From 01f14089a5737c0fec5ccbd0612f63ff7132fa3e Mon Sep 17 00:00:00 2001
From: Sandeep Satheesh <sinchan4x496@gmail.com>
Date: Mon, 24 Sep 2018 23:14:44 +0530
Subject: [PATCH 1/5] Add linear prime sieve
---
.../linear-prime-sieve/linearPrimeSieve.js | 24 +++++++++++++++++++
1 file changed, 24 insertions(+)
create mode 100644 src/algorithms/math/linear-prime-sieve/linearPrimeSieve.js
diff --git a/src/algorithms/math/linear-prime-sieve/linearPrimeSieve.js b/src/algorithms/math/linear-prime-sieve/linearPrimeSieve.js
new file mode 100644
index 0000000000..5d600ebe30
--- /dev/null
+++ b/src/algorithms/math/linear-prime-sieve/linearPrimeSieve.js
@@ -0,0 +1,24 @@
+/**
+ * @param {number} maxNumber
+ * @return {number[]}
+ */
+export default function linearPrimeSieve(maxNumber) {
+ const primes =[];
+ const leastPrimeFactor = new Array(maxNumber + 1).fill(0); // leastPrimeFactor[i] gives us the least prime factor of 'i'
+
+ for(let i=2; i<=maxNumber; i++ ){
+ if(!leastPrimeFactor[i]){ // leastPrimeFactor[i] = 0 means 'i' itself is its least prime factor, i.e 'i' is prime
+ leastPrimeFactor[i] = i;
+ primes.push(i);
+ }
+
+ /*
+ * start setting leastPrimeFactor[] for numbers 'x', where x = p * i, p is x's least prime factor and p <= leastPrimeFactor[i]
+ * x = p*i, this representation will be unique for any number, therefore leastPrimeFactor[x] will be set only once.
+ */
+ for(let j=0; j<primes.length && primes[j]*i <=maxNumber && primes[j]<=leastPrimeFactor[i]; j++)
+ leastPrimeFactor[primes[j]*i] = primes[j];
+ }
+
+ return primes;
+}
From 786cd38228078b37cb09e87082c33d5f2d412ddd Mon Sep 17 00:00:00 2001
From: Sandeep Satheesh <sinchan4x496@gmail.com>
Date: Mon, 24 Sep 2018 23:18:06 +0530
Subject: [PATCH 2/5] Adds README.md
---
.../math/linear-prime-sieve/README.md | 44 +++++++++++++++++++
1 file changed, 44 insertions(+)
create mode 100644 src/algorithms/math/linear-prime-sieve/README.md
diff --git a/src/algorithms/math/linear-prime-sieve/README.md b/src/algorithms/math/linear-prime-sieve/README.md
new file mode 100644
index 0000000000..b83aa39c77
--- /dev/null
+++ b/src/algorithms/math/linear-prime-sieve/README.md
@@ -0,0 +1,44 @@
+# Linear Prime Sieve
+
+Linear Prime Sieve is an algorithm for finding all prime numbers up to some limit `n` in linear time.
+
+
+
+## How it works
+
+1. Create an array `leastPrimeFactor` of `n + 1` positions filled with `0`(to represent the least prime factors of numbers `0` through `n`)
+2. Start at position `i = 2` (the first prime number)
+3. If the least prime factor of `i` is `0` it means `i` is a prime number, so we add it to our list of `primes`.
+4. We start setting the least prime factor for numbers `x`, where x = `p` * `i`, `p` is `x`'s least prime factor and `p` <= `leastPrimeFactor(i)`, we do
+ this by traversing the list of primes accumulated till now.
+5. Increment the value of `i`.
+6. We repeat steps 3 through 5 until `i` exceeds `n`.
+
+
+When the algorithm terminates, all the prime numbers from `0` through `n` will be added to our primes list
+
+
+
+## Complexity
+
+The algorithm has a complexity of `O(n)`.
+
+
+
+## Proof Of Correctness
+
+In order to prove the time complexity, we need to show that the alogrithm sets all leastPrimeFactor[] correctly, and that every value will be set only once.
+If so the algorithm will have linear time complexity, since all the remaining work of the algorithm is of `O(n)`.
+We Observe that every number `i` has exactly one representation of form:
+ `i = leastPrimeFactor[i] * x`, where leastPrimeFactor[i] <= leastPrimeFactor[x].
+
+In our Alogrith, for every `x`, it goes through all prime numbers it could be multiplied by, i.e. all prime numbers up to leastPrimeFactor[x] inclusive,
+in order to get the numbers in the form given above.
+Hence, the algorithm will go through every composite number exactly once, setting the correct values for leastPrimeFactor[] there.
+
+
+
+## References
+
+- David Gries, Jayadev Misra. A Linear Sieve Algorithm for Finding Prime Numbers [1978].
+- [CP-Algorithms/Linear-Sieve](https://cp-algorithms.com/algebra/prime-sieve-linear.html).
