# Segment Tree | Set 2 (Range Maximum Query with Node Update)

• Difficulty Level : Hard
• Last Updated : 25 May, 2021

Given an array arr[0 . . . n-1]. Find the maximum of elements from index l to r where 0 <= l <= r <= n-1. Also, change the value of a specified element of the array to a new value x. We need to do arr[i] = x where 0 <= i <= n-1 and then find the maximum element of given range with updated values.
Example :

```Input : {1, 3, 5, 7, 9, 11}
Maximum Query : L = 1, R = 3
update : set arr = 8
Output :
Max of values in given range = 7
Updated max of values in given range = 8```

A simple solution is to run a loop from l to r and calculate the maximum of elements in given range. To update a value, simply do arr[i] = x. The first operation takes O(n) time and the second operation takes O(1) time.

Efficient Approach : Here, we need to perform operations in O(Logn) time so we can use Segment Treeto do both operations in O(Logn) time.
Representation of Segment trees
1. Leaf Nodes are the elements of the input array.
2. Each internal node represents the maximum of all of its child.
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 index 2*i+2 and the parent is at index (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 maximum 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 into two halves at every level. Since the constructed tree is always a full binary tree with n leaves, there will be n-1 internal nodes. So total nodes will be 2*n – 1. Height of the segment tree will be log2n. 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^ceil(log2n) ) – 1.
Query for maximum value of given range : Once the tree is constructed, below is the algorithm to find maximum of given range.

```node--> node number, l -->
query start index, r --> query end index;

int getMax(node, l, r)
{
if range of node is within l and r
return value of node
else if range of node is completely outside l and r
return -1
else
return max(getMax(node's left child, l, r),
getMax(node's right child, l, r))
}```

Below is the implementation of above approach :

## C++

 `// CPP code for range maximum query and updates``#include ``using` `namespace` `std;` `// A utility function to get the``// middle index of given range.``int` `getMid(``int` `s, ``int` `e)``{``    ``return` `s + (e - s) / 2;``}` `/*  A recursive function to get the sum of``    ``values in given range of the array.``    ``The following are parameters for this``    ``function.` `    ``st       -> Pointer to segment tree``    ``node     -> Index of current node in``                ``the segment tree .``    ``ss & se  -> Starting and ending indexes``                ``of the segment represented``                ``by current node, i.e., st[node]``    ``l & r    -> Starting and ending indexes``                ``of range query */``int` `MaxUtil(``int``* st, ``int` `ss, ``int` `se, ``int` `l,``            ``int` `r, ``int` `node)``{``    ``// If segment of this node is completely``    ``// part of given range, then return``    ``// the max of segment``    ``if` `(l <= ss && r >= se)``        ``return` `st[node];` `    ``// If segment of this node does not``    ``// belong to given range``    ``if` `(se < l || ss > r)``        ``return` `-1;` `    ``// If segment of this node is partially``    ``// the part of given range``    ``int` `mid = getMid(ss, se);``    ` `    ``return` `max(MaxUtil(st, ss, mid, l, r,``                       ``2 * node + 1),``               ``MaxUtil(st, mid + 1, se, l,``                       ``r, 2 * node + 2));``}` `/* A recursive function to update the nodes``   ``which have the given index in their range.