algorithm - 如何有效地计算奇怪的数字
问题描述
我正在尝试打印 n 个奇怪的数字,其中 n 是非常大的数字(例如:10000)。
如果我有一些错误,我发现这个网站可以检查 n 600 的算法: http ://www.numbersaplenty.com/set/weird_number/more.php
但是,我的算法在更大的数字上真的很慢:
import java.util.ArrayList;
import java.util.List;
public class Test {
public static void main(String[] args) {
int n = 2;
for ( int count = 1 ; count <= 15000 ; n += 2 ) {
if (n % 6 == 0) {
continue;
}
List<Integer> properDivisors = getProperDivisors(n);
int divisorSum = properDivisors.stream().mapToInt(i -> i.intValue()).sum();
if ( isDeficient(divisorSum, n) ) {
continue;
}
if ( isWeird(n, properDivisors, divisorSum) ) {
System.out.printf("w(%d) = %d%n", count, n);
count++;
}
}
}
private static boolean isWeird(int n, List<Integer> divisors, int divisorSum) {
return isAbundant(divisorSum, n) && ! isSemiPerfect(divisors, n);
}
private static boolean isDeficient(int divisorSum, int n) {
return divisorSum < n;
}
private static boolean isAbundant(int divisorSum, int n) {
return divisorSum > n;
}
private static boolean isSemiPerfect(List<Integer> divisors, int sum) {
int size = divisors.size();
// The value of subset[i][j] will be true if there is a subset of divisors[0..j-1] with sum equal to i
boolean subset[][] = new boolean[sum+1][size+1];
// If sum is 0, then answer is true
for (int i = 0; i <= size; i++) {
subset[0][i] = true;
}
// If sum is not 0 and set is empty, then answer is false
for (int i = 1; i <= sum; i++) {
subset[i][0] = false;
}
// Fill the subset table in bottom up manner
for ( int i = 1 ; i <= sum ; i++ ) {
for ( int j = 1 ; j <= size ; j++ ) {
subset[i][j] = subset[i][j-1];
int test = divisors.get(j-1);
if ( i >= test ) {
subset[i][j] = subset[i][j] || subset[i - test][j-1];
}
}
}
return subset[sum][size];
}
private static final List<Integer> getProperDivisors(int number) {
List<Integer> divisors = new ArrayList<Integer>();
long sqrt = (long) Math.sqrt(number);
for ( int i = 1 ; i <= sqrt ; i++ ) {
if ( number % i == 0 ) {
divisors.add(i);
int div = number / i;
if ( div != i && div != number ) {
divisors.add(div);
}
}
}
return divisors;
}
}
我有三个简单的突破:
以上几点基于https://mathworld.wolfram.com/DeficientNumber.html
- 如果 aa 数字是奇数,至少对于 10^21 个数字来说它不会很奇怪(这对我想要获得的数字很有用)。
我使用的另一个优化是查找数字的所有除数的优化。我们不是循环到 n,而是循环到 SQRT(n)。
但是,我仍然需要优化: 1. isSemiPerfect 因为它真的很慢 2. 如果我可以进一步优化 getProperDivisors 它也会很好。
欢迎提出任何建议,因为我找不到更多优化来在合理的时间内找到 10000 个奇怪的数字。
PS:Java、C#、PHP 和 JavaScript 中的任何代码对我来说都可以。
编辑:我找到了这个主题并修改了 isSemiPerfect 看起来像这样。但是,它看起来并没有优化而是减慢了计算速度:
private static boolean isSemiPerfect(List<Integer> divisors, int n) {
BigInteger combinations = BigInteger.valueOf(2).pow(divisors.size());
for (BigInteger i = BigInteger.ZERO; i.compareTo(combinations) < 0; i = i.add(BigInteger.ONE)) {
int sum = 0;
for (int j = 0; j < i.bitLength(); j++) {
sum += i.testBit(j) ? divisors.get(j) : 0;
}
if (sum == n) {
return true;
}
}
return false;
}
解决方案
问题确实出在函数 isSemiPerfect 上。我用 C++ 转换了你的代码,它仍然很慢。
然后我通过使用回溯修改了这个函数。我现在在大约 15 秒内获得了前 15000 个奇怪的值。我的解释是,在几乎所有情况下,该值都是半完美的,并且回溯函数快速收敛。
还要注意,在我的回溯实现中,我对除数进行了排序,这样可以减少要检查的案例数量。
编辑 1:在 getProperDivisors 中更正了一个错误。最终结果似乎没有被修改!
