# Primes Without 1's

Requirement primes numbers which do not have digit 1 in it.

Input

The first line contains T, the number of test cases. Followed by T lines each contain two numbers M and N

Output

For every test case print the total count of prime numbers which does not contain 1 as a digit.

If all prime number with digit 1 between M and N(Inclusive). then print -1.

Separate the answers for each test case by an empty line.

Constraints

1<=T<=10000 1<=M<=N<=1000000

My code:

final static int MAX = 1000001;
final static int BASE = 10;

final static boolean[] isPrime = generatePrime();

public static void main(String args[] ) throws Exception {
System.in));
StringBuilder output = new StringBuilder(noOfTestCaseT);

while (noOfTestCaseT != 0) {
noOfTestCaseT--;

int n = Integer.parseInt(tempInt[0].trim());
int m = Integer.parseInt(tempInt[1].trim());
int primesWithoutOnes = getPrimesWithoutOneCount(n, m);
output.append(primesWithoutOnes);
output.append("\n");

}
System.out.println(output);
}

private static int getPrimesWithoutOneCount(int n, int m) {
int totalCount = 0;
for ( int i = n ; i <= m ; i++){
if(isPrime[i] && ! isDigitOnePresent(i) ){
totalCount++;
}
}

if (totalCount == 0) totalCount = -1;
}

private static boolean isDigitOnePresent(int i) {
while(i !=0 ){
int temp = i % BASE;
if( temp == 1)
return true;
i /= BASE;
}
return false;
}

private static boolean[] generatePrime() {
int root = (int) Math.sqrt(MAX);
boolean[] isPrime = new boolean[MAX];
Arrays.fill(isPrime, true);
isPrime[0] = false; isPrime[1] = false;
for (int i = 0; i < root; i++) {
if (isPrime[i]) {
for (int j = i * i ; j < MAX; j = j + i) {
isPrime[j] = false;
}
}

}
return isPrime;
}


While the code is giving the desired output, it's taking around 2.6 sec for all inputs combined. While the desired time limit is 1 sec(s) for all input files combined. What are the various way I can optimize it in Java?

(I tried with the Sieve of Sundaram but was not able to reduce the time too much).

Submission and all inputs given with time taken for various inputs

• Java Mission Control's Flight Recorder might show you exactly where's room for improvement. Jun 8, 2015 at 16:06
• As there are many more numbers with 1s in than primes wouldn't it make sense to check that first? Jun 8, 2015 at 20:07
• As @OldCurmudgeon said, I'd consider eliminating all numbers containing the digit 1 first, then check the primality of each. Jun 8, 2015 at 21:47

## Java is handicapped at HackerEarth

I think that you are handicapped by the fact that Java submissions appear to have a lot higher overhead than C submissions at your HackerEarth site. I checked the submissions for that problem and no one who used Java had a passing time. Then I checked a passing C++ submission and it used the same algorithm as yours. However, you might still be able to beat the requirement if you make some optimizations.

## Don't find all primes

Instead of finding primes to MAX every time, you should read the whole input and find out the maximum m value in the input. Then generate only primes up to the maximum m. Many of the input files only requested primes up to 10000 or so.

Also, you can speed up both your prime generating function and your prime counting function by only checking odd numbers.

## Rule out the numbers containing 1 once

Currently, you generate all the primes and then for each (n,m) pair, you search the primes list and rule out primes containing the digit 1. Instead of doing it this way, you should:

1. Generate the primes.
2. Iterate through all the primes and rule out primes containing the digit 1 by setting isPrime[i] = false for any prime containing the digit 1.
3. For each (n,m) pair, count the number of primes using isPrime[i] without having to check for the digit 1 again.

This way, you will only check each prime for the digit 1 once instead of once per range.

## Finding counts for ranges in O(1) time

Since your input contains many ranges to check, you can make an array to keep track of the cumulative prime count up to each number. Once you generate this array, you can find the count for each (n,m) range in $O(1)$ time by just doing:

count = cumulativeCount[m] - cumulativeCount[n-1];


## The final program

Here is the program with all the changes. Note that with Java, I couldn't get the runtime below 0.14 seconds even when I removed everything but the parsing code. With an equivalent C program, the runtime was 0.03 seconds. This was using one of the large input files with 5000 test cases.

