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This algorithm efficiently locates all instances of a specific pattern within a text in linear time. If the length of the text is "n" and the length of the pattern is "m," then the total time taken is O(m + n), with a linear auxiliary space. It is worth noting that the time and auxiliary space of this algorithm is the same as the KMP algorithm, but this particular algorithm is simpler to comprehend. In this approach, we create a Z array as part of the process.

What is Z Array?

For a string str[0..n-1], Z array is of same length as string. An element Z[i] of Z array stores length of the longest substring starting from str[i] which is also a prefix of str[0..n-1]. The first entry of Z array is meaning less as complete string is always prefix of itself. 

Example:
Index 0 1 2 3 4 5 6 7 8 9 10 11
Text a a b c a a b x a a a z
Z values X 1 0 0 3 1 0 0 2 2 1 0

More Examples:
str = "aaaaaa"
Z[] = {x, 5, 4, 3, 2, 1}
str = "aabaacd"
Z[] = {x, 1, 0, 2, 1, 0, 0}
str = "abababab"
Z[] = {x, 0, 6, 0, 4, 0, 2, 0}

How is Z array helpful in Searching Pattern in Linear time? 
The idea is to concatenate pattern and text, and create a string "P$T" where P is pattern, $ is a special character should not be present in pattern and text, and T is text. Build the Z array for concatenated string. In Z array, if Z value at any point is equal to pattern length, then pattern is present at that point. 

Example:
Pattern P = "aab", Text T = "baabaa"
The concatenated string is = "aab$baabaa"
Z array for above concatenated string is {x, 1, 0, 0, 0, 3, 1, 0, 2, 1}.
Since length of pattern is 3, the value 3 in Z array indicates presence of pattern.

How to construct Z array? 
     A Simple Solution is to run two nested loops, the outer loop goes to every index and the inner loop finds length of the longest prefix that matches the substring starting at the current index. The time complexity of this solution is O(n²).
      We can construct Z array in linear time. 

The idea is to maintain an interval [L, R] which is the interval with max R such that [L,R] is prefix substring (substring which is also prefix).
Steps for maintaining this interval are as follows –
1) If i > R then there is no prefix substring that starts before i and ends after i, so we reset L and R and compute new [L,R] by comparing str[0..] to str[i..] and get Z[i] (= R-L+1).
2) If i <= R then let K = i-L, now Z[i] >= min(Z[K], R-i+1) because str[i..] matches with str[K..] for atleast R-i+1 characters (they are in [L,R] interval which we know is a prefix substring). Now two sub cases arise –
a) If Z[K] < R-i+1 then there is no prefix substring starting at str[i] (otherwise Z[K] would be larger) so Z[i] = Z[K] and interval [L,R] remains same.
b) If Z[K] >= R-i+1 then it is possible to extend the [L,R] interval thus we will set L as i and start matching from str[R] onwards and get new R then we will update interval [L,R] and calculate Z[i] (=R-L+1).

The algorithm runs in linear time because we never compare character less than R and with matching we increase R by one so there are at most T comparisons. In mismatch case, mismatch happen only once for each i (because of which R stops), that’s another at most T comparison making overall linear complexity.

Below is the implementation of Z algorithm for pattern searching. 

// A C++ program that implements Z algorithm for pattern searching
#include<iostream>
using namespace std;

void getZarr(string str, int Z[]);

// prints all occurrences of pattern in text using Z algo
void search(string &text, string &pattern)
{
    // Create concatenated string "P$T"
    string concat = pattern + "$" + text;
    int l = concat.length();

    // Construct Z array
    int Z[l];
    getZarr(concat, Z);

    // now looping through Z array for matching condition
    for (int i = 0; i < l; ++i)
    {
        // if Z[i] (matched region) is equal to pattern
        // length we got the pattern
        if (Z[i] == pattern.length())
            cout << "Pattern found at index "
                << i - pattern.length() -1 << endl;
    }
}

// Fills Z array for given string str[]
void getZarr(string str, int Z[])
{
    int n = str.length();
    int L, R, k;

    // [L,R] make a window which matches with prefix of s
    L = R = 0;
    for (int i = 1; i < n; ++i)
    {
        // if i>R nothing matches so we will calculate.
        // Z[i] using naive way.
        if (i > R)
        {
            L = R = i;

            // R-L = 0 in starting, so it will start
            // checking from 0'th index. For example,
            // for "ababab" and i = 1, the value of R
            // remains 0 and Z[i] becomes 0. For string
            // "aaaaaa" and i = 1, Z[i] and R become 5
            while (R<n && str[R-L] == str[R])
                R++;
            Z[i] = R-L;
            R--;
        }
        else
        {
            // k = i-L so k corresponds to number which
            // matches in [L,R] interval.
            k = i-L;

