Segment Tree | Range Minimum Query

We have introduced segment tree with a simple example in the previous post. In this post, Range Minimum Query problem is discussed as another example where Segment Tree can be used. The following is the problem statement:
We have an array arr[0 . . . n-1]. We should be able to efficiently find the minimum value from index qs (query start) to qe (query end) where 0 <= qs <= qe <= n-1. 

A simple solution is to run a loop from qs to qe and find minimum element in given range. This solution takes O(n) time in worst case. 
Another solution is to create a 2D array where an entry [i, j] stores the minimum value in range arr[i..j]. Minimum of a given range can now be calculated in O(1) time, but preprocessing takes O(n^2) time. Also, this approach needs O(n^2) extra space which may become huge for large input arrays.
Segment tree can be used to do preprocessing and query in moderate time. With a segment tree, preprocessing time is O(n) and  the time complexity for a range minimum query is O(Logn). The extra space required is O(n) to store the segment tree.
Representation of Segment trees 
1. Leaf Nodes are the elements of the input array. 
2. Each internal node represents minimum of all leaves under it.
An array representation of tree is used to represent Segment Trees. For each node at index i, the left child is at index 2*i+1, right child at 2*i+2 and the parent is at ?(i – 1) / 2?.

Construction of Segment Tree from given array
We start with a segment arr[0 . . . n-1]. and every time we divide the current segment into two halves(if it has not yet become a segment of length 1), and then call the same procedure on both halves, and for each such segment, we store the minimum value in a segment tree node. 
All levels of the constructed segment tree will be completely filled except the last level. Also, the tree will be a Full Binary Tree because we always divide segments in two halves at every level. Since the constructed tree is always full binary tree with n leaves, there will be n-1 internal nodes. So total number of nodes will be 2*n – 1. 
Height of the segment tree will be ?log?n?. Since the tree is represented using array and relation between parent and child indexes must be maintained, size of memory allocated for segment tree will be  2 * 2^?log₂n?  – 1. 

Query for minimum value of given range
Once the tree is constructed, how to do range minimum query using the constructed segment tree. Following is algorithm to get the minimum. 

// qs --> query start index, qe --> query end index
int RMQ(node, qs, qe) 
{
   if range of node is within qs and qe
        return value in node
   else if range of node is completely outside qs and qe
        return INFINITE
   else
    return min( RMQ(node's left child, qs, qe), RMQ(node's right child, qs, qe) )
}

Implementation:

C++

// C++ program for range minimum
// query using segment tree
#include <bits/stdc++.h>
using namespace std;

// A utility function to get minimum of two numbers
int minVal(int x, int y) { return (x < y)? x: y; }

// A utility function to get the
// middle index from corner indexes.
int getMid(int s, int e) { return s + (e -s)/2; }

/* A recursive function to get the
minimum value in a given range
of array indexes. The following
are parameters for this function.

    st --> Pointer to segment tree
    index --> Index of current node in the
           segment tree. Initially 0 is
           passed as root is always at index 0
    ss & se --> Starting and ending indexes
                of the segment represented
                by current node, i.e., st[index]
    qs & qe --> Starting and ending indexes of query range */
int RMQUtil(int *st, int ss, int se, int qs, int qe, int index)
{
    // If segment of this node is a part
    // of given range, then return
    // the min of the segment
    if (qs <= ss && qe >= se)
        return st[index];

    // If segment of this node
    // is outside the given range
    if (se < qs || ss > qe)
        return INT_MAX;

    // If a part of this segment
    // overlaps with the given range
    int mid = getMid(ss, se);
    return minVal(RMQUtil(st, ss, mid, qs, qe, 2*index+1),
                RMQUtil(st, mid+1, se, qs, qe, 2*index+2));
}