import java.util.Arrays;
import java.lang.StringBuilder;

public class p4 {
final static int BASE = 10;
static boolean[] isPrime;

public static void main(String args[] ) throws Exception {
System.in));
StringBuilder output = new StringBuilder(noOfTestCaseT);
int [][] testCases = new int[noOfTestCaseT][2];
int maxNum = 0;

for (int i=0;i<noOfTestCaseT;i++) {
testCases[i][0] = Integer.parseInt(tempInt[0].trim());
testCases[i][1] = Integer.parseInt(tempInt[1].trim());
if (testCases[i][1] > maxNum)
maxNum = testCases[i][1];
}

isPrime = generatePrime(maxNum+1);

int [] cumulativeCount = new int[maxNum+1];
int count = 1;
cumulativeCount[2] = 1;
for (int i=3;i<=maxNum;i++) {
if (((i&1) == 1) && isPrime[i] && !isDigitOnePresent(i))
count++;
cumulativeCount[i] = count;
}

for (int i=0;i<noOfTestCaseT;i++) {
int n = testCases[i][0];
int m = testCases[i][1];
if (n > 0)
n--;
int primesWithoutOnes = cumulativeCount[m] - cumulativeCount[n];
if (primesWithoutOnes == 0) {
output.append("-1\n");
} else {
output.append(primesWithoutOnes);
output.append("\n");
}
}
System.out.print(output);
}

private static boolean isDigitOnePresent(int i) {
while(i !=0 ){
int temp = i % BASE;
if( temp == 1)
return true;
i /= BASE;
}
return false;
}

private static boolean[] generatePrime(int maxNum) {
int root = (int) Math.sqrt(maxNum) + 1;
boolean[] isPrime = new boolean[maxNum];
Arrays.fill(isPrime, true);
isPrime[0] = false; isPrime[1] = false;
for (int i = 3; i < root; i+=2) {
if (isPrime[i]) {
int increment = i+i;
for (int j = i * i ; j < maxNum; j = j + increment) {
isPrime[j] = false;
}
}
}
return isPrime;
}
}


## Bonus code

I have each iteration of the program here at Github, so you can test each improvement one by one. I also have a C equivalent of the final program as well.

As a special bonus, I also implemented the cumulative count idea using a binary indexed tree, although that was probably less efficient than using a simple array. It's an interesting data structure though, which is often useful when you need to track cumulative counts.

• if (((i&1) == 1) && isPrime[i] && !isDigitOnePresent(i)) arrayIndex out of bound exception Jun 9, 2015 at 4:28
• @AnkurAnand I fixed that just a little while ago by generating maxNum+1 primes. I had so many versions that some of them didn't have all the fixes, sorry.
– JS1
Jun 9, 2015 at 4:47
• Not a problem .. until i learns and i learned so many things out of this :) Jun 9, 2015 at 5:15
• ideone.com/dbKicE for input between 276 297 the primes are 277 281 283 293 the output should be 3 but i'm getting it as 4 means it's counting all the primes Jun 9, 2015 at 5:17
• @AnkurAnand The problem was actually with the prime generator function. The square root needed to be rounded up. If you had changed your MAX to 297 or 298, you would have seen the same problem. I fixed it by adding one to root. So the problem was that 289 was being considered a prime because the loop never iterated to i = 17, because i did not reach 17 in the loop to root.
– JS1
Jun 9, 2015 at 6:06

The most obvious culprit for the slowness seems to be that you find all the primes in the maximum allowed range, when in the test file there might be significantly smaller sub-ranges. You could try to change the logic to find primes within ranges on-demand as needed.

A small speed improvement is possible in the way you find if a number contains a 1. You could replace the modulo with checking the last bit using the & operator.

Not related to speed, but many of the names are very tedious. I recommend some renames:

• "getPrimesWithoutOneCount" to "countPrimesWithoutOne"
• "isDigitOnePresent" to "containsOne"

Lastly, the input parsing looks really tedious. You could rewrite using a Scanner elegantly.

• Scanner is the unsung hero of beginner Java assignments. It's also super powerful for throwing together one-off data manipulation and analysis scripts where you want the familiarity of Java but want something out the door immediately. Jun 8, 2015 at 21:45

From my perspective, the biggest issue is the loop inside the getPrimesWithoutOneCount call. This makes your complexity significant.

With a little more pre-processing, you can do a much better log(n) lookup later on. If you extend your preprocessing with:

private static final int[] oneless = getOnelessPrimes();

private static final int[] getOnelessPrimes() {
int[] oneless = new int[isPrime.length];
int size = 0;
for (int i = 0; i < isPrime.length; i++) {
if (isPrime[i] && !isDigitOnePresent(i)) {
oneless[size++] = i;
}
}
return Arrays.copyOf(oneless, 0, size);
}


That returns an array of prime values that have no 1 in them.

