# Ashton and String Hackerrank

Online coding challenge Hacker Rank.

I tried to solve it using the naive appraoch first but its failing on some of the inputs and rest its getting timed out. How to get started in problem like these so that I could reduce the time complexity involved?

public class Solution {
private static Set<String> substrings(String text) {
int length = text.length();
Set<String> set = new TreeSet<>();

for (int i = 0; i < length; i++) {
for (int j = i + 1; j <= length; j++) {
}
}
return set;
}

public static void main(String[] args) {
Scanner scanner =  new Scanner(System.in);
int T = scanner.nextInt();
String s = scanner.next();
int K = scanner.nextInt();
StringBuilder sb = new StringBuilder();
for (String sub : substrings(s)) {
sb.append(sub);
}
System.out.println(sb.charAt(K - 1));
}
}

• Failing on some of the inputs as in gives the wrong result? Or failing only by taking too long? – Simon Forsberg Jan 8 '17 at 18:46
• That's essentially a brute-force approach, hence the time outs. This is quite a complicated problem, but probably you need to look into suffix arrays. – Tunaki Jan 8 '17 at 18:57
• @SimonForsberg its failing on some of the inputs too but the test cases are not disclosed but I think that my logic is correct although its brute force. – CodeYogi Jan 8 '17 at 18:59
• @Tunaki how you decided to use suffix array that's my question here. – CodeYogi Jan 8 '17 at 18:59

### Program only handles one testcase

Your program reads in T but doesn't iterate T times, so it only handles the first test case. This could be the cause of the "wrong result" error.

### Memory limit exceeded

Your program solves the problem correctly, but it uses $O(n^3)$ memory, so on large inputs it will run out of memory (possibly giving you a wrong answer instead of a timeout). For a string of size 100000 (which is the maximum according to the problem description), the total length of all substrings is: 166671666700000, or $1.7 * 10^{14}$, which would require 170 terabytes of memory.

### Solution using: Suffix array + LCP array

In the problem description, you will notice on the "Need Help?" section on the right hand side that there are links to two topics:

You probably will want to read those links later, because they will give useful information on how to generate the two arrays in the most efficient way possible. But for now, I will explain how, if you have already generated the suffix array and LCP array, you can solve the rest of the problem in $O(n)$ time and no additional space other than the $O(n)$ space used for the two arrays.

### Definitions

First, some quick definitions of my own (if you haven't read through those above links yet):

1. The suffix array for a string of length $n$ is just a sorted array of the $n$ suffixes of the string. To save space, the array contains integers instead of strings, where the integer is the index of the starting character of the suffix in the original string. For example, if suffix[3] = 5, that is equivalent to suffix[3] = original_string.substring(5).

2. The LCP array holds the length of the longest common prefix between two successive strings of the suffix array. For example, if suffix[5] = "abcd" and suffix[6] = "abyz", then LCP[6] = 2 because the two strings have a common prefix of length 2.

### Generating substrings in order

The key to finding the kth character in the concatenated substrings is that you can generate all substrings in lexicographical order by iterating across the suffix array. Let's take an example of the string "abcd". The suffix array for this string is:

suffix[0] = "abcd"
suffix[1] = "bcd"
suffix[2] = "cd"
suffix[3] = "d"


If you take each possible prefix string of each possible suffix string, you will have generated all possible substrings of the original string. For example, if you take suffix[0] = "abcd", there are four possible prefix strings:

"a", "ab", "abc", "abcd"


If you take suffix[1] = "bcd", there are three possible prefix strings:

"b", "bc", "bcd"


Now, if you notice, the order in which I just generated the substrings are the first 7 substrings of the original string in correct lexicographical order. If you continue through the suffix array like this, all of the strings you generate will be in the order that you are supposed to concatenate to the final string (except for duplicate substrings, which I will talk about later).

You will also notice that for a particular length suffix, it is easy to compute how long all of its substrings will be. For example, if the length of the suffix is 4 (such as "abcd"), then the length of the substrings it generates is 1 + 2 + 3 + 4. In general, if a suffix has length n, then the length of the substrings it generates is n * (n+1) / 2.

