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Bug in range sieve when left boundry is 0 #1471

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Bug in range sieve when left boundry is 0 #1471

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21 changes: 13 additions & 8 deletions src/algebra/sieve-of-eratosthenes.md
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Original file line number Diff line number Diff line change
Expand Up @@ -207,7 +207,7 @@ To solve such a problem, we can use the idea of the Segmented sieve.
We pre-generate all prime numbers up to $\sqrt R$, and use those primes to mark all composite numbers in the segment $[L, R]$.

```cpp
vector<char> segmentedSieve(long long L, long long R) {
vector<char> segmented_sieve(long long L, long long R) {
// generate all primes up to sqrt(R)
long long lim = sqrt(R);
vector<char> mark(lim + 1, false);
Expand All @@ -222,10 +222,12 @@ vector<char> segmentedSieve(long long L, long long R) {

vector<char> isPrime(R - L + 1, true);
for (long long i : primes)
for (long long j = max(i * i, (L + i - 1) / i * i); j <= R; j += i)
for (long long j = max(i, (L + i - 1) / i) * i; j <= R; j += i)
isPrime[j - L] = false;
if (L == 1)
isPrime[0] = false;
if (L == 0)
isPrime[L] = false;
if (L <= 1)
isPrime[min(1 - L, R - L)] = false;
return isPrime;
}
```
Expand All @@ -234,14 +236,16 @@ Time complexity of this approach is $O((R - L + 1) \log \log (R) + \sqrt R \log
It's also possible that we don't pre-generate all prime numbers:

```cpp
vector<char> segmentedSieveNoPreGen(long long L, long long R) {
vector<char> segmented_sieve_no_pre_gen(long long L, long long R) {
vector<char> isPrime(R - L + 1, true);
long long lim = sqrt(R);
for (long long i = 2; i <= lim; ++i)
for (long long j = max(i * i, (L + i - 1) / i * i); j <= R; j += i)
for (long long j = max(i, (L + i - 1) / i) * i; j <= R; j += i)
isPrime[j - L] = false;
if (L == 1)
isPrime[0] = false;
if (L == 0)
isPrime[L] = false;
if (L <= 1)
isPrime[min(1 - L, R - L)] = false;
return isPrime;
}
```
Expand All @@ -258,6 +262,7 @@ However, this algorithm also has its own weaknesses.

* [Leetcode - Four Divisors](https://leetcode.com/problems/four-divisors/)
* [Leetcode - Count Primes](https://leetcode.com/problems/count-primes/)
* [Leetcode - Closest Prime Numbers in Range](https://leetcode.com/problems/closest-prime-numbers-in-range/)
* [SPOJ - Printing Some Primes](http://www.spoj.com/problems/TDPRIMES/)
* [SPOJ - A Conjecture of Paul Erdos](http://www.spoj.com/problems/HS08PAUL/)
* [SPOJ - Primal Fear](http://www.spoj.com/problems/VECTAR8/)
Expand Down