Consideremos el siguiente problema para comprender los árboles de segmentos.
Tenemos una array arr[0 . . . n-1]. Deberíamos poder
1 Encontrar el xor de los elementos del índice l a r donde 0 <= l <= r <= n-1.
2 Cambiar el valor de un elemento específico de la array a un nuevo valor x. Necesitamos hacer arr[i] = x donde 0 <= i <= n-1.
Similar a Suma del rango dado.
Una solución simple es ejecutar un ciclo de l a r y calcular xor de elementos en un rango dado. Para actualizar un valor, simplemente haga arr[i] = x. La primera operación toma el tiempo O(n) y la segunda operación toma el tiempo O(1).
Enfoque eficiente:
si el número de consultas y actualizaciones es igual, podemos realizar ambas operaciones en tiempo O (log n). Podemos usar un árbol de segmentos para hacer ambas operaciones en tiempo O (Iniciar sesión).
Representación de árboles de segmentos
1. Los Nodes hoja son los elementos del arreglo de entrada.
2. Cada Node interno representa alguna fusión de los Nodes hoja. La fusión puede ser diferente para diferentes problemas. Para este problema, fusionar es Xor de hojas debajo de un Node.
Se utiliza una representación de array de árbol para representar árboles de segmento. Para cada Node en el índice i, el hijo izquierdo está en el índice 2*i+1, el hijo derecho en 2*i+2 y el padre está en (i-1)/2.
Consulta de producto de rango dado
Una vez que se construye el árbol, cómo obtener el Xor usando el árbol de segmento construido. El siguiente es un algoritmo para obtener el xor de elementos.
int getXor(node, l, r)
{
if range of node is within l and r
return value in node
else if range of node is completely outside l and r
return 0
else
return getXor(node's left child, l, r) ^
getXor(node's right child, l, r)
}
C++
// C++ program to show segment tree operations
// like construction, query and update
#include <bits/stdc++.h>
#include <math.h>
using namespace std;
// 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 Xor of
values in given range of the array. The
following are parameters for this function.
st --> Pointer to segment tree
si --> 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[si]
qs & qe --> Starting and ending indexes of
query range */
int getXorUtil(int *st, int ss, int se, int qs,
int qe, int si)
{
// If segment of this node is a part of given
// range, then return the Xor of the segment
if (qs <= ss && qe >= se)
return st[si];
// If segment of this node is outside
// the given range
if (se < qs || ss > qe)
return 0;
// If a part of this segment overlaps
// with the given range
int mid = getMid(ss, se);
return getXorUtil(st, ss, mid, qs, qe, 2*si+1) ^
getXorUtil(st, mid+1, se, qs, qe, 2*si+2);
}
/* A recursive function to update the nodes
which have the given index in their range.
The following are parameters
st, si, ss and se are same as getXorUtil()
i --> index of the element to be updated.
This index is in input array.*/
void updateValueUtil(int *st, int ss, int se, int i,
int prev_val, int new_val, int si)
{
// Base Case: If the input index lies outside
// the range of this segment
if (i < ss || i > se)
return;
// If the input index is in range of this node,
// then update the value of the node and its children
st[si] = (st[si]^prev_val)^new_val;
if (se != ss)
{
int mid = getMid(ss, se);
updateValueUtil(st, ss, mid, i, prev_val,
new_val, 2*si + 1);
updateValueUtil(st, mid+1, se, i, prev_val,
new_val, 2*si + 2);
}
}
// The function to update a value in input
// array and segment tree. It uses updateValueUtil()
// to update the value in segment tree
void updateValue(int arr[], int *st, int n,
int i, int new_val)
{
// Check for erroneous input index
if (i < 0 || i > n-1)
{
printf("Invalid Input");
return;
}
int temp = arr[i];
// Update the value in array
arr[i] = new_val;
// Update the values of nodes in segment tree
updateValueUtil(st, 0, n-1, i, temp, new_val, 0);
}
// Return Xor of elements in range from index qs (query start)
// to qe (query end). It mainly uses getXorUtil()
int getXor(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 0;
}
return getXorUtil(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 Xor of values in this node
int mid = getMid(ss, se);
st[si] = 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;
// 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 program to test above functions
int main()
{
int arr[] = {1, 3, 5, 7, 9, 11};
int n = sizeof(arr)/sizeof(arr[0]);
// Build segment tree from given array
int *st = constructST(arr, n);
// Print Xor of values in array from index 1 to 3
printf("Xor of values in given range = %d\n",
getXor(st, n, 0, 2));
// Update: set arr[1] = 10 and update corresponding
// segment tree nodes
updateValue(arr, st, n, 1, 10);
// Find Xor after the value is updated
printf("Updated Xor of values in given range = %d\n",
getXor(st, n, 0, 2));
return 0;
}
Java
// Java implementation of the approach
import java.util.*;
class GFG {
// A utility function to get the middle
// index from corner indexes.
static int getMid(int s, int e) {
return s + (e - s) / 2;
}
/*
* A recursive function to get the Xor of
* values in given range of the array. The
* following are parameters for this function.