``   ``The following are parameters st, ss and``   ``se are same as defined``   ``above index -> index of the element``   ``to be updated.*/``void` `updateValue(``int` `arr[], ``int``* st, ``int` `ss, ``int` `se,``                 ``int` `index, ``int` `value, ``int` `node)``{``    ``if` `(index < ss || index > se)``    ``{``        ``cout << ``"Invalid Input"` `<< endl;``        ``return``;``    ``}``    ` `    ``if` `(ss == se)``    ``{  ``        ``// update value in array and in segment tree``        ``arr[index] = value;``        ``st[node] = value;``    ``}``    ``else` `{``            ``int` `mid = getMid(ss, se);``            ` `            ``if` `(index >= ss && index <= mid)``                ``updateValue(arr, st,``                            ``ss, mid, index,``                            ``value, 2 * node + 1);``            ``else``                ``updateValue(arr,``                            ``st, mid + 1, se,``                            ``index,``                            ``value, 2 * node + 2);``            ` `            ``st[node] = max(st[2 * node + 1],``                       ``st[2 * node + 2]);``    ``}``    ``return``;``}` `// Return max of elements in range from``// index l (query start) to r (query end).``int` `getMax(``int``* st, ``int` `n, ``int` `l, ``int` `r)``{``    ``// Check for erroneous input values``    ``if` `(l < 0 || r > n - 1 || l > r)``    ``{``        ``printf``(``"Invalid Input"``);``        ``return` `-1;``    ``}` `    ``return` `MaxUtil(st, 0, n - 1, l, r, 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 max of values in this node``    ``int` `mid = getMid(ss, se);``    ` `    ``st[si] = max(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.*/``int``* constructST(``int` `arr[], ``int` `n)``{``    ``// Height of segment tree``    ``int` `x = (``int``)(``ceil``(log2(n)));` `    ``// Maximum size of segment tree``    ``int` `max_size = 2 * (``int``)``pow``(2, x) - 1;` `    ``// Allocate memory``    ``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 code``int` `main()``{``    ``int` `arr[] = { 1, 3, 5, 7, 9, 11 };``    ``int` `n = ``sizeof``(arr) / ``sizeof``(arr);` `    ``// Build segment tree from given array``    ``int``* st = constructST(arr, n);` `    ``// Print max of values in array``    ``// from index 1 to 3``    ``cout << ``"Max of values in given range = "``         ``<< getMax(st, n, 1, 3) << endl;` `    ``// Update: set arr = 8 and update``    ``// corresponding segment tree nodes.``    ``updateValue(arr, st, 0, n - 1, 1, 8, 0);` `    ``// Find max after the value is updated``    ``cout << ``"Updated max of values in given range = "``         ``<< getMax(st, n, 1, 3) << endl;``    ` `    ``return` `0;``}`