// Return minimum of elements in range
// from index qs (query start) to
// qe (query end). It mainly uses RMQUtil()
int RMQ(int *st, int n, int qs, int qe)
{
    // Check for erroneous input values
    if (qs < 0 || qe > n-1 || qs > qe)
    {
        cout<<"Invalid Input";
        return -1;
    }

    return RMQUtil(st, 0, n-1, qs, qe, 0);
}

// A recursive function that constructs
// Segment Tree for array[ss..se].
// si is index of current node in segment tree st
int constructSTUtil(int arr[], int ss, int se,
                                int *st, int si)
{
    // If there is one element in array,
    // store it in current node of
    // segment tree and return
    if (ss == se)
    {
        st[si] = arr[ss];
        return arr[ss];
    }

    // If there are more than one elements,
    // then recur for left and right subtrees
    // and store the minimum of two values in this node
    int mid = getMid(ss, se);
    st[si] = minVal(constructSTUtil(arr, ss, mid, st, si*2+1),
                    constructSTUtil(arr, mid+1, se, st, si*2+2));
    return st[si];
}

/* Function to construct segment tree
from given array. This function allocates
memory for segment tree and calls constructSTUtil() to
fill the allocated memory */
int *constructST(int arr[], int n)
{
    // Allocate memory for segment tree

    //Height of segment tree
    int x = (int)(ceil(log2(n)));

    // Maximum size of segment tree
    int max_size = 2*(int)pow(2, x) - 1;

    int *st = new int[max_size];

    // Fill the allocated memory st
    constructSTUtil(arr, 0, n-1, st, 0);

    // Return the constructed segment tree
    return st;
}

// Driver program to test above functions
int main()
{
    int arr[] = {1, 3, 2, 7, 9, 11};
    int n = sizeof(arr)/sizeof(arr[0]);

    // Build segment tree from given array
    int *st = constructST(arr, n);

    int qs = 1; // Starting index of query range
    int qe = 5; // Ending index of query range

    // Print minimum value in arr[qs..qe]
    cout<<"Minimum of values in range ["<<qs<<", "<<qe<<"] "<<
    "is = "<<RMQ(st, n, qs, qe)<<endl;

    return 0;
}

// This code is contributed by rathbhupendra

C

// C program for range minimum query using segment tree
#include <stdio.h>
#include <math.h>
#include <limits.h>

// A utility function to get minimum of two numbers
int minVal(int x, int y) { return (x < y)? x: y; }

// A utility function to get the middle index from corner indexes.
int getMid(int s, int e) {  return s + (e -s)/2;  }

/*  A recursive function to get the minimum value in a given range
     of array indexes. The following are parameters for this function.

    st    --> Pointer to segment tree
    index --> Index of current node in the segment tree. Initially
              0 is passed as root is always at index 0
    ss & se  --> Starting and ending indexes of the segment represented
                  by current node, i.e., st[index]
    qs & qe  --> Starting and ending indexes of query range */
int RMQUtil(int *st, int ss, int se, int qs, int qe, int index)
{
    // If segment of this node is a part of given range, then return
    //  the min of the segment
    if (qs <= ss && qe >= se)
        return st[index];

    // If segment of this node is outside the given range
    if (se < qs || ss > qe)
        return INT_MAX;

    // If a part of this segment overlaps with the given range
    int mid = getMid(ss, se);
    return minVal(RMQUtil(st, ss, mid, qs, qe, 2*index+1),
                  RMQUtil(st, mid+1, se, qs, qe, 2*index+2));
}

// Return minimum of elements in range from index qs (query start) to
// qe (query end).  It mainly uses RMQUtil()
int RMQ(int *st, int n, int qs, int qe)
{
    // Check for erroneous input values
    if (qs < 0 || qe > n-1 || qs > qe)
    {
        printf("Invalid Input");
        return -1;
    }

    return RMQUtil(st, 0, n-1, qs, qe, 0);
}

// A recursive function that constructs Segment Tree for array[ss..se].
// si is index of current node in segment tree st
int constructSTUtil(int arr[], int ss, int se, int *st, int si)
{
    // If there is one element in array, store it in current node of
    // segment tree and return
    if (ss == se)
    {
        st[si] = arr[ss];
        return arr[ss];
    }

