Hemos introducido el árbol de segmentos con un ejemplo simple en la publicación anterior. En esta publicación, el problema de consulta de rango mínimo se analiza como otro ejemplo en el que se puede usar el árbol de segmentos. El siguiente es el enunciado del problema:
Tenemos un arreglo arr[0 . . . n-1]. Deberíamos poder encontrar de manera eficiente el valor mínimo desde el índice qs (inicio de la consulta) hasta qe (final de la consulta) donde 0 <= qs <= qe <= n-1 .
Una solución simple es ejecutar un ciclo de qs a qe y encontrar el elemento mínimo en el rango dado. Esta solución toma tiempo O(n) en el peor de los casos.
Otra solución es crear una array 2D donde una entrada [i, j] almacene el valor mínimo en el rango arr[i..j]. El mínimo de un rango dado ahora se puede calcular en tiempo O(1), pero el preprocesamiento toma tiempo O(n^2). Además, este enfoque necesita O (n ^ 2) espacio adicional que puede volverse enorme para arrays de entrada grandes.
El árbol de segmentos se puede utilizar para realizar preprocesamiento y consultas en un tiempo moderado. Con un árbol de segmentos, el tiempo de preprocesamiento es O(n) y la complejidad del tiempo para una consulta de rango mínimo es O(Logn). El espacio adicional requerido es O(n) para almacenar el árbol de segmentos.
Representación de árboles de segmentos
1. Los Nodes hoja son los elementos del arreglo de entrada.
2. Cada Node interno representa el mínimo de todas las hojas debajo de él.
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⌋ .
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>
Publicación traducida automáticamente
Artículo escrito por GeeksforGeeks-1 y traducido por Barcelona Geeks. The original can be accessed here. Licence: CCBY-SA