* st --> Pointer to segment tree
* si --> 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[si]
* qs & qe --> Starting and ending indexes of
* query range
*/
static int getXorUtil(int[] st, int ss, int se,
int qs, int qe, int si)
{
// If segment of this node is a part of given
// range, then return the Xor of the segment
if (qs <= ss && qe >= se)
return st[si];
// If segment of this node is outside
// the given range
if (se < qs || ss > qe)
return 0;
// If a part of this segment overlaps
// with the given range
int mid = getMid(ss, se);
return getXorUtil(st, ss, mid, qs, qe, 2 * si + 1) ^
getXorUtil(st, mid + 1, se, qs, qe, 2 * si + 2);
}
/*
* A recursive function to update the nodes
* which have the given index in their range.
* The following are parameters
* st, si, ss and se are same as getXorUtil()
* i --> index of the element to be updated.
* This index is in input array.
*/
static void updateValueUtil(int[] st, int ss, int se, int i,
int prev_val, int new_val, int si)
{
// Base Case: If the input index lies outside
// the range of this segment
if (i < ss || i > se)
return;
// If the input index is in range of this node,
// then update the value of the node and its children
st[si] = (st[si] ^ prev_val) ^ new_val;
if (se != ss) {
int mid = getMid(ss, se);
updateValueUtil(st, ss, mid, i, prev_val,
new_val, 2 * si + 1);
updateValueUtil(st, mid + 1, se, i, prev_val,
new_val, 2 * si + 2);
}
}
// The function to update a value in input
// array and segment tree. It uses updateValueUtil()
// to update the value in segment tree
static void updateValue(int arr[], int[] st, int n,
int i, int new_val)
{
// Check for erroneous input index
if (i < 0 || i > n - 1) {
System.out.printf("Invalid Input\n");
return;
}
int temp = arr[i];
// Update the value in array
arr[i] = new_val;
// Update the values of nodes in segment tree
updateValueUtil(st, 0, n - 1, i, temp, new_val, 0);
}
// Return Xor of elements in range from index qs (query start)
// to qe (query end). It mainly uses getXorUtil()
static int getXor(int[] st, int n, int qs, int qe)
{
// Check for erroneous input values
if (qs < 0 || qe > n - 1 || qs > qe)
{
System.out.printf("Invalid Input\n");
return 0;
}
return getXorUtil(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
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 Xor of values in this node
int mid = getMid(ss, se);
st[si] = 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
*/
static int[] 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;
// 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 Xor of values in array from index 1 to 3
System.out.printf("Xor of values in given range = %d\n",
getXor(st, n, 0, 2));
// Update: set arr[1] = 10 and update corresponding
// segment tree nodes
updateValue(arr, st, n, 1, 10);
// Find Xor after the value is updated
System.out.printf("Updated Xor of values in given range = %d\n",
getXor(st, n, 0, 2));
}
}
// This code is contributed by
// sanjeev2552
Python3
# Python3 program to show segment tree operations
# like construction, query and update
from math import ceil, log2;
# 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 Xor of
values in given range of the array. The
following are parameters for this function.
st --> Pointer to segment tree
si --> 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[si]
qs & qe --> Starting and ending indexes of
query range """
def getXorUtil(st, ss, se, qs, qe, si) :
# If segment of this node is a part of given
# range, then return the Xor of the segment
if (qs <= ss and qe >= se) :
return st[si];
# If segment of this node is outside
# the given range
if (se < qs or ss > qe) :
return 0;
# If a part of this segment overlaps
# with the given range
mid = getMid(ss, se);
return getXorUtil(st, ss, mid, qs, qe, 2 * si + 1) ^ \
getXorUtil(st, mid + 1, se, qs, qe, 2 * si + 2);
""" A recursive function to update the nodes
which have the given index in their range.
The following are parameters
st, si, ss and se are same as getXorUtil()
i --> index of the element to be updated.