## Java

 `// Java code for range maximum query and updates``import` `java.io.*;``import` `java.util.*;` `class` `GFG {` `    ``// A utility function to get the``    ``// middle index of given range.``    ``static` `int` `getMid(``int` `s, ``int` `e)``    ``{``        ``return` `s + (e - s) / ``2``;``    ``}` `    ``/*``    ``* A recursive function to get the sum``    ``of values in given range of the array.``    ``* The following are parameters``      ``for this function.``    ``*``    ``* st -> Pointer to segment tree``    ``* node -> Index of current node in``    ``*         the segment tree.``    ``* ss & se -> Starting and ending indexes``    ``*         of the segment represented``    ``*         by current node, i.e., st[node]``    ``* l & r -> Starting and ending indexes``    ``*         of range query``    ``*/``    ``static` `int` `MaxUtil(``int``[] st, ``int` `ss,``                       ``int` `se, ``int` `l,``                       ``int` `r, ``int` `node)``    ``{` `        ``// If segment of this node is completely``        ``// part of given range, then return``        ``// the max of segment``        ``if` `(l <= ss && r >= se)``            ``return` `st[node];` `        ``// If segment of this node does not``        ``// belong to given range``        ``if` `(se < l || ss > r)``            ``return` `-``1``;` `        ``// If segment of this node is partially``        ``// the part of given range``        ``int` `mid = getMid(ss, se);` `        ``return` `Math.max(``            ``MaxUtil(st, ss, mid, l, r,``                     ``2` `* node + ``1``),``            ``MaxUtil(st, mid + ``1``, se, l, r,``                    ``2` `* node + ``2``));``    ``}` `    ``/*``    ``* A recursive function to update the``    ``nodes which have the given index in their``    ``* range. The following are parameters``    ``st, ss and se are same as defined above``    ``* index -> index of the element to be updated.``    ``*/``    ``static` `void` `updateValue(``int` `arr[], ``int``[]``                            ``st, ``int` `ss,``                            ``int` `se, ``int` `index,``                            ``int` `value,``                            ``int` `node)``    ``{``        ``if` `(index < ss || index > se) {``            ``System.out.println(``"Invalid Input"``);``            ``return``;``        ``}` `        ``if` `(ss == se) {` `            ``// update value in array and in``            ``// segment tree``            ``arr[index] = value;``            ``st[node] = value;``        ``}``        ``else` `{``            ``int` `mid = getMid(ss, se);` `            ``if` `(index >= ss && index <= mid)``                ``updateValue(arr, st, ss, mid,``                            ``index, value,``                            ``2` `* node + ``1``);``            ``else``                ``updateValue(arr, st, mid + ``1``, se, index,``                            ``value, ``2` `* node + ``2``);` `            ``st[node] = Math.max(st[``2` `* node + ``1``],``                                ``st[``2` `* node + ``2``]);``        ``}``        ``return``;``    ``}` `    ``// Return max of elements in range from``    ``// index l (query start) to r (query end).``    ``static` `int` `getMax(``int``[] st, ``int` `n, ``int` `l, ``int` `r)``    ``{` `        ``// Check for erroneous input values``        ``if` `(l < ``0` `|| r > n - ``1` `|| l > r) {``            ``System.out.printf(``"Invalid Input\n"``);``            ``return` `-``1``;``        ``}` `        ``return` `MaxUtil(st, ``0``, n - ``1``, l, r, ``0``);``    ``}` `    ``// A recursive function that constructs Segment``    ``// Tree for array[ss..se]. si is index of``    ``// current node in segment tree st``    ``static` `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 max of values in this node``        ``int` `mid = getMid(ss, se);` `        ``st[si] = Math.max(``            ``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.``    ``*/``    ``static` `int``[] constructST(``int` `arr[], ``int` `n)``    ``{` `        ``// 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``;` `        ``// Allocate memory``        ``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 Code``    ``public` `static` `void` `main(String[] args)``    ``{` `        ``int``[] arr = { ``1``, ``3``, ``5``, ``7``, ``9``, ``11` `};``        ``int` `n = arr.length;` `        ``// Build segment tree from given array``        ``int``[] st = constructST(arr, n);` `        ``// Print max of values in array``        ``// from index 1 to 3``        ``System.out.println(``"Max of values in given range = "``                           ``+ getMax(st, n, ``1``, ``3``));` `        ``// Update: set arr = 8 and update``        ``// corresponding segment tree nodes.``        ``updateValue(arr, st, ``0``, n - ``1``, ``1``, ``8``, ``0``);` `        ``// Find max after the value is updated``        ``System.out.println(``            ``"Updated max of values in given range = "``            ``+ getMax(st, n, ``1``, ``3``));``    ``}``}` `// This code is contributed by``// sanjeev2552`

