给定一个由N个不同的正整数组成的数组。任务是找到对给定数组进行排序的最低成本。交换两个元素X和Y的成本为X * Y。
例子：
Input: arr[] = {8, 4, 5, 3, 2, 7}
Output: 57
Explanation:
Swap element at index 4 with index 5 – cost(arr[4]*arr[5]) = (2*7) = 14,
Array becomes {8, 4, 5, 3, 7, 2}
then, swap element at index 0 with 5 – cost(arr[0]*arr[5]) = (8*2) = 16,