This index is in input array."""
def updateValueUtil(st, ss, se, i,
prev_val, new_val, si) :
# Base Case: If the input index lies
# outside the range of this segment
if (i < ss or i > se) :
return;
# If the input index is in range of this node,
# then update the value of the node and its children
st[si] = (st[si] ^ prev_val) ^ new_val;
if (se != ss) :
mid = getMid(ss, se);
updateValueUtil(st, ss, mid, i, prev_val,
new_val, 2 * si + 1);
updateValueUtil(st, mid + 1, se, i,
prev_val, new_val, 2 * si + 2);
# The function to update a value in input
# array and segment tree. It uses updateValueUtil()
# to update the value in segment tree
def updateValue(arr, st, n, i, new_val) :
# Check for erroneous input index
if (i < 0 or i > n - 1) :
print("Invalid Input");
return;
temp = arr[i];
# Update the value in array
arr[i] = new_val;
# Update the values of nodes in segment tree
updateValueUtil(st, 0, n - 1, i, temp, new_val, 0);
# Return Xor of elements in range from
# index qs (query start) to qe (query end).
# It mainly uses getXorUtil()
def getXor(st, n, qs, qe) :
# Check for erroneous input values
if (qs < 0 or qe > n - 1 or qs > qe) :
print("Invalid Input");
return 0;
return getXorUtil(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 Xor of values in this node
mid = getMid(ss, se);
st[si] = 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;
# 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);
# Print Xor of values in array from index 1 to 3
print("Xor of values in given range =",
getXor(st, n, 0, 2));
# Update: set arr[1] = 10 and update
# corresponding segment tree nodes
updateValue(arr, st, n, 1, 10);
# Find Xor after the value is updated
print("Updated Xor of values in given range =",
getXor(st, n, 0, 2));
# This code is contributed by AnkitRai01
C#
// C# implementation of the approach
using System;
class GFG
{
// A utility function to get the middle
// index from corner indexes.
static int getMid(int s, int e)
{
return s + (e - s) / 2;
}
/*
* A recursive function to get the Xor of
* values in given range of the array. The
* following are parameters for this function.
* st --> Pointer to segment tree
* si --> 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[si]
* qs & qe --> Starting and ending indexes of
* query range
*/
static int getXorUtil(int[] st, int ss, int se,
int qs, int qe, int si)
{
// If segment of this node is a part of given
// range, then return the Xor of the segment
if (qs <= ss && qe >= se)
return st[si];
// If segment of this node is outside
// the given range
if (se < qs || ss > qe)
return 0;
// If a part of this segment overlaps
// with the given range
int mid = getMid(ss, se);
return getXorUtil(st, ss, mid, qs, qe, 2 * si + 1) ^
getXorUtil(st, mid + 1, se, qs, qe, 2 * si + 2);
}
/*
* A recursive function to update the nodes
* which have the given index in their range.
* The following are parameters
* st, si, ss and se are same as getXorUtil()
* i --> index of the element to be updated.
* This index is in input array.
*/
static void updateValueUtil(int[] st, int ss, int se, int i,
int prev_val, int new_val, int si)
{
// Base Case: If the input index lies outside
// the range of this segment
if (i < ss || i > se)
return;
// If the input index is in range of this node,
// then update the value of the node and its children
st[si] = (st[si] ^ prev_val) ^ new_val;
if (se != ss)
{
int mid = getMid(ss, se);
updateValueUtil(st, ss, mid, i, prev_val,
new_val, 2 * si + 1);
updateValueUtil(st, mid + 1, se, i, prev_val,
new_val, 2 * si + 2);
}
}
// The function to update a value in input
// array and segment tree. It uses updateValueUtil()
// to update the value in segment tree
static void updateValue(int[] arr, int[] st, int n,
int i, int new_val)
{
// Check for erroneous input index
if (i < 0 || i > n - 1)
{
Console.WriteLine("Invalid Input");
return;
}
int temp = arr[i];
// Update the value in array
arr[i] = new_val;
// Update the values of nodes in segment tree
updateValueUtil(st, 0, n - 1, i, temp, new_val, 0);
}
// Return Xor of elements in range from index qs (query start)
// to qe (query end). It mainly uses getXorUtil()
static int getXor(int[] st, int n, int qs, int qe)
{
// Check for erroneous input values
if (qs < 0 || qe > n - 1 || qs > qe)
{
Console.WriteLine("Invalid Input");
return 0;
}
return getXorUtil(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
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 Xor of values in this node
int mid = getMid(ss, se);
st[si] = 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
*/
static int[] 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;
// 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()
{
int[] arr = { 1, 3, 5, 7, 9, 11 };
int n = arr.Length;
// Build segment tree from given array
int[] st = constructST(arr, n);
// Print Xor of values in array from index 1 to 3
Console.WriteLine("Xor of values in given range = " +
getXor(st, n, 0, 2));
// Update: set arr[1] = 10 and update corresponding
// segment tree nodes
updateValue(arr, st, n, 1, 10);
// Find Xor after the value is updated
Console.Write("Updated Xor of values in given range = " +
getXor(st, n, 0, 2));
}
}
// This code is contributed by
// Saurabh Jaiswal
Producción:
Xor of values in given range = 7 Updated Xor of values in given range = 14
Publicación traducida automáticamente
Artículo escrito por sahilkhoslaa y traducido por Barcelona Geeks. The original can be accessed here. Licence: CCBY-SA