main (String[] args)
{
int []arr = { 2, 8, 4, 10, 6, 7 };
int n = arr.length;
if (hasValidNum(arr, n))
System.out.println("Yes");
else
System.out.println("No");
}
}
// This code is contributed by chandan_jnu
Python3
# Python3 implementation of the approach
MAXN = 1000001
# Stores smallest prime factor for
# every number
spf = [i for i in range(MAXN)]
# Hash to store prime factors count
hash1 = [0 for i in range(MAXN)]
# Function to calculate SPF (Smallest
# Prime Factor) for every number till MAXN
def sieve():
# Separately marking spf for
# every even number as 2
for i in range(4, MAXN, 2):
spf[i] = 2
# Checking if i is prime
for i in range(3, MAXN):
if i * i >= MAXN:
break
# Marking SPF for all numbers
# divisible by i
if (spf[i] == i):
for j in range(i * i, MAXN, i):
# Marking spf[j] if it is not
# previously marked
if (spf[j] == j):
spf[j] = i
# Function to store the prime factors
# after dividing by the smallest prime
# factor at every step
def getFactorization(x):
while (x != 1):
temp = spf[x]
if (x % temp == 0):
# Storing the count of
# prime factors in hash
hash1[spf[x]] += 1
x = x // spf[x]
while (x % temp == 0):
x = x // temp
# Function that returns true if there
# are no common prime factors between x
# and other numbers of the array
def check(x):
while (x != 1):
temp = spf[x]
# Checking whether it common
# prime factor with other numbers
if (x % temp == 0 and hash1[temp] > 1):
return False
while (x % temp == 0):
x = x //temp
return True
# Function that returns true if there is
# an element in the array which is coprime
# with all the other elements of the array
def hasValidNum(arr, n):
# Using sieve for generating
# prime factors
sieve()
for i in range(n):
getFactorization(arr[i])
# Checking the common prime factors
# with other numbers
for i in range(n):
if (check(arr[i])):
return True
return False
# Driver code
arr = [2, 8, 4, 10, 6, 7]
n = len(arr)
if (hasValidNum(arr, n)):
print("Yes")
else:
print("No")
# This code is contributed by mohit kumar
C#
// C# implementation of the approach
using System;
class GFG
{
static int MAXN=1000001;
// Stores smallest prime factor for every number
static int[] spf = new int[MAXN];
// Hash to store prime factors count
static int[] hash1 = new int[MAXN];
// Function to calculate SPF (Smallest Prime Factor)
// for every number till MAXN
static void sieve()
{
spf[1] = 1;
for (int i = 2; i < MAXN; i++)
// Marking smallest prime factor for every
// number to be itself
spf[i] = i;
// Separately marking spf for every even
// number as 2
for (int i = 4; i < MAXN; i += 2)
spf[i] = 2;
// Checking if i is prime
for (int i = 3; i * i < MAXN; i++)
{
// Marking SPF for all numbers divisible by i
if (spf[i] == i)
{
for (int j = i * i; j < MAXN; j += i)
// Marking spf[j] if it is not
// previously marked
if (spf[j] == j)
spf[j] = i;
}
}
}
// Function to store the prime factors after dividing
// by the smallest prime factor at every step
static void getFactorization(int x)
{
int temp;
while (x != 1)
{
temp = spf[x];
if (x % temp == 0)
{
// Storing the count of
// prime factors in hash
hash1[spf[x]]++;
x = x / spf[x];
}
while (x % temp == 0)
x = x / temp;
}
}
// Function that returns true if there are
// no common prime factors between x
// and other numbers of the array
static bool check(int x)
{
int temp;
while (x != 1)
{
temp = spf[x];
// Checking whether it common
// prime factor with other numbers
if (x % temp == 0 && hash1[temp] > 1)
return false;
while (x % temp == 0)
x = x / temp;
}
return true;
}
// Function that returns true if there is
// an element in the array which is coprime
// with all the other elements of the array
static bool hasValidNum(int []arr, int n)
{
// Using sieve for generating prime factors
sieve();
for (int i = 0; i < n; i++)
getFactorization(arr[i]);
// Checking the common prime factors
// with other numbers
for (int i = 0; i < n; i++)
if (check(arr[i]))
return true;
return false;
}
// Driver code
static void Main()
{
int []arr = { 2, 8, 4, 10, 6, 7 };
int n = arr.Length;
if (hasValidNum(arr, n))
Console.WriteLine("Yes");
else
Console.WriteLine("No");
}
}
// This code is contributed by chandan_jnu
PHP
<?php
// PHP implementation of the approach
$MAXN = 10001;
// Stores smallest prime factor for every number
$spf = array_fill(0, $MAXN, 0);