    // If there are more than one elements, then recur for left and
    // right subtrees and store the minimum of two values in this node
    int mid = getMid(ss, se);
    st[si] =  minVal(constructSTUtil(arr, ss, mid, st, si*2+1),
                     constructSTUtil(arr, mid+1, se, st, si*2+2));
    return st[si];
}

/* Function to construct segment tree from given array. This function
   allocates memory for segment tree and calls constructSTUtil() to
   fill the allocated memory */
int *constructST(int arr[], int n)
{
    // Allocate memory for segment tree

    //Height of segment tree
    int x = (int)(ceil(log2(n)));

    // Maximum size of segment tree
    int max_size = 2*(int)pow(2, x) - 1;

    int *st = new int[max_size];

    // Fill the allocated memory st
    constructSTUtil(arr, 0, n-1, st, 0);

    // Return the constructed segment tree
    return st;
}

// Driver program to test above functions
int main()
{
    int arr[] = {1, 3, 2, 7, 9, 11};
    int n = sizeof(arr)/sizeof(arr[0]);

    // Build segment tree from given array
    int *st = constructST(arr, n);

    int qs = 1;  // Starting index of query range
    int qe = 5;  // Ending index of query range

    // Print minimum value in arr[qs..qe]
    printf("Minimum of values in range [%d, %d] is = %d\n",
                           qs, qe, RMQ(st, n, qs, qe));

    return 0;
}

Java

// Program for range minimum query using segment tree
class SegmentTreeRMQ
{
    int st[]; //array to store segment tree

    // A utility function to get minimum of two numbers
    int minVal(int x, int y) {
        return (x < y) ? x : y;
    }

    // A utility function to get the middle index from corner
    // indexes.
    int getMid(int s, int e) {
        return s + (e - s) / 2;
    }

    /*  A recursive function to get the minimum value in a given
        range of array indexes. The following are parameters for
        this function.

        st    --> Pointer to segment tree
        index --> Index of current node in the segment tree. Initially
                   0 is passed as root is always at index 0
        ss & se  --> Starting and ending indexes of the segment
                     represented by current node, i.e., st[index]
        qs & qe  --> Starting and ending indexes of query range */
    int RMQUtil(int ss, int se, int qs, int qe, int index)
    {
        // If segment of this node is a part of given range, then
        // return the min of the segment
        if (qs <= ss && qe >= se)
            return st[index];

        // If segment of this node is outside the given range
        if (se < qs || ss > qe)
            return Integer.MAX_VALUE;

        // If a part of this segment overlaps with the given range
        int mid = getMid(ss, se);
        return minVal(RMQUtil(ss, mid, qs, qe, 2 * index + 1),
                RMQUtil(mid + 1, se, qs, qe, 2 * index + 2));
    }

    // Return minimum of elements in range from index qs (query
    // start) to qe (query end).  It mainly uses RMQUtil()
    int RMQ(int n, int qs, int qe)
    {
        // Check for erroneous input values
        if (qs < 0 || qe > n - 1 || qs > qe) {
            System.out.println("Invalid Input");
            return -1;
        }

        return RMQUtil(0, n - 1, qs, qe, 0);
    }

    // A recursive function that constructs Segment Tree for
    // array[ss..se]. si is index of current node in segment tree st
    int constructSTUtil(int arr[], int ss, int se, int si)
    {
        // If there is one element in array, store it in current
        //  node of segment tree and return
        if (ss == se) {
            st[si] = arr[ss];
            return arr[ss];
        }

        // If there are more than one elements, then recur for left and
        // right subtrees and store the minimum of two values in this node
        int mid = getMid(ss, se);
        st[si] = minVal(constructSTUtil(arr, ss, mid, si * 2 + 1),
                constructSTUtil(arr, mid + 1, se, si * 2 + 2));
        return st[si];
    }

    /* Function to construct segment tree from given array. This function
       allocates memory for segment tree and calls constructSTUtil() to
       fill the allocated memory */
    void constructST(int arr[], int n)
    {
        // Allocate memory for segment tree

        //Height of segment tree
        int x = (int) (Math.ceil(Math.log(n) / Math.log(2)));