## Python3

 `# Python3 code for range maximum query and updates``from` `math ``import` `ceil, log` `# A utility function to get the``# middle index of given range.`  `def` `getMid(s, e):``    ``return` `s ``+` `(e ``-` `s) ``/``/` `2` `# /* A recursive function to get the sum of``    ``# values in given range of the array.``    ``# The following are parameters for this``    ``# function.``    ``#``    ``# st     -> Pointer to segment tree``    ``# node     -> Index of current node in``    ``#             the segment tree .``    ``# ss & se -> Starting and ending indexes``    ``#             of the segment represented``    ``#             by current node, i.e., st[node]``    ``# l & r -> Starting and ending indexes``    ``#             of range query */`  `def` `MaxUtil(st, ss, se, l, r, node):` `    ``# If segment of this node is completely``    ``# part of given range, then return``    ``# the max of segment``    ``if` `(l <``=` `ss ``and` `r >``=` `se):``        ``return` `st[node]` `    ``# If segment of this node does not``    ``# belong to given range``    ``if` `(se < l ``or` `ss > r):``        ``return` `-``1` `    ``# If segment of this node is partially``    ``# the part of given range``    ``mid ``=` `getMid(ss, se)` `    ``return` `max``(MaxUtil(st, ss, mid, l, r,``                       ``2` `*` `node ``+` `1``),``               ``MaxUtil(st, mid ``+` `1``, se, l,``                       ``r, ``2` `*` `node ``+` `2``))` `#``# /* A recursive function to update the nodes which``# have the given index in their range. The following``# are parameters st, ss and se are same as defined``# above index -> index of the element to be updated.*/`  `def` `updateValue(arr, st, ss, se, index, value, node):``    ``if` `(index < ss ``or` `index > se):``        ``print``(``"Invalid Input"``)``        ``return` `    ``if` `(ss ``=``=` `se):` `        ``# update value in array and in segment tree``        ``arr[index] ``=` `value``        ``st[node] ``=` `value``    ``else``:``        ``mid ``=` `getMid(ss, se)` `        ``if` `(index >``=` `ss ``and` `index <``=` `mid):``            ``updateValue(arr, st, ss, mid, index,``                        ``value, ``2` `*` `node ``+` `1``)``        ``else``:``            ``updateValue(arr, st, mid ``+` `1``, se,``                        ``index, value, ``2` `*` `node ``+` `2``)` `        ``st[node] ``=` `max``(st[``2` `*` `node ``+` `1``],``                       ``st[``2` `*` `node ``+` `2``])``    ``return` `# Return max of elements in range from``# index l (query start) to r (query end).`  `def` `getMax(st, n, l, r):` `    ``# Check for erroneous input values``    ``if` `(l < ``0` `or` `r > n ``-` `1` `or` `l > r):``        ``printf(``"Invalid Input"``)``        ``return` `-``1` `    ``return` `MaxUtil(st, ``0``, n ``-` `1``, l, r, ``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 max of values in this node``    ``mid ``=` `getMid(ss, se)` `    ``st[si] ``=` `max``(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.*/`  `def` `constructST(arr, n):` `    ``# Height of segment tree``    ``x ``=` `ceil(log(n, ``2``))` `    ``# Maximum size of segment tree``    ``max_size ``=` `2` `*` `pow``(``2``, x) ``-` `1` `    ``# Allocate memory``    ``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``, ``5``, ``7``, ``9``, ``11``]``    ``n ``=` `len``(arr)` `    ``# Build segment tree from given array``    ``st ``=` `constructST(arr, n)` `    ``# Prmax of values in array``    ``# from index 1 to 3``    ``print``(``"Max of values= "``, getMax(st, n, ``1``, ``3``))` `    ``# Update: set arr = 8 and update``    ``# corresponding segment tree nodes.``    ``updateValue(arr, st, ``0``, n ``-` `1``, ``1``, ``8``, ``0``)` `    ``# Find max after the value is updated``    ``print``(``"Updated values = "``, getMax(st, n, ``1``, ``3``))` `# This code is contributed by mohit kumar 29`