\ No newline at end of file
From 65037f32ddb5f42d9bda4ae180a867971ec807c7 Mon Sep 17 00:00:00 2001
From: Sandeep Satheesh <sinchan4x496@gmail.com>
Date: Mon, 24 Sep 2018 23:24:04 +0530
Subject: [PATCH 3/5] Adds linearPrimeSieve tests
---
.../__test__/linearPrimeSieve.test.js | 12 ++++++++++++
1 file changed, 12 insertions(+)
create mode 100644 src/algorithms/math/linear-prime-sieve/__test__/linearPrimeSieve.test.js
diff --git a/src/algorithms/math/linear-prime-sieve/__test__/linearPrimeSieve.test.js b/src/algorithms/math/linear-prime-sieve/__test__/linearPrimeSieve.test.js
new file mode 100644
index 0000000000..7a419c5691
--- /dev/null
+++ b/src/algorithms/math/linear-prime-sieve/__test__/linearPrimeSieve.test.js
@@ -0,0 +1,12 @@
+import linearPrimeSieve from '../linearPrimeSieve';
+
+describe('linearPrimeSieve', () => {
+ it('should find all primes less than or equal to n', () => {
+ expect(linearPrimeSieve(5)).toEqual([2, 3, 5]);
+ expect(linearPrimeSieve(10)).toEqual([2, 3, 5, 7]);
+ expect(linearPrimeSieve(100)).toEqual([
+ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41,
+ 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97,
+ ]);
+ });
+});
\ No newline at end of file
From b43738839e18f4f8a797cb5ad671192b244d04b4 Mon Sep 17 00:00:00 2001
From: Sandeep Satheesh <sinchan4x496@gmail.com>
Date: Mon, 1 Oct 2018 22:35:42 +0530
Subject: [PATCH 4/5] Refactor linearPrimeSieve.js
---
.../linear-prime-sieve/linearPrimeSieve.js | 47 ++++++++++++-------
1 file changed, 29 insertions(+), 18 deletions(-)
diff --git a/src/algorithms/math/linear-prime-sieve/linearPrimeSieve.js b/src/algorithms/math/linear-prime-sieve/linearPrimeSieve.js
index 5d600ebe30..611c469ae1 100644
--- a/src/algorithms/math/linear-prime-sieve/linearPrimeSieve.js
+++ b/src/algorithms/math/linear-prime-sieve/linearPrimeSieve.js
@@ -3,22 +3,33 @@
* @return {number[]}
*/
export default function linearPrimeSieve(maxNumber) {
- const primes =[];
- const leastPrimeFactor = new Array(maxNumber + 1).fill(0); // leastPrimeFactor[i] gives us the least prime factor of 'i'
-
- for(let i=2; i<=maxNumber; i++ ){
- if(!leastPrimeFactor[i]){ // leastPrimeFactor[i] = 0 means 'i' itself is its least prime factor, i.e 'i' is prime
- leastPrimeFactor[i] = i;
- primes.push(i);
- }
-
- /*
- * start setting leastPrimeFactor[] for numbers 'x', where x = p * i, p is x's least prime factor and p <= leastPrimeFactor[i]
- * x = p*i, this representation will be unique for any number, therefore leastPrimeFactor[x] will be set only once.
- */
- for(let j=0; j<primes.length && primes[j]*i <=maxNumber && primes[j]<=leastPrimeFactor[i]; j++)
- leastPrimeFactor[primes[j]*i] = primes[j];
- }
-
- return primes;
+ const primes = [];
+ // leastPrimeFactor[i] gives us the least prime factor of 'i'
+ const leastPrimeFactor = new Array(maxNumber + 1).fill(0);
+
+ for (let i = 2; i <= maxNumber; i += 1) {
+ if (!leastPrimeFactor[i]) {
+ /* leastPrimeFactor[i] = 0 means 'i' itself is its least prime factor,
+ * i.e 'i' is prime
+ */
+ leastPrimeFactor[i] = i;
+ primes.push(i);
+ }
+
+ /*
+ * start setting leastPrimeFactor[] for numbers 'x', where x = p * i, p is x's
+ * least prime factor and p <= leastPrimeFactor[i]. x = p*i, this representation will
+ * be unique for any number, therefore leastPrimeFactor[x] will be
+ * set only once.
+ */
+
+ let j = 0;
+ while (j < primes.length) {
+ if ((primes[j] * i > maxNumber) || (primes[j] > leastPrimeFactor[i])) { break; }
+ leastPrimeFactor[primes[j] * i] = primes[j];
+ j += 1;
+ }
+ }
+
+ return primes;
}
From 49bdb355c93daa4fab9c5b8716e35e911e034a98 Mon Sep 17 00:00:00 2001
From: Sandeep Satheesh <sinchan4x496@gmail.com>
Date: Mon, 1 Oct 2018 22:38:59 +0530
Subject: [PATCH 5/5] refactor linearPrimSieve.test.js as per eslint
---
.../math/linear-prime-sieve/__test__/linearPrimeSieve.test.js | 2 +-
1 file changed, 1 insertion(+), 1 deletion(-)
diff --git a/src/algorithms/math/linear-prime-sieve/__test__/linearPrimeSieve.test.js b/src/algorithms/math/linear-prime-sieve/__test__/linearPrimeSieve.test.js
index 7a419c5691..2e84028320 100644
--- a/src/algorithms/math/linear-prime-sieve/__test__/linearPrimeSieve.test.js
+++ b/src/algorithms/math/linear-prime-sieve/__test__/linearPrimeSieve.test.js
@@ -9,4 +9,4 @@ describe('linearPrimeSieve', () => {
43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97,
]);
});
-});
\ No newline at end of file
+});