        //Maximum size of segment tree
        int max_size = 2 * (int) Math.pow(2, x) - 1;
        st = new int[max_size]; // allocate memory

        // Fill the allocated memory st
        constructSTUtil(arr, 0, n - 1, 0);
    }

    // Driver program to test above functions
    public static void main(String args[])
    {
        int arr[] = {1, 3, 2, 7, 9, 11};
        int n = arr.length;
        SegmentTreeRMQ tree = new SegmentTreeRMQ();

        // Build segment tree from given array
        tree.constructST(arr, n);

        int qs = 1;  // Starting index of query range
        int qe = 5;  // Ending index of query range

        // Print minimum value in arr[qs..qe]
        System.out.println("Minimum of values in range [" + qs + ", "
                           + qe + "] is = " + tree.RMQ(n, qs, qe));
    }
}
// This code is contributed by Ankur Narain Verma

Python3

# Python3 program for range minimum
# query using segment tree
import sys;
from math import ceil,log2;

INT_MAX = sys.maxsize;

# A utility function to get
# minimum of two numbers
def minVal(x, y) :
    return x if (x < y) else y;

# A utility function to get the
# middle index from corner indexes.
def getMid(s, e) :
    return s + (e - s) // 2;

""" A recursive function to get the
minimum value in a given range
of array indexes. The following
are parameters for this function.

    st --> Pointer to segment tree
    index --> Index of current node in the
        segment tree. Initially 0 is
        passed as root is always at index 0
    ss & se --> Starting and ending indexes
                of the segment represented
                by current node, i.e., st[index]
    qs & qe --> Starting and ending indexes of query range """
def RMQUtil( st, ss, se, qs, qe, index) :

    # If segment of this node is a part
    # of given range, then return
    # the min of the segment
    if (qs <= ss and qe >= se) :
        return st[index];

    # If segment of this node
    # is outside the given range
    if (se < qs or ss > qe) :
        return INT_MAX;

    # If a part of this segment
    # overlaps with the given range
    mid = getMid(ss, se);
    return minVal(RMQUtil(st, ss, mid, qs,
                          qe, 2 * index + 1),
                  RMQUtil(st, mid + 1, se,
                          qs, qe, 2 * index + 2));

# Return minimum of elements in range
# from index qs (query start) to
# qe (query end). It mainly uses RMQUtil()
def RMQ( st, n, qs, qe) :

    # Check for erroneous input values
    if (qs < 0 or qe > n - 1 or qs > qe) :

        print("Invalid Input");
        return -1;

    return RMQUtil(st, 0, n - 1, qs, qe, 0);

# A recursive function that constructs
# Segment Tree for array[ss..se].
# si is index of current node in segment tree st
def constructSTUtil(arr, ss, se, st, si) :

    # If there is one element in array,
    # store it in current node of
    # segment tree and return
    if (ss == se) :

        st[si] = arr[ss];
        return arr[ss];

    # If there are more than one elements,
    # then recur for left and right subtrees
    # and store the minimum of two values in this node
    mid = getMid(ss, se);
    st[si] = minVal(constructSTUtil(arr, ss, mid,
                                    st, si * 2 + 1),
                    constructSTUtil(arr, mid + 1, se,
                                    st, si * 2 + 2));

    return st[si];

"""Function to construct segment tree
from given array. This function allocates
memory for segment tree and calls constructSTUtil()
to fill the allocated memory """
def constructST( arr, n) :

    # Allocate memory for segment tree

    # Height of segment tree
    x = (int)(ceil(log2(n)));

    # Maximum size of segment tree
    max_size = 2 * (int)(2**x) - 1;

    st = [0] * (max_size);

    # Fill the allocated memory st
    constructSTUtil(arr, 0, n - 1, st, 0);

    # Return the constructed segment tree
    return st;

# Driver Code
if __name__ == "__main__" :

    arr = [1, 3, 2, 7, 9, 11];
    n = len(arr);

    # Build segment tree from given array
    st = constructST(arr, n);

    qs = 1; # Starting index of query range
    qe = 5; # Ending index of query range

