## Amazon Interview Question

Country: United States
Interview Type: Phone Interview

Comment hidden because of low score. Click to expand.
3
of 3 vote

I. At first find found all primes <= N (sieve of Eratosthenes). Getting the sum will be easy then.
Follow-up:
Cache the sums for any given N to save time. {N:SUM}
Optimization: Don't have to store sums for every N.
When N = 7, N = 8, N = 9, N = 10, the prime sum remains 17.
For N between 11 to 12, the prime sum is 28.
For N between 13 to 16, the sum is 41.
Use a BST structure as the cache. For N = 16, cache:
{2:3, 4:6, 6:11, 10:17, 12:28, 16:41}

For a given N, call cache.ceilingKey(N) to find the bucket for N.

N/log(n) * log(N)

Complexity
Time:
sieve of Eratosthenes takes O(NloglogN) time.
Insert an element into BST takes O(logN), there are N/logN primes in total to be added.
So building the cache takes logN * N / LogN = O(N) time
requesting primeSum(N) takes O(logN)

Space:
sieve of Eratosthenes takes O(N) extra space which will later be release after the cache is created.
Cache: O(N/logN)

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``````import java.util.TreeMap;
public class PrimeSum {

TreeMap<Integer, Integer> sums;

public PrimeSum(int n) { //input the upper limit for all Ns
sums = new TreeMap<>();
// init an array to track prime numbers
boolean[] primes = new boolean[n + 1];
for (int i = 2; i < n; i++)
primes[i] = true;
for (int i = 2; i <= Math.sqrt(n); i++) {
if (primes[i]) {
for (int j = i + i; j < n; j += i)
primes[j] = false;
}
}
// insert sums into cache
int sum = 0;
for(int i = 2; i <= n; i++) {
if(primes[i]) {
sums.put(i - 1, sum);
sum += i;
}
}
if(primes[n]) {
sums.put(n, sum);
}
}

public int primeSum(int N) {
Integer ceiling = sums.ceilingKey(N);
//if(ceiling == null) {
//Exception("input value overflows");
//}
return sums.get(ceiling);
}

}``````

Comment hidden because of low score. Click to expand.
1
of 1 vote

With O(Max_N) preprocessing time and O(Max_N) storage with subsequent queries in O(1) time -

``````class Primes:
__primeUptoN = [0, 0]
__primeSum = [0]

def __init__(self):
self.__generatePrimesIfNotGeneratedBefore()

def primeSum(self, N):
return self.__primeSum[self.__primeUptoN[N]]

@classmethod
def __generatePrimesIfNotGeneratedBefore(Primes):
if len(Primes.__primeSum) > 1:
return
MAX_N = 10 ** 6
non_primes = set()
for i in range(2, MAX_N + 1):
if i not in non_primes:
Primes.__primeSum.append(Primes.__primeSum[-1] + i)
j = i + i
while j <= MAX_N:
j += i
Primes.__primeUptoN.append(len(Primes.__primeSum) - 1)``````

Comment hidden because of low score. Click to expand.
0
of 0 vote

``````package general;

import java.util.Arrays;
import java.util.TreeMap;

public class SieveOfEratosthenese {

public static void getPrimesBySeieveOfEraosthenes(int n, boolean primes[]) {

for(int i=2; i<= Math.sqrt(n); i++){
if(primes[i]){
for(int j=i*i;j<=n; j+=i){
primes[j] = false;
}
}
}
}

public static void main(String[] args) {
// TODO Auto-generated method stub

int n = 16;
boolean[] primes = new boolean[n+1];
Arrays.fill(primes, true);
getPrimesBySeieveOfEraosthenes(n, primes);

for (int i = 2; i <= n; i++) {
if (primes[i])
System.out.print((i) + " ");
}

TreeMap <Integer, Integer> sums = new TreeMap<>();

int sum =0;

for (int i = 2; i <= n; i++) {
if (primes[i]){
sums.put(i-1, sum);
sum +=i;
}

}

if(primes[n]){
sums.put(n,sum);
}

System.out.println( "\n"+  sums.get(sums.ceilingKey(4)));
}

}``````

Comment hidden because of low score. Click to expand.
0
of 0 vote

``````public class SumOfPrimeNumbers {
static List<Integer> cachePrime = new ArrayList<>();
static List<Integer> cacheSumPrime = new ArrayList<>();
static {
}

static private void add(int n, int sum) {
}

static private int nextPrime(int prime) {
while( true ) {
++prime;
int i = 0;
for(  ; i < cachePrime.size() ; ++i )
if( prime % cachePrime.get(i) == 0 )
break;
if( i == cachePrime.size() )
return prime;
}
}

static public int sumPrime(int n) {
if( cachePrime.contains(n) ) {
return cacheSumPrime.get( cachePrime.indexOf(n) );
}

int lastPrime = cachePrime.size()-1;
if( cachePrime.get(lastPrime) > n ) {
while (cachePrime.get(lastPrime) > n) {
--lastPrime;
}
}
else {
int nextPrime = nextPrime(cachePrime.get(lastPrime));
while (cachePrime.get(lastPrime) < n && nextPrime < n) {
add(nextPrime, nextPrime + cacheSumPrime.get(lastPrime));
lastPrime = cachePrime.size() - 1;
nextPrime = nextPrime(cachePrime.get(lastPrime));
}
}
return cacheSumPrime.get(lastPrime);
}
}``````

Comment hidden because of low score. Click to expand.
0
of 0 vote

``````import java.util.*;
public class Primes {

Map<Integer, Integer> cache = new HashMap<Integer, Integer>();
public static void main(String[] args) {
Primes primes = new Primes();
primes.allPrimesTo(20);
System.out.println(primes.cache.get(13));
System.out.println(primes.cache.get(5));
}

int allPrimesTo(int n) {
int sum = 0;
for(int i=2; i<=n; i++) {
if(isPrime(i)) {
sum+=i;
cache.put(i, sum);
}
}
return sum;
}

boolean isPrime(int n) {
if(n ==2 ) return false;
for(int i=2; i<n; i++) {
if(n%i==0) return false;
}
return true;
}

}``````

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