// Hash to store prime factors count
$hash1 = array_fill(0, $MAXN, 0);
// Function to calculate SPF (Smallest Prime Factor)
// for every number till MAXN
function sieve()
{
global $spf, $MAXN, $hash1;
$spf[1] = 1;
for ($i = 2; $i < $MAXN; $i++)
// Marking smallest prime factor for every
// number to be itself
$spf[$i] = $i;
// Separately marking spf for every even
// number as 2
for ($i = 4; $i < $MAXN; $i += 2)
$spf[$i] = 2;
// Checking if i is prime
for ($i = 3; $i * $i < $MAXN; $i++)
{
// Marking SPF for all numbers divisible by i
if ($spf[$i] == $i)
{
for ($j = $i * $i; $j < $MAXN; $j += $i)
// Marking spf[j] if it is not
// previously marked
if ($spf[$j] == $j)
$spf[$j] = $i;
}
}
}
// Function to store the prime factors after dividing
// by the smallest prime factor at every step
function getFactorization($x)
{
global $spf,$MAXN,$hash1;
while ($x != 1)
{
$temp = $spf[$x];
if ($x % $temp == 0)
{
// Storing the count of
// prime factors in hash
$hash1[$spf[$x]]++;
$x = (int)($x / $spf[$x]);
}
while ($x % $temp == 0)
$x = (int)($x / $temp);
}
}
// Function that returns true if there are
// no common prime factors between x
// and other numbers of the array
function check($x)
{
global $spf,$MAXN,$hash1;
while ($x != 1)
{
$temp = $spf[$x];
// Checking whether it common
// prime factor with other numbers
if ($x % $temp == 0 && $hash1[$temp] > 1)
return false;
while ($x % $temp == 0)
$x = (int)($x / $temp);
}
return true;
}
// Function that returns true if there is
// an element in the array which is coprime
// with all the other elements of the array
function hasValidNum($arr, $n)
{
global $spf,$MAXN,$hash1;
// Using sieve for generating prime factors
sieve();
for ($i = 0; $i < $n; $i++)
getFactorization($arr[$i]);
// Checking the common prime factors
// with other numbers
for ($i = 0; $i < $n; $i++)
if (check($arr[$i]))
return true;
return false;
}
// Driver code
$arr = array( 2, 8, 4, 10, 6, 7 );
$n = count($arr);
if (hasValidNum($arr, $n))
echo "Yes";
else
echo "No";
// This code is contributed by chandan_jnu
?>
Javascript
<script>
// Javascript implementation of the approach
let MAXN = 1000001;
// Stores smallest prime factor for every number
let spf = new Array(MAXN);
// Hash to store prime factors count
let hash1 = new Array(MAXN);
// Function to calculate SPF (Smallest Prime Factor)
// for every number till MAXN
function sieve()
{
spf[1] = 1;
for(let i = 2; i < MAXN; i++)
// Marking smallest prime factor for
// every number to be itself
spf[i] = i;
// Separately marking spf for every even
// number as 2
for(let i = 4; i < MAXN; i += 2)
spf[i] = 2;
// Checking if i is prime
for(let i = 3; i * i < MAXN; i++)
{
// Marking SPF for all numbers divisible by i
if (spf[i] == i)
{
for(let j = i * i; j < MAXN; j += i)
// Marking spf[j] if it is not
// previously marked
if (spf[j] == j)
spf[j] = i;
}
}
}
// Function to store the prime factors
// after dividing by the smallest prime
// factor at every step
function getFactorization(x)
{
let temp;
while (x != 1)
{
temp = spf[x];
if (x % temp == 0)
{
// Storing the count of
// prime factors in hash
hash1[spf[x]]++;
x = x / spf[x];
}
while (x % temp == 0)
x = x / temp;
}
}
// Function that returns true if there are
// no common prime factors between x
// and other numbers of the array
function check(x)
{
let temp;
while (x != 1)
{
temp = spf[x];
// Checking whether it common
// prime factor with other numbers
if (x % temp == 0 && hash1[temp] > 1)
return false;
while (x % temp == 0)
x = x / temp;
}
return true;
}
// Function that returns true if there is
// an element in the array which is coprime
// with all the other elements of the array
function hasValidNum(arr, n)
{
// Using sieve for generating prime factors
sieve();
for(let i = 0; i < n; i++)
getFactorization(arr[i]);
// Checking the common prime factors
// with other numbers
for(let i = 0; i < n; i++)
if (check(arr[i]))
return true;
return false;
}
// Driver code
let arr = [ 2, 8, 4, 10, 6, 7 ];
let n = arr.length;
if (hasValidNum(arr, n))
document.write("Yes");
else
document.write("No");
// This code is contributed by unknown2108
</script>
Producción:
Yes
Publicación traducida automáticamente
Artículo escrito por Sairahul Jella y traducido por Barcelona Geeks. The original can be accessed here. Licence: CCBY-SA