    # Print minimum value in arr[qs..qe]
    print("Minimum of values in range [", qs,
          ",", qe, "]", "is =", RMQ(st, n, qs, qe));

# This code is contributed by AnkitRai01

C#

// C# Program for range minimum
// query using segment tree
using System;

public class SegmentTreeRMQ
{
    int []st; //array to store segment tree

    // A utility function to
    // get minimum of two numbers
    int minVal(int x, int y)
    {
        return (x < y) ? x : y;
    }

    // A utility function to get the
    //  middle index from corner indexes.
    int getMid(int s, int e)
    {
        return s + (e - s) / 2;
    }

    /* A recursive function to get
        the minimum value in a given
        range of array indexes.
        The following are parameters for
        this function.

        st --> Pointer to segment tree
        index --> Index of current node in the
        segment tree. Initially 0 is passed
        as root is always at index 0
        ss & se --> Starting and ending indexes of the segment
                    represented by current node, i.e., st[index]
        qs & qe --> Starting and ending indexes of query range */
    int RMQUtil(int ss, int se, int qs, int qe, int index)
    {
        // If segment of this node is a
        // part of given range, then
        // return the min of the segment
        if (qs <= ss && qe >= se)
            return st[index];

        // If segment of this node
        // is outside the given range
        if (se < qs || ss > qe)
            return int.MaxValue;

        // If a part of this segment
        // overlaps with the given range
        int mid = getMid(ss, se);
        return minVal(RMQUtil(ss, mid, qs, qe, 2 * index + 1),
                RMQUtil(mid + 1, se, qs, qe, 2 * index + 2));
    }

    // Return minimum of elements
    // in range from index qs (query
    // start) to qe (query end).
    // It mainly uses RMQUtil()
    int RMQ(int n, int qs, int qe)
    {
        // Check for erroneous input values
        if (qs < 0 || qe > n - 1 || qs > qe)
        {
            Console.WriteLine("Invalid Input");
            return -1;
        }

        return RMQUtil(0, n - 1, qs, qe, 0);
    }

    // A recursive function that
    // constructs Segment Tree for
    // array[ss..se]. si is index
    // of current node in segment tree st
    int constructSTUtil(int []arr, int ss, int se, int si)
    {
        // If there is one element in array,
        // store it in current node of
        // segment tree and return
        if (ss == se)
        {
            st[si] = arr[ss];
            return arr[ss];
        }

        // If there are more than one elements,
        // then recur for left and right subtrees
        // and store the minimum of two values in this node
        int mid = getMid(ss, se);
        st[si] = minVal(constructSTUtil(arr, ss, mid, si * 2 + 1),
                constructSTUtil(arr, mid + 1, se, si * 2 + 2));
        return st[si];
    }

    /* Function to construct segment
    tree from given array. This function
    allocates memory for segment tree
    and calls constructSTUtil() to
    fill the allocated memory */
    void constructST(int []arr, int n)
    {
        // Allocate memory for segment tree

        // Height of segment tree
        int x = (int) (Math.Ceiling(Math.Log(n) / Math.Log(2)));

        // Maximum size of segment tree
        int max_size = 2 * (int) Math.Pow(2, x) - 1;
        st = new int[max_size]; // allocate memory

        // Fill the allocated memory st
        constructSTUtil(arr, 0, n - 1, 0);
    }

    // Driver code
    public static void Main()
    {
        int []arr = {1, 3, 2, 7, 9, 11};
        int n = arr.Length;
        SegmentTreeRMQ tree = new SegmentTreeRMQ();

        // Build segment tree from given array
        tree.constructST(arr, n);

        int qs = 1; // Starting index of query range
        int qe = 5; // Ending index of query range

        // Print minimum value in arr[qs..qe]
        Console.WriteLine("Minimum of values in range [" + qs + ", "
                        + qe + "] is = " + tree.RMQ(n, qs, qe));
    }
}

/* This code contributed by PrinciRaj1992 */

Javascript

<script>

      // JavaScript Program for range minimum
      // query using segment tree
      class SegmentTreeRMQ {
        constructor() {
          this.st = []; //array to store segment tree
        }