#include <iostream>
#include <vector>
#include <cmath>
#include <numeric>
#include <algorithm>
// return true if sum is obtained
bool test_sum (std::vector<int>& arr, int amount) {
int n = arr.size();
std::sort(arr.begin(), arr.end(), std::greater<int>());
std::vector<int> bound (n);
std::vector<int> select (n);
bound[n-1] = arr[n-1];
for (int i = n-2; i >= 0; --i) {
bound[i] = bound[i+1] + arr[i];
}
int sum = 0; // current sum
int i = 0; // index of the coin being examined
bool up_down = true;
while (true) {
if (up_down) {
if (i == n || sum + bound[i] < amount) {
up_down = false;
i--;
continue;
}
sum += arr[i];
select[i] = 1;
if (sum == amount) return true;
if (sum < amount) {
i++;
continue;
}
up_down = false;
if (select[i] == 0) i--;
} else { // DOWN
if (i < 0) break;
if (select[i] == 0) {
i--;
} else {
sum -= arr[i];
select[i] = 0;
i++;
up_down = true;
}
}
}
return false;
}
bool isDeficient(int divisorSum, int n) {
return divisorSum < n;
}
bool isAbundant(int divisorSum, int n) {
return divisorSum > n;
}
bool isSemiPerfect(std::vector<int> &divisors, int sum) {
int size = divisors.size();
// The value of subset[i][j] will be true if there is a subset of divisors[0..j-1] with sum equal to i
//bool subset[sum+1][size+1];
std::vector<std::vector<bool>> subset(sum+1, std::vector<bool> (size+1));
// If sum is 0, then answer is true
for (int i = 0; i <= size; i++) {
subset[0][i] = true;
}
// If sum is not 0 and set is empty, then answer is false
for (int i = 1; i <= sum; i++) {
subset[i][0] = false;
}
// Fill the subset table in bottom up manner
for ( int i = 1 ; i <= sum ; i++ ) {
for ( int j = 1 ; j <= size ; j++ ) {
subset[i][j] = subset[i][j-1];
int test = divisors[j-1];
if ( i >= test ) {
subset[i][j] = subset[i][j] || subset[i - test][j-1];
}
}
}
return subset[sum][size];
}
bool isWeird(int n, std::vector<int> &divisors, int divisorSum) {
//return isAbundant(divisorSum, n) && !isSemiPerfect(divisors, n);
return isAbundant(divisorSum, n) && !test_sum(divisors, n);
}
std::vector<int> getProperDivisors_old(int number) {
std::vector<int> divisors;
long sqrtn = sqrt(number);
for ( int i = 1 ; i <= sqrtn ; i++ ) {
if ( number % i == 0 ) {
divisors.push_back(i);
int div = number / i;
if (div != i && div != number) {
divisors.push_back(div);
}
}
}
return divisors;
}
std::vector<int> getProperDivisors(int number) {
std::vector<int> divisors;
long sqrtn = sqrt(number);
divisors.push_back(1);
for ( int i = 2 ; i <= sqrtn ; i++ ) {
if (number % i == 0) {
divisors.push_back(i);
int div = number/i;
if (div != i) divisors.push_back(div);
}
}
return divisors;
}
int main() {
int n = 2, count;
std::vector<int> weird;
int Nweird = 15000;
for (count = 0; count < Nweird; n += 2) {
if (n % 6 == 0) continue;
auto properDivisors = getProperDivisors(n);
int divisorSum = std::accumulate (properDivisors.begin(), properDivisors.end(), 0);
if (isDeficient(divisorSum, n) ) {
continue;
}
if (isWeird(n, properDivisors, divisorSum)) {
//std::cout << count << " " << n << "\n";
weird.push_back (n);
count++;
}
}
for (int i = Nweird - 10; i < Nweird; ++i) {
std::cout << weird.at(i) << " ";
}
std::cout << "\n";
}
编辑 2除数的生成被完全重新定义。它现在使用素数分解。更复杂,但全球时间除以 7.5。在我的电脑上生成奇怪的数字现在需要 2 秒。
#include <iostream>
#include <vector>
#include <cmath>
#include <numeric>
#include <algorithm>
template <typename T>
struct factor {T val = 0; T mult = 0;};
template <typename T>
class decompo {
private:
std::vector<T> memory = {2, 3, 5, 7, 11, 13, 17, 19, 23, 31, 37, 39, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97};
T index = 0;
public:
decompo () {};
void reset () {index = 0;};
T pop () {index = memory.size() - 1; return memory[index];};
T get_next ();
std::vector<T> find_all_primes (T n);
std::vector<factor<T>> decomp (T n);
std::vector<T> GetDivisors (T n);
void complete (T n);
};
template <typename T>
T decompo<T>::get_next () {
++index;
if (index <= memory.size()) {
return memory[index-1];
}
T n = memory.size();
T candidate = memory[n-1] + 2;
while (1) {
bool found = true;
for (T i = 1; memory[i] * memory[i] <= candidate; ++i) {
if (candidate % memory[i] == 0) {
found = false;
break;
}
}
if (found) {
memory.push_back (candidate);
return candidate;
}
candidate += 2;
}
}
template <typename T>
std::vector<T> decompo<T>::find_all_primes (T n) {
reset();
std::vector<T> result;
while (1) {
T candidate = get_next();
if (candidate <= n) {
result.push_back (candidate);
} else {
return result;
