Sparse Table

We have briefly discussed sparse table in Range Minimum Query (Square Root Decomposition and Sparse Table)

Sparse table concept is used for fast queries on a set of static data (elements do not change). It does preprocessing so that the queries can be answered efficiently.

Example Problem 1 : Range Minimum Query

We have an array arr[0 . . . n-1]. We need to efficiently find the minimum value from index L (query start) to R (query end) where 0 <= L <= R <= n-1. Consider a situation when there are many range queries.

Example:

Input:  arr[]   = {7, 2, 3, 0, 5, 10, 3, 12, 18};
query[] = [0, 4], [4, 7], [7, 8]

Output: Minimum of [0, 4] is 0
Minimum of [4, 7] is 3
Minimum of [7, 8] is 12

The idea is to precompute minimum of all subarrays of size 2j where j varies from 0 to Log n. We make a table lookup[i][j] such that lookup[i][j] contains minimum of range starting from i and of size 2j. For example lookup contains minimum of range [0, 7] (starting with 0 and of size 23)

How to fill this lookup or sparse table?
The idea is simple, fill in a bottom-up manner using previously computed values. We compute ranges with current power of 2 using values of lower power of two. For example, to find a minimum of range [0, 7] (Range size is a power of 3), we can use the minimum of following two.
a) Minimum of range [0, 3] (Range size is a power of 2)
b) Minimum of range [4, 7] (Range size is a power of 2)

Based on above example, below is formula,

// Minimum of single element subarrays is same
// as the only element.
lookup[i] = arr[i]

// If lookup <=  lookup,
// then lookup = lookup
If lookup[i][j-1] <= lookup[i+2j-1-1][j-1]
lookup[i][j] = lookup[i][j-1]

// If lookup >  lookup,
// then lookup = lookup
Else
lookup[i][j] = lookup[i+2j-1-1][j-1] For any arbitrary range [l, R], we need to use ranges which are in powers of 2. The idea is to use the closest power of 2. We always need to do at most one comparison (compare the minimum of two ranges which are powers of 2). One range starts with L and ends with “L + closest-power-of-2”. The other range ends at R and starts with “R – same-closest-power-of-2 + 1”. For example, if the given range is (2, 10), we compare the minimum of two ranges (2, 9) and (3, 10).

Based on above example, below is formula,

// For (2, 10), j = floor(Log2(10-2+1)) = 3
j = floor(Log(R-L+1))

// If lookup <=  lookup,
// then min(2, 10) = lookup
If lookup[L][j] <= lookup[R-(int)pow(2, j)+1][j]
min(L, R) = lookup[L][j]

// If lookup >  arr[lookup,
// then min(2, 10) = lookup
Else
min(L, R) = lookup[i+2j-1-1][j-1]

Since we do only one comparison, the time complexity of query is O(1).

Below is the implementation of the above idea.

C++

 // C++ program to do range minimum query // using sparse table #include using namespace std; #define MAX 500    // lookup[i][j] is going to store minimum // value in arr[i..j]. Ideally lookup table // size should not be fixed and should be // determined using n Log n. It is kept // constant to keep code simple. int lookup[MAX][MAX];    // Fills lookup array lookup[][] in bottom up manner. void buildSparseTable(int arr[], int n) {     // Initialize M for the intervals with length 1     for (int i = 0; i < n; i++)         lookup[i] = arr[i];        // Compute values from smaller to bigger intervals     for (int j = 1; (1 << j) <= n; j++) {            // Compute minimum value for all intervals with         // size 2^j         for (int i = 0; (i + (1 << j) - 1) < n; i++) {                // For arr, we compare arr[lookup]              // and arr[lookup]             if (lookup[i][j - 1] <                          lookup[i + (1 << (j - 1))][j - 1])                 lookup[i][j] = lookup[i][j - 1];             else                 lookup[i][j] =                           lookup[i + (1 << (j - 1))][j - 1];         }     } }    // Returns minimum of arr[L..R] int query(int L, int R) {     // Find highest power of 2 that is smaller     // than or equal to count of elements in given     // range. For [2, 10], j = 3     int j = (int)log2(R - L + 1);        // Compute minimum of last 2^j elements with first     // 2^j elements in range.     // For [2, 10], we compare arr[lookup] and     // arr[lookup],     if (lookup[L][j] <= lookup[R - (1 << j) + 1][j])         return lookup[L][j];        else         return lookup[R - (1 << j) + 1][j]; }    // Driver program int main() {     int a[] = { 7, 2, 3, 0, 5, 10, 3, 12, 18 };     int n = sizeof(a) / sizeof(a);     buildSparseTable(a, n);     cout << query(0, 4) << endl;     cout << query(4, 7) << endl;     cout << query(7, 8) << endl;     return 0; }