            // if Z[k] is less than remaining interval
            // then Z[i] will be equal to Z[k].
            // For example, str = "ababab", i = 3, R = 5
            // and L = 2
            if (Z[k] < R-i+1)
                Z[i] = Z[k];

            // For example str = "aaaaaa" and i = 2, R is 5,
            // L is 0
            else
            {
                // else start from R and check manually
                L = i;
                while (R<n && str[R-L] == str[R])
                    R++;
                Z[i] = R-L;
                R--;
            }
        }
    }
}

// Driver program
int main()
{
    string text = "GEEKS FOR GEEKS";
    string pattern = "GEEK";
    search(text, pattern);
    return 0;
}

// A Java program that implements Z algorithm for pattern
// searching
class GFG { 

    //  prints all occurrences of pattern in text using
    // Z algo
    public static void search(String text, String pattern)
    {

        // Create concatenated string "P$T"
        String concat = pattern + "$" + text;

        int l = concat.length();

        int Z[] = new int[l];

        // Construct Z array
        getZarr(concat, Z);

        // now looping through Z array for matching condition
        for(int i = 0; i < l; ++i){

            // if Z[i] (matched region) is equal to pattern
            // length we got the pattern

            if(Z[i] == pattern.length()){
                System.out.println("Pattern found at index "
                              + (i - pattern.length() - 1));
            }
        }
    }

    // Fills Z array for given string str[]
    private static void getZarr(String str, int[] Z) {

        int n = str.length();
        
        // [L,R] make a window which matches with 
        // prefix of s
        int L = 0, R = 0;

        for(int i = 1; i < n; ++i) {

            // if i>R nothing matches so we will calculate.
            // Z[i] using naive way.
            if(i > R){

                L = R = i;

                // R-L = 0 in starting, so it will start
                // checking from 0'th index. For example,
                // for "ababab" and i = 1, the value of R
                // remains 0 and Z[i] becomes 0. For string
                // "aaaaaa" and i = 1, Z[i] and R become 5

                while(R < n && str.charAt(R - L) == str.charAt(R))
                    R++;
                
                Z[i] = R - L;
                R--;

            }
            else{

                // k = i-L so k corresponds to number which
                // matches in [L,R] interval.
                int k = i - L;

                // if Z[k] is less than remaining interval
                // then Z[i] will be equal to Z[k].
                // For example, str = "ababab", i = 3, R = 5
                // and L = 2
                if(Z[k] < R - i + 1)
                    Z[i] = Z[k];

                // For example str = "aaaaaa" and i = 2, R is 5,
                // L is 0
                else{


                // else start from R and check manually
                    L = i;
                    while(R < n && str.charAt(R - L) == str.charAt(R))
                        R++;
                    
                    Z[i] = R - L;
                    R--;
                }
            }
        }
    }
    
    // Driver program
    public static void main(String[] args) 
    {
        String text = "GEEKS FOR GEEKS";
        String pattern = "GEEK";

        search(text, pattern);
    }
}

// This code is contributed by PavanKoli.

# Python3 program that implements Z algorithm
# for pattern searching

# Fills Z array for given string str[]
def getZarr(string, z):
    n = len(string)

    # [L,R] make a window which matches
    # with prefix of s
    l, r, k = 0, 0, 0
    for i in range(1, n):

        # if i>R nothing matches so we will calculate.
        # Z[i] using naive way.
        if i > r:
            l, r = i, i

            # R-L = 0 in starting, so it will start
            # checking from 0'th index. For example,
            # for "ababab" and i = 1, the value of R
            # remains 0 and Z[i] becomes 0. For string
            # "aaaaaa" and i = 1, Z[i] and R become 5
            while r < n and string[r - l] == string[r]:
                r += 1
            z[i] = r - l
            r -= 1
        else:

            # k = i-L so k corresponds to number which
            # matches in [L,R] interval.
            k = i - l

            # if Z[k] is less than remaining interval
            # then Z[i] will be equal to Z[k].
            # For example, str = "ababab", i = 3, R = 5
            # and L = 2
            if z[k] < r - i + 1:
                z[i] = z[k]

            # For example str = "aaaaaa" and i = 2, 
            # R is 5, L is 0
            else:

                # else start from R and check manually
                l = i
                while r < n and string[r - l] == string[r]:
                    r += 1
                z[i] = r - l
                r -= 1

# prints all occurrences of pattern 
# in text using Z algo
def search(text, pattern):

    # Create concatenated string "P$T"
    concat = pattern + "$" + text
    l = len(concat)

    # Construct Z array
    z = [0] * l
    getZarr(concat, z)

    # now looping through Z array for matching condition
    for i in range(l):

        # if Z[i] (matched region) is equal to pattern
        # length we got the pattern
        if z[i] == len(pattern):
            print("Pattern found at index", 
                      i - len(pattern) - 1)