        // A utility function to
        // get minimum of two numbers
        minVal(x, y) {
          return x < y ? x : y;
        }

        // A utility function to get the
        //  middle index from corner indexes.
        getMid(s, e) {
          return parseInt(s + (e - s) / 2);
        }

        /* A recursive function to get
            the minimum value in a given
            range of array indexes.
            The following are parameters for
            this function.

            st --> Pointer to segment tree
            index --> Index of current node in the
            segment tree. Initially 0 is passed
            as root is always at index 0
            ss & se --> Starting and ending indexes of the segment
                        represented by current node, i.e., st[index]
            qs & qe --> Starting and ending indexes of query range */
        RMQUtil(ss, se, qs, qe, index) {
          // If segment of this node is a
          // part of given range, then
          // return the min of the segment
          if (qs <= ss && qe >= se) return this.st[index];

          // If segment of this node
          // is outside the given range
          if (se < qs || ss > qe) return 2147483647;

          // If a part of this segment
          // overlaps with the given range
          var mid = this.getMid(ss, se);
          return this.minVal(
            this.RMQUtil(ss, mid, qs, qe, 2 * index + 1),
            this.RMQUtil(mid + 1, se, qs, qe, 2 * index + 2)
          );
        }

        // Return minimum of elements
        // in range from index qs (query
        // start) to qe (query end).
        // It mainly uses RMQUtil()
        RMQ(n, qs, qe) {
          // Check for erroneous input values
          if (qs < 0 || qe > n - 1 || qs > qe) {
            document.write("Invalid Input");
            return -1;
          }

          return this.RMQUtil(0, n - 1, qs, qe, 0);
        }

        // A recursive function that
        // constructs Segment Tree for
        // array[ss..se]. si is index
        // of current node in segment tree st
        constructSTUtil(arr, ss, se, si) {
          // If there is one element in array,
          // store it in current node of
          // segment tree and return
          if (ss == se) {
            this.st[si] = arr[ss];
            return arr[ss];
          }

          // If there are more than one elements,
          // then recur for left and right subtrees
          // and store the minimum of two values in this node
          var mid = this.getMid(ss, se);
          this.st[si] = this.minVal(
            this.constructSTUtil(arr, ss, mid, si * 2 + 1),
            this.constructSTUtil(arr, mid + 1, se, si * 2 + 2)
          );
          return this.st[si];
        }

        /* Function to construct segment
        tree from given array. This function
        allocates memory for segment tree
        and calls constructSTUtil() to
        fill the allocated memory */
        constructST(arr, n) {
          // Allocate memory for segment tree

          // Height of segment tree
          var x = parseInt(Math.ceil(Math.log(n) / Math.log(2)));

          // Maximum size of segment tree
          var max_size = 2 * parseInt(Math.pow(2, x) - 1);
          this.st = new Array(max_size).fill(0); // allocate memory

          // Fill the allocated memory st
          this.constructSTUtil(arr, 0, n - 1, 0);
        }
      }
      // Driver code
      var arr = [1, 3, 2, 7, 9, 11];
      var n = arr.length;
      var tree = new SegmentTreeRMQ();

      // Build segment tree from given array
      tree.constructST(arr, n);

      var qs = 1; // Starting index of query range
      var qe = 5; // Ending index of query range

      // Print minimum value in arr[qs..qe]
      document.write(
        "Minimum of values in range [" +
          qs +
          ", " +
          qe +
          "] is = " +
          tree.RMQ(n, qs, qe) +
          "<br>"
      );
</script>

Output:

 Minimum of values in range [1, 5] is = 2

Time Complexity:
Time Complexity for tree construction is O(n). There are total 2n-1 nodes, and value of every node is calculated only once in tree construction.
Time complexity to query is O(Logn). To query a range minimum, we process at most two nodes at every level and number of levels is O(Logn). 
Auxiliary Space: O(n),  since n extra space has been taken.

Please refer following links for more solutions to range minimum query problem. 
https://www.geeksforgeeks.org/range-minimum-query-for-static-array/

Related Topic: Segment Tree