### Solving the problem (no duplicate substrings)

This gives us the following code to find the kth character, assuming that there are no duplicate substrings. For each suffix in the suffix array, we find out if K falls within the substrings for that suffix. If it does, we find out which substring it is and print out the required character. If it does not, we quickly move on to the next suffix.

    int     n         = s.length();
Integer [] suffix = GenerateSuffixArray(s);

for (int i=0;i<n;i++) {
long len   = n - suffix[i];
long count = len * (len + 1) / 2;

// If K is not found within the substrings generated by this
// suffix, then go on to the next suffix.
if (K > count) {
K -= count;
continue;
}

// K must be found within this suffix.  Now iterate through its
// substring to find the exact substring and character.
for (long j=1;j<=len;j++) {
if (K <= j) {
System.out.println(s.charAt((int) (suffix[i] + j - 1)));
break;
}
K -= j;
}
break;
}


### Handling duplicate substrings

Duplicate substrings are handled by skipping past them. For example, suppose you had two suffixes in your suffix array like this:

suffix[0] = "abcd"
suffix[1] = "abxyabcd"


Notice that the substrings generated would be:

"a", "ab", "abc", "abcd"
"a", "ab", "abx", "abxy", "abxya", "abxyab", "abxyabc", "abxyabcd"


Because there is a common prefix of length 2, the first 2 substrings generated are the same. So when we handle the second suffix, we should skip those because they were already handled previously.

### Sample solution

Here is a sample solution to the problem, that can get the correct answer for a string of 100000 length, but not in the most time efficient manner. The reason is that I generated the suffix array and the LCP array in the simplest way possible, requiring $O(n^2)$ time. To really solve the problem, you should read those links above and generate the suffix array and LCP array in a more efficient manner. I'll leave that as an exercise to the reader.

import java.util.*;

public class Solution {

public static Integer[] GenerateSuffixArray(final String s)
{
int        len = s.length();
Integer [] ret = new Integer[len];

for (int i=0; i < len; i++) {
ret[i] = i;
}
Arrays.sort(ret, new Comparator<Integer>() {
public int compare(Integer i1, Integer i2) {
return s.substring(i1).compareTo(s.substring(i2));
}
});
return ret;
}

public static int[] GenerateLCP(String s, Integer [] suffix)
{
int    len     = s.length();
int [] ret     = new int[len];
String prevStr = s;

ret[0] = 0;
for (int i=1; i < len; i++) {
String str  = s.substring(i);
int    j    = 0;
int    jMax = Math.min(prevStr.length(), str.length());

// Find the longest common prefix of prevStr and str.
for (j=0; j < jMax; j++) {
if (prevStr.charAt(j) != str.charAt(j))
break;
}
ret[i] = j;
prevStr = str;
}
return ret;
}

public static void main(String[] args) {
Scanner scanner =  new Scanner(System.in);
int T = scanner.nextInt();

while (T-- > 0) {
String s = scanner.next();
long K = scanner.nextLong();

int     n         = s.length();
Integer [] suffix = GenerateSuffixArray(s);
int     [] lcp    = GenerateLCP(s, suffix);

for (int i=0;i<n;i++) {
long len    = n - suffix[i];
long count  = len * (len + 1) / 2;
long prefix = lcp[i];

count -= prefix * (prefix + 1) / 2;

// If K is not found within the substrings generated by this
// suffix, then go on to the next suffix.
if (K > count) {
K -= count;
continue;
}

// K must be found within this suffix.  Now iterate through its
// substring to find the exact substring and character.
for (long j=prefix+1;j<=len;j++) {
if (K <= j) {
System.out.println(s.charAt((int) (suffix[i] + K - 1)));
break;
}
K -= j;
}
break;
}
}
}
}

• A very thoughtful and detailed answers, it would be great if you could elaborate the space complexity part too. – CodeYogi Jan 9 '17 at 16:53
• @CodeYogi The space complexity is $O(n)$, due to the two arrays being of length $n$. I briefly mentioned it in my review but there is more detail in the links that describe how to construct each type of array. As you can see, the main loop doesn't allocate any more memory, so all of the memory used is in the construction of the two arrays. – JS1 Jan 9 '17 at 18:45