# Driver Code
if __name__ == "__main__":
    text = "GEEKS FOR GEEKS"
    pattern = "GEEK"
    search(text, pattern)

# This code is contributed by
# sanjeev2552

// A C# program that implements Z 
// algorithm for pattern searching 
using System;

class GFG
{

// prints all occurrences of 
// pattern in text using Z algo 
public static void search(string text,
                          string pattern)
{

    // Create concatenated string "P$T" 
    string concat = pattern + "$" + text;

    int l = concat.Length;

    int[] Z = new int[l];

    // Construct Z array 
    getZarr(concat, Z);

    // now looping through Z array
    // for matching condition 
    for (int i = 0; i < l; ++i)
    {

        // if Z[i] (matched region) is equal 
        // to pattern length we got the pattern 

        if (Z[i] == pattern.Length)
        {
            Console.WriteLine("Pattern found at index " + 
                             (i - pattern.Length - 1));
        }
    }
}

// Fills Z array for given string str[] 
private static void getZarr(string str,
                            int[] Z)
{

    int n = str.Length;

    // [L,R] make a window which 
    // matches with prefix of s 
    int L = 0, R = 0;

    for (int i = 1; i < n; ++i)
    {

        // if i>R nothing matches so we will 
        // calculate. Z[i] using naive way. 
        if (i > R)
        {
            L = R = i;

            // R-L = 0 in starting, so it will start 
            // checking from 0'th index. For example, 
            // for "ababab" and i = 1, the value of R 
            // remains 0 and Z[i] becomes 0. For string 
            // "aaaaaa" and i = 1, Z[i] and R become 5 
            while (R < n && str[R - L] == str[R])
            {
                R++;
            }

            Z[i] = R - L;
            R--;

        }
        else
        {

            // k = i-L so k corresponds to number 
            // which matches in [L,R] interval. 
            int k = i - L;

            // if Z[k] is less than remaining interval 
            // then Z[i] will be equal to Z[k]. 
            // For example, str = "ababab", i = 3, 
            // R = 5 and L = 2 
            if (Z[k] < R - i + 1)
            {
                Z[i] = Z[k];
            }

            // For example str = "aaaaaa" and 
            // i = 2, R is 5, L is 0 
            else
            {


                // else start from R and 
                // check manually 
                L = i;
                while (R < n && str[R - L] == str[R])
                {
                    R++;
                }

                Z[i] = R - L;
                R--;
            }
        }
    }
}

// Driver Code 
public static void Main(string[] args)
{
    string text = "GEEKS FOR GEEKS";
    string pattern = "GEEK";

    search(text, pattern);
}
}

// This code is contributed by Shrikant13

// A JavaScript program that implements Z algorithm for
// pattern searching

//  prints all occurrences of pattern in text using algo
function search(text, pattern)
{
    // Create concatenated string "P$T"
    let concat = pattern + "$" + text;

    let l = concat.length;

    let Z = new Array(l);

    // Construct Z array
    getZarr(concat, Z);

    // now looping through Z array for matching condition
    for (let i = 0; i < l; ++i) {

        // if Z[i] (matched region) is equal to pattern
        // length we got the pattern

        if (Z[i] == pattern.length) {
            console.log("Pattern found at index "
                        + (i - pattern.length - 1));
        }
    }
}

// Fills Z array for given string str[]
function getZarr(str, Z)
{
    let n = str.length;

    // [L,R] make a window which matches with
    // prefix of s
    let L = 0, R = 0;

    for (let i = 1; i < n; ++i) {

        // if i>R nothing matches so we will calculate.
        // Z[i] using naive way.
        if (i > R) {

            L = R = i;

            // R-L = 0 in starting, so it will start
            // checking from 0'th index. For example,
            // for "ababab" and i = 1, the value of R
            // remains 0 and Z[i] becomes 0. For string
            // "aaaaaa" and i = 1, Z[i] and R become 5

            while (R < n && str[R - L] == str[R])
                R++;

            Z[i] = R - L;
            R--;
        }
        else {

            // k = i-L so k corresponds to number which
            // matches in [L,R] interval.
            let k = i - L;

            // if Z[k] is less than remaining interval
            // then Z[i] will be equal to Z[k].
            // For example, str = "ababab", i = 3, R = 5
            // and L = 2
            if (Z[k] < R - i + 1)
                Z[i] = Z[k];

            // For example str = "aaaaaa" and i = 2, R is 5,
            // L is 0
            else {

                // else start from R and check manually
                L = i;
                while (R < n && str[R - L] == str[R])
                    R++;

                Z[i] = R - L;
                R--;
            }
        }
    }
}

// Driver program
let text = "GEEKS FOR GEEKS";
let pattern = "GEEK";

search(text, pattern);

Pattern found at index 0
Pattern found at index 10

Time Complexity: O(m+n), where m is length of pattern and n is length of text.
Auxiliary Space: O(n)