给定一个由N个整数和Q个查询组成的数组arr [] ，其中每个查询由一个索引范围[L，R]组成。对于每个查询，任务是在数组中查找除给定索引范围内的元素以外的最小元素。
例子：
Input: arr[] = {3, 2, 1, 4, 5}, Q[][] = {{1, 2}, {2, 3}}
Output:
3
2
Query 1: min(arr[0], arr[3..4]) = min(3, 4, 5) = 3
Query 2: min(arr[0..1], arr[4]) = min(3, 2, 5) = 2
Input: arr[] = {1, 2, 3, 4, 5}, Q[][] = {{0, 2}}
Output:
4
方法：在有多个查询的情况下，可以构建一个段树来查找任何索引范围内的最小元素。现在，对于每个查询[L，R]，不包括此范围的最小元素将是min(min(arr [0…L-1])，min(arr [R + 1 … N-1]))
下面是上述方法的实现：
C++
// C++ implementation of the approach
#include
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[] = { 3, 2, 1, 4, 5 };
int n = sizeof(arr) / sizeof(arr[0]);
int queries[][2] = { { 1, 2 }, { 2, 3 } };
int q = sizeof(queries) / sizeof(queries[0]);
// Build segment tree from given array
int* st = constructST(arr, n);
// Perform queries
for (int i = 0; i < q; i++) {
// Current index range to be exluded
int L = queries[i][0];
int R = queries[i][1];
// Minimum in the left part
int left = ((L - 1) < 0)
? INT_MAX
: RMQ(st, n, 0, L - 1);
// Minimum in the right part
int right = ((R + 1) >= n)
? INT_MAX
: RMQ(st, n, R + 1, n - 1);
// Minimum in the array excluding the given range
int minn = min(left, right);
// Complete array has been excluded
if (minn == INT_MAX)
cout << -1 << endl;
else
cout << minn << endl;
}
return 0;
} Java
// Java implementation of the approach
import java.util.*;
class GFG{
// A utility function to get minimum of two numbers
static int minVal(int x, int y) {
return (x < y) ? x : y;
}
// 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 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
*/
static 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 Integer.MAX_VALUE;
// 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()
static int RMQ(int[] st, 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(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 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
*/
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;
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 = { 3, 2, 1, 4, 5 };
int n = arr.length;
int[][] queries = { { 1, 2 }, { 2, 3 } };
int q = queries.length;
// Build segment tree from given array
int[] st = constructST(arr, n);
// Perform queries
for (int i = 0; i < q; i++)
{
// Current index range to be exluded
int L = queries[i][0];
int R = queries[i][1];
// Minimum in the left part
int left = ((L - 1) < 0) ? Integer.MAX_VALUE : RMQ(st, n, 0, L - 1);
// Minimum in the right part
int right = ((R + 1) >= n) ? Integer.MAX_VALUE : RMQ(st, n, R + 1, n - 1);
// Minimum in the array excluding the given range
int minn = Math.min(left, right);
// Complete array has been excluded
if (minn == Integer.MAX_VALUE)
System.out.println(-1);
else
System.out.println(minn);
}
}
}
// This code is contributed by
// sanjeev2552Python3
# Python3 implementation of the approach
from math import ceil,log,floor,sqrt
# A utility function to get minimum of two numbers
def minVal(x, y):
return min(x, 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 10**9
# 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 = ceil(log(n, 2))
# Maximum size of segment tree
max_size = 2 * pow(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 program to test above functions
if __name__ == '__main__':
arr = [3, 2, 1, 4, 5]
n = len(arr)
queries = [[1, 2], [2, 3]]
q = len(queries)
# Build segment tree from given array
st = constructST(arr, n)
# Perform queries
for i in range(q):
# Current index range to be exluded
L = queries[i][0]
R = queries[i][1]
# Minimum in the left part
if( (L - 1) < 0):
left = 10**9
else:
left = RMQ(st, n, 0, L - 1)
# Minimum in the right part
if (R + 1) >= n:
right = 10**9
else:
right = RMQ(st, n, R + 1, n - 1)
# Minimum in the array excluding the given range
minn = min(left, right)
# Complete array has been excluded
if (minn == 10**9):
print(-1)
else:
print(minn)
# This code is contributed by mohit kumar 29C#
// C# implementation of the approach
using System;
class GFG{
// A utility function to get minimum
// of two numbers
static int minVal(int x, int y)
{
return (x < y) ? x : y;
}
// 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 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
*/
static 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 Int32.MaxValue;
// 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()
static int RMQ(int[] st, int n, int qs, int qe)
{
// Check for erroneous input values
if (qs < 0 || qe > n - 1 || qs > qe)
{
Console.Write("Invalid Input" + "\n");
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
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 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
*/
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;
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 = { 3, 2, 1, 4, 5 };
int n = arr.Length;
int[,] queries = { { 1, 2 }, { 2, 3 } };
int q = queries.GetLength(0);
// Build segment tree from given array
int[] st = constructST(arr, n);
// Perform queries
for(int i = 0; i < q; i++)
{
// Current index range to be exluded
int L = queries[i, 0];
int R = queries[i, 1];
// Minimum in the left part
int left = ((L - 1) < 0) ? Int32.MaxValue :
RMQ(st, n, 0, L - 1);
// Minimum in the right part
int right = ((R + 1) >= n) ? Int32.MaxValue :
RMQ(st, n, R + 1, n - 1);
// Minimum in the array excluding
// the given range
int minn = Math.Min(left, right);
// Complete array has been excluded
if (minn == Int32.MaxValue)
Console.Write(-1 + "\n");
else
Console.Write(minn + "\n");
}
}
}
// This code is contributed by rutvik_56输出：
3
2
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