}
}
}
template <typename T>
void decompo<T>::complete (T n) {
T last = pop();
while (last < n) {
last = get_next();
}
return;
}
template <typename T>
std::vector<factor<T>> decompo<T>::decomp (T n) {
reset();
std::vector<factor<T>> result;
if (n < 2) return result;
T candidate = get_next();
T last_prime = 0;
while (candidate*candidate <= n) {
if (n % candidate == 0) {
if (candidate == last_prime) {
result[result.size()-1].mult ++;
} else {
result.push_back ({candidate, 1});
last_prime = candidate;
}
n /= candidate;
} else {
candidate = get_next();
}
}
if (n > 1) {
if (n != last_prime) result.push_back ({n, 1});
else result[result.size()-1].mult ++;
}
return result;
}
template <typename T>
std::vector<T> decompo<T>::GetDivisors (T n) {
std::vector<T> div;
auto primes = decomp (n);
int n_primes = primes.size();
std::vector<int> exponent (n_primes, 0);
div.push_back(1);
int current_index = 0;
int product = 1;
std::vector<int> product_partial(n_primes, 1);;
while (true) {
current_index = 0;
while (current_index < n_primes && exponent[current_index] == primes[current_index].mult) current_index++;
if (current_index == n_primes) break;
for (int index = 0; index < current_index; ++index) {
exponent[index] = 0;
product /= product_partial[index];
product_partial[index] = 1;
}
exponent[current_index]++;
product *= primes[current_index].val;
product_partial[current_index] *= primes[current_index].val;
if (product != n && product != 1) div.push_back (product);
}
return div;
}
// return true if sum is obtained
bool test_sum (std::vector<int>& arr, int amount) {
int n = arr.size();
std::sort(arr.begin(), arr.end(), std::greater<int>());
std::vector<int> bound (n);
std::vector<int> select (n);
bound[n-1] = arr[n-1];
for (int i = n-2; i >= 0; --i) {
bound[i] = bound[i+1] + arr[i];
}
int sum = 0; // current sum
int i = 0; // index of the coin being examined
bool up_down = true;
while (true) {
if (up_down) {
if (i == n || sum + bound[i] < amount) {
up_down = false;
i--;
continue;
}
sum += arr[i];
select[i] = 1;
if (sum == amount) return true;
if (sum < amount) {
i++;
continue;
}
up_down = false;
if (select[i] == 0) i--;
} else { // DOWN
if (i < 0) break;
if (select[i] == 0) {
i--;
} else {
sum -= arr[i];
select[i] = 0;
i++;
up_down = true;
}
}
}
return false;
}
bool isDeficient(int divisorSum, int n) {
return divisorSum < n;
}
bool isAbundant(int divisorSum, int n) {
return divisorSum > n;
}
bool isSemiPerfect(std::vector<int> &divisors, int sum) {
int size = divisors.size();
// The value of subset[i][j] will be true if there is a subset of divisors[0..j-1] with sum equal to i
//bool subset[sum+1][size+1];
std::vector<std::vector<bool>> subset(sum+1, std::vector<bool> (size+1));
// If sum is 0, then answer is true
for (int i = 0; i <= size; i++) {
subset[0][i] = true;
}
// If sum is not 0 and set is empty, then answer is false
for (int i = 1; i <= sum; i++) {
subset[i][0] = false;
}
// Fill the subset table in bottom up manner
for ( int i = 1 ; i <= sum ; i++ ) {
for ( int j = 1 ; j <= size ; j++ ) {
subset[i][j] = subset[i][j-1];
int test = divisors[j-1];
if ( i >= test ) {
subset[i][j] = subset[i][j] || subset[i - test][j-1];
}
}
}
return subset[sum][size];
}
bool isWeird(int n, std::vector<int> &divisors, int divisorSum) {
//return isAbundant(divisorSum, n) && !isSemiPerfect(divisors, n);
return isAbundant(divisorSum, n) && !test_sum(divisors, n);
}
std::vector<int> getProperDivisors(int number) {
std::vector<int> divisors;
long sqrtn = sqrt(number);
divisors.push_back(1);
for ( int i = 2 ; i <= sqrtn ; i++ ) {
if (number % i == 0) {
divisors.push_back(i);
int div = number/i;
if (div != i) divisors.push_back(div);
}
}
return divisors;
}
int main() {
decompo <int> decomposition;
decomposition.complete (1e3); // not relly useful
int n = 2, count;
std::vector<int> weird;
int Nweird = 15000;
for (count = 0; count < Nweird; n += 2) {
if (n % 6 == 0) continue;
//auto properDivisors = getProperDivisors(n);
auto properDivisors = decomposition.GetDivisors(n);
int divisorSum = std::accumulate (properDivisors.begin(), properDivisors.end(), 0);
if (isDeficient(divisorSum, n) ) {
continue;
}
if (isWeird(n, properDivisors, divisorSum)) {
//std::cout << count << " " << n << "\n";
weird.push_back (n);
count++;
}
}
for (int i = Nweird - 10; i < Nweird; ++i) {
std::cout << weird.at(i) << " ";
}
std::cout << "\n";
}
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