## C#

 `// C# code for range maximum query and updates``using` `System;``class` `GFG``{` `  ``// A utility function to get the``  ``// middle index of given range.``  ``static` `int` `getMid(``int` `s, ``int` `e)``  ``{``    ``return` `s + (e - s) / 2;``  ``}` `  ``/*``    ``* A recursive function to get the sum``    ``of values in given range of the array.``    ``* The following are parameters``      ``for this function.``    ``*``    ``* st -> Pointer to segment tree``    ``* node -> Index of current node in``    ``*         the segment tree.``    ``* ss & se -> Starting and ending indexes``    ``*         of the segment represented``    ``*         by current node, i.e., st[node]``    ``* l & r -> Starting and ending indexes``    ``*         of range query``    ``*/``  ``static` `int` `MaxUtil(``int``[] st, ``int` `ss, ``int` `se,``                     ``int` `l, ``int` `r, ``int` `node)``  ``{` `    ``// If segment of this node is completely``    ``// part of given range, then return``    ``// the max of segment``    ``if` `(l <= ss && r >= se)``    ``{``      ``return` `st[node];``    ``}` `    ``// If segment of this node does not``    ``// belong to given range``    ``if` `(se < l || ss > r)``    ``{``      ``return` `-1;``    ``}` `    ``// If segment of this node is partially``    ``// the part of given range``    ``int` `mid = getMid(ss, se);``    ``return` `Math.Max(MaxUtil(st, ss, mid, l, r, 2 * node + 1),``                    ``MaxUtil(st, mid + 1, se, l, r,2 * node + 2));` `  ``}` `  ``/*``    ``* A recursive function to update the``    ``nodes which have the given index in their``    ``* range. The following are parameters``    ``st, ss and se are same as defined above``    ``* index -> index of the element to be updated.``    ``*/``  ``static` `void` `updateValue(``int``[] arr,``int``[] st, ``int` `ss,``                          ``int` `se, ``int` `index,``                          ``int` `value,``int` `node)``  ``{``    ``if` `(index < ss || index > se)``    ``{``      ``Console.WriteLine(``"Invalid Input"``);``      ``return` `;``    ``}``    ``if` `(ss == se)``    ``{` `      ``// update value in array and in``      ``// segment tree``      ``arr[index] = value;``      ``st[node] = value;``    ``}``    ``else``    ``{``      ``int` `mid = getMid(ss, se);``      ``if` `(index >= ss && index <= mid)``      ``{``        ``updateValue(arr, st, ss, mid, index,``                    ``value, 2 * node + 1);``      ``}``      ``else``      ``{``        ``updateValue(arr, st, mid + 1, se,``                    ``index,value, 2 * node + 2);``      ``}``      ``st[node] = Math.Max(st[2 * node + 1],``                          ``st[2 * node + 2]);``    ``}``    ``return``;``  ``}` `  ``// Return max of elements in range from``  ``// index l (query start) to r (query end).``  ``static` `int` `getMax(``int``[] st, ``int` `n, ``int` `l, ``int` `r)``  ``{` `    ``// Check for erroneous input values``    ``if``(l < 0 || r > n - 1 || l > r)``    ``{``      ``Console.WriteLine(``"Invalid Input"``);``      ``return` `-1;``    ``}``    ``return` `MaxUtil(st, 0, n - 1, l, r, 0);``  ``}` `  ``// A recursive function that constructs Segment``  ``// Tree for array[ss..se]. si is index of``  ``// current node in segment tree st``  ``static` `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 max of values in this node``    ``int` `mid = getMid(ss, se);``    ``st[si] = Math.Max(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.``    ``*/``  ``static` `int``[] constructST(``int``[] arr, ``int` `n)``  ``{` `    ``// 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;` `    ``// Allocate memory``    ``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 Code``  ``static` `public` `void` `Main ()``  ``{``    ``int``[] arr = { 1, 3, 5, 7, 9, 11 };``    ``int` `n = arr.Length;` `    ``// Build segment tree from given array``    ``int``[] st = constructST(arr, n);` `    ``// Print max of values in array``    ``// from index 1 to 3``    ``Console.WriteLine(``"Max of values in given range = "``                      ``+ getMax(st, n, 1, 3));` `    ``// Update: set arr = 8 and update``    ``// corresponding segment tree nodes.``    ``updateValue(arr, st, 0, n - 1, 1, 8, 0);` `    ``// Find max after the value is updated``    ``Console.WriteLine(``"Updated max of values in given range = "``                      ``+ getMax(st, n, 1, 3));``  ``}``}` `// This code is contributed by avanitrachhadiya2155`

## Javascript

 ``

Output

```Max of values in given range = 7
Updated max of values in given range = 8```

My Personal Notes arrow_drop_up