更新时间:2023-02-10 16:59:29
是的,CalculateAmicable
是效率低的-O(N)
复杂性-因此,您可以进行N
测试1e5 * 1e5 == 1e10
- 百亿操作很慢.
将复杂度降低到O(log(N))
的方案是
Yes, CalculateAmicable
is inefficient - O(N)
complexity - and so you have fo N
tests 1e5 * 1e5 == 1e10
- ten billions operations which is slow.
The scheme which reduces complexity to O(log(N))
is
int n = 100000;
//TODO: implement IList<int> GetPrimesUpTo(int) yourself
var primes = GetPrimesUpTo((int)(Math.Sqrt(n + 1) + 1));
// Key - number itself, Value - divisors' sum
var direct = Enumerable
.Range(1, n)
.AsParallel()
.ToDictionary(index => index,
index => GetDivisorsSum(index, primes) - index);
var result = Enumerable
.Range(1, n)
.Where(x => x < direct[x] &&
direct.ContainsKey((int) direct[x]) &&
direct[(int) direct[x]] == x)
.Select(x => $"{x,5}, {direct[x],5}");
Console.Write(string.Join(Environment.NewLine, result));
结果不到一秒钟(Core i7 3.2Ghz .Net 4.6 IA-64):
The outcome is within a second (Core i7 3.2Ghz .Net 4.6 IA-64):
220, 284
1184, 1210
2620, 2924
5020, 5564
6232, 6368
10744, 10856
12285, 14595
17296, 18416
63020, 76084
66928, 66992
67095, 71145
69615, 87633
79750, 88730
详细信息GetDivisorsSum
:
private static long GetDivisorsSum(long value, IList<int> primes) {
HashSet<long> hs = new HashSet<long>();
IList<long> divisors = GetPrimeDivisors(value, primes);
ulong n = (ulong) 1;
n = n << divisors.Count;
long result = 1;
for (ulong i = 1; i < n; ++i) {
ulong v = i;
long p = 1;
for (int j = 0; j < divisors.Count; ++j) {
if ((v % 2) != 0)
p *= divisors[j];
v = v / 2;
}
if (hs.Contains(p))
continue;
result += p;
hs.Add(p);
}
return result;
}
和GetPrimeDivisors
:
private static IList<long> GetPrimeDivisors(long value, IList<int> primes) {
List<long> results = new List<long>();
int v = 0;
long threshould = (long) (Math.Sqrt(value) + 1);
for (int i = 0; i < primes.Count; ++i) {
v = primes[i];
if (v > threshould)
break;
if ((value % v) != 0)
continue;
while ((value % v) == 0) {
value = value / v;
results.Add(v);
}
threshould = (long) (Math.Sqrt(value) + 1);
}
if (value > 1)
results.Add(value);
return results;
}