Now, when you are looking for the count of primes... you can:

int left = Arrays.binarySearch(oneless, first);
if (left < 0) {
left = -left - 1;
}
int right = Arrays.binarySearch(oneless, left, oneless.length, last);
if (right < 0) {
right = -right - 1 - 1;
}

int count = right < left ? -1 : (right - left + 1);


That makes your search a lot faster, especially in cases where the difference between M and N is large... e.g. for you, find the number of 1-less primes between 1 and 1000000 would require a lot of iteration. For the above algorithm, though it would require just a little.

It doesn't appear that you use BASE anywhere else in the code other than inside your isDigitOnePresent method, personally I would create that variable inside that method, but that is just my opinion.

• That was a very nice observation :) .. I tried with your idea .. it was fractional change only :( Jun 8, 2015 at 16:05
• someone pointed out to me the error in my thoughts about the order of the calls. it shouldn't make much of a difference if any.
– Malachi
Jun 8, 2015 at 16:06
• Even though it won't make any difference i will remember this all time while writing any code in future ! Jun 8, 2015 at 16:08

As @OldCurmudgeon said, I'd consider eliminating all numbers containing the digit 1 first, then check the primality of each. Also, don't iterate through each number, that's inefficient.

Example: The number 10000 contains a one. Since it is the first digit that is a one, you know that the next 9999 numbers also contain a one, so you can eliminate those immediately.

Do this elimination process in much faster time by simply eliminating the numbers based on the most significant digit that is a 1. At least do the first group (largest) ranges as that will save lots of time. If you don't eliminate every number with a one this way, you can still use the function you already have for the numbers you didn't eliminate this way.

This saves thousands of iterations at best!

### First digit is 1:

• 1,
• 10-19, 100-199, 1000-1999, 10000-19999,

### Second digit is 1:

• 21, 210-219, 2100-2199, 21000-21999
• 31, 310-319, ...
• ...
• 91, 910-919, ...

### Third digit is 1:

• 201, 2010-2019, 20100-20199
• ...

It's unclear why you're using a StringBuilder in main rather than just printing as you go. Printing as you go is more responsive and easier to write, so it seems the obvious choice.

You do no checking for validity on the input. This is fine for the example, but a bad idea if the code is actually meant to be used.

Instead of counting down the number of test cases, it's cannonical to count up the number seen. Plus, why noOfTestCaseT? What's the T mean? Why include the Of? I'd also use num instead of no to prevent ambiguity with the word "no".

Your String[] tempInt = reader.readLine().split(" "); won't work for leading or multiple spaces, although you look like you want it to. Perhaps build a Scanner?

To top it off, try low and high instead of n and m and change getPrimesWithoutOneCount to the equally descriptive but easier to read countPrimeWithoutOnes.

public static void main(String args[] ) throws Exception {

for (int i = 0; i < numTestCases; i++) {

System.out.println(countPrimeWithoutOnes(low, high));
}
}


This also allows using nextLine to check that the whole line was used.

countPrimeWithoutOnes needs reformatting. Further, the action

if (totalCount == 0) totalCount = -1;


should not be part of countPrimeWithoutOnes. This is not what the function claims to do! Let main do that, since it's responsible for output.

The whole function can be simplified to

private static int countPrimeWithoutOnes(int low, int high) {
return (int)IntStream.rangeClosed(low, high)
.filter(i -> isPrime[i] && !isDigitOnePresent(i))
.count();
}


then, but this is unfortunately way slower in this microbenchmark. I'll leave it as a hypothetical, then.

The problem mostly revolves around having a really large number of inputs, so the overhead from generatePrime is largely insignificant. Other people have explained this already, though, so there's not much point me repeating it.

• The more responsive direct output is too slow in some cases. Jun 8, 2015 at 18:19
• As said by @maaartinus here is another one to show how slow calling println make a code for large no of loops codereview.stackexchange.com/questions/92009/… Jun 9, 2015 at 4:09
• @AnkurAnand That will only matter if the other work is small, though (like a couple of parseInts in your example). Nevertheless, I suggest you do whatever makes sense as long as you can demonstrate that it matters. If you can show that directly printing in this case is too slow, feel free to speed it up. Jun 9, 2015 at 4:31

An alternative, which may be faster (I haven't tested it):

private static boolean isDigitOnePresent(int i) {
return Integer.toString(i).contains("1");
}


It uses only library methods, and so should be reasonably fast.

• no it's not .. Not every library functions are fast .. Jun 10, 2015 at 3:37