Java

 // Java program to do range minimum query // using sparse table import java.io.*;    class GFG {        static int MAX =500;            // lookup[i][j] is going to store minimum     // value in arr[i..j]. Ideally lookup table     // size should not be fixed and should be     // determined using n Log n. It is kept     // constant to keep code simple.     static int [][]lookup = new int[MAX][MAX];            // Fills lookup array lookup[][] in bottom up manner.     static void buildSparseTable(int arr[], int n)     {                    // Initialize M for the intervals with length 1         for (int i = 0; i < n; i++)             lookup[i] = arr[i];                // Compute values from smaller to bigger intervals         for (int j = 1; (1 << j) <= n; j++) {                    // Compute minimum value for all intervals with             // size 2^j             for (int i = 0; (i + (1 << j) - 1) < n; i++) {                        // For arr, we compare arr[lookup]                  // and arr[lookup]                 if (lookup[i][j - 1] <                              lookup[i + (1 << (j - 1))][j - 1])                     lookup[i][j] = lookup[i][j - 1];                 else                     lookup[i][j] =                              lookup[i + (1 << (j - 1))][j - 1];             }         }     }            // Returns minimum of arr[L..R]     static int query(int L, int R)     {                    // Find highest power of 2 that is smaller         // than or equal to count of elements in given         // range. For [2, 10], j = 3         int j = (int)Math.log(R - L + 1);                // Compute minimum of last 2^j elements with first         // 2^j elements in range.         // For [2, 10], we compare arr[lookup] and         // arr[lookup],         if (lookup[L][j] <= lookup[R - (1 << j) + 1][j])             return lookup[L][j];                else             return lookup[R - (1 << j) + 1][j];     }            // Driver program     public static void main (String[] args)     {         int a[] = { 7, 2, 3, 0, 5, 10, 3, 12, 18 };         int n = a.length;                    buildSparseTable(a, n);                    System.out.println(query(0, 4));         System.out.println(query(4, 7));         System.out.println(query(7, 8));            } }    // This code is contributed by vt_m.

Python3

 # Python3 program to do range minimum  # query using sparse table  import math    # Fills lookup array lookup[][] in  # bottom up manner.  def buildSparseTable(arr, n):        # Initialize M for the intervals     # with length 1      for i in range(0, n):          lookup[i] = arr[i]             j = 1            # Compute values from smaller to      # bigger intervals      while (1 << j) <= n:             # Compute minimum value for all          # intervals with size 2^j         i = 0         while (i + (1 << j) - 1) < n:                 # For arr, we compare arr[lookup]              # and arr[lookup]              if (lookup[i][j - 1] <                  lookup[i + (1 << (j - 1))][j - 1]):                  lookup[i][j] = lookup[i][j - 1]              else:                 lookup[i][j] = \                         lookup[i + (1 << (j - 1))][j - 1]                             i += 1         j += 1            # Returns minimum of arr[L..R]  def query(L, R):         # Find highest power of 2 that is smaller      # than or equal to count of elements in      # given range. For [2, 10], j = 3      j = int(math.log2(R - L + 1))         # Compute minimum of last 2^j elements      # with first 2^j elements in range.      # For [2, 10], we compare arr[lookup]      # and arr[lookup],      if lookup[L][j] <= lookup[R - (1 << j) + 1][j]:          return lookup[L][j]         else:         return lookup[R - (1 << j) + 1][j]     # Driver Code if __name__ == "__main__":        a = [7, 2, 3, 0, 5, 10, 3, 12, 18]      n = len(a)      MAX = 500            # lookup[i][j] is going to store minimum      # value in arr[i..j]. Ideally lookup table      # size should not be fixed and should be      # determined using n Log n. It is kept      # constant to keep code simple.      lookup = [[0 for i in range(MAX)]                  for j in range(MAX)]        buildSparseTable(a, n)      print(query(0, 4))      print(query(4, 7))      print(query(7, 8))         # This code is contributed by Rituraj Jain

C#

 // C# program to do range minimum query // using sparse table using System;    public class GFG {            static int MAX= 500;            // lookup[i][j] is going to store minimum     // value in arr[i..j]. Ideally lookup table     // size should not be fixed and should be     // determined using n Log n. It is kept     // constant to keep code simple.     static int [,]lookup = new int[MAX, MAX];            // Fills lookup array lookup[][] in bottom up manner.     static void buildSparseTable(int []arr, int n)     {         // Initialize M for the intervals with length 1         for (int i = 0; i < n; i++)             lookup[i, 0] = arr[i];                // Compute values from smaller to bigger intervals         for (int j = 1; (1 << j) <= n; j++) {                    // Compute minimum value for all intervals with             // size 2^j             for (int i = 0; (i + (1 << j) - 1) < n; i++) {                        // For arr, we compare arr[lookup]                  // and arr[lookup]                 if (lookup[i, j - 1] <                              lookup[i + (1 << (j - 1)), j - 1])                     lookup[i, j] = lookup[i, j - 1];                 else                     lookup[i, j] =                              lookup[i + (1 << (j - 1)), j - 1];             }         }     }            // Returns minimum of arr[L..R]     static int query(int L, int R)     {                    // Find highest power of 2 that is smaller         // than or equal to count of elements in given         // range. For [2, 10], j = 3         int j = (int)Math.Log(R - L + 1);                // Compute minimum of last 2^j elements with first         // 2^j elements in range.         // For [2, 10], we compare arr[lookup] and         // arr[lookup],         if (lookup[L, j] <= lookup[R - (1 << j) + 1, j])             return lookup[L, j];                else             return lookup[R - (1 << j) + 1, j];     }            // Driver program     static public void Main ()     {         int []a = { 7, 2, 3, 0, 5, 10, 3, 12, 18 };         int n = a.Length;                    buildSparseTable(a, n);                    Console.WriteLine(query(0, 4));         Console.WriteLine(query(4, 7));         Console.WriteLine(query(7, 8));     } }    // This code is contributed by vt_m.

Output:

Minimum of [0, 4] is 0
Minimum of [4, 7] is 3
Minimum of [7, 8] is 12

So sparse table method supports query operation in O(1) time with O(n Log n) preprocessing time and O(n Log n) space.

Example Problem 2 : Range GCD Query

We have an array arr[0 . . . n-1]. We need to find the greatest common divisor in the range L and R where 0 <= L <= R <= n-1. Consider a situation when there are many range queries
Examples:

Input : arr[] = {2, 3, 5, 4, 6, 8}
queries[] = {(0, 2), (3, 5), (2, 3)}
Output : 1
2
1

We use below properties of GCD:

• GCD function is associative [ GCD(a, b, c) = GCD(GCD(a, b), c) = GCD(a, GCD(b, c))], we can compute GCD of a range using GCDs of subranges.
• If we take GCD of an overlapping range more than once, then it does not change answer. For example GCD(a, b, c) = GCD(GCD(a, b), GCD(b, c)). Therefore like minimum range query problem, we need to do only one comparison to find GCD of given range.

We build a sparse table using the same logic as above. After building the sparse table, we can find all GCDs by breaking given range in powers of 2 and add GCD of every piece to the current answer.

C++

 // C++ program to do range minimum query // using sparse table #include using namespace std; #define MAX 500    // lookup[i][j] is going to store GCD of // arr[i..j]. Ideally lookup table // size should not be fixed and should be // determined using n Log n. It is kept // constant to keep code simple. int table[MAX][MAX];    // it builds sparse table. void buildSparseTable(int arr[], int n) {     // GCD of single element is element itself     for (int i = 0; i < n; i++)         table[i] = arr[i];        // Build sparse table     for (int j = 1; j <= n; j++)         for (int i = 0; i <= n - (1 << j); i++)             table[i][j] = __gcd(table[i][j - 1],                     table[i + (1 << (j - 1))][j - 1]); }    // Returns GCD of arr[L..R] int query(int L, int R) {     // Find highest power of 2 that is smaller     // than or equal to count of elements in given     // range.For [2, 10], j = 3     int j = (int)log2(R - L + 1);        // Compute GCD of last 2^j elements with first     // 2^j elements in range.     // For [2, 10], we find GCD of arr[lookup] and     // arr[lookup],     return __gcd(table[L][j], table[R - (1 << j) + 1][j]); }    // Driver program int main() {     int a[] = { 7, 2, 3, 0, 5, 10, 3, 12, 18 };     int n = sizeof(a) / sizeof(a);     buildSparseTable(a, n);     cout << query(0, 2) << endl;     cout << query(1, 3) << endl;     cout << query(4, 5) << endl;     return 0; }

Python3

 # Python3 program to do range minimum  # query using sparse table  import math    # Fills lookup array lookup[][] in  # bottom up manner.  def buildSparseTable(arr, n):        # GCD of single element is element itself      for i in range(0, n):          table[i] = arr[i]         # Build sparse table     j = 1     while (1 << j) <= n:          i = 0         while i <= n - (1 << j):              table[i][j] = math.gcd(table[i][j - 1],                                     table[i + (1 << (j - 1))][j - 1])                            i += 1         j += 1    # Returns minimum of arr[L..R]  def query(L, R):         # Find highest power of 2 that is smaller      # than or equal to count of elements in      # given range. For [2, 10], j = 3      j = int(math.log2(R - L + 1))         # Compute GCD of last 2^j elements with      # first 2^j elements in range.      # For [2, 10], we find GCD of arr[lookup]      # and arr[lookup],      return math.gcd(table[L][j],                      table[R - (1 << j) + 1][j])        # Driver Code  if __name__ == "__main__":        a = [7, 2, 3, 0, 5, 10, 3, 12, 18]      n = len(a)      MAX = 500            # lookup[i][j] is going to store minimum      # value in arr[i..j]. Ideally lookup table      # size should not be fixed and should be      # determined using n Log n. It is kept      # constant to keep code simple.      table = [[0 for i in range(MAX)]                  for j in range(MAX)]        buildSparseTable(a, n)      print(query(0, 2))      print(query(1, 3))      print(query(4, 5))         # This code is contributed by Rituraj Jain

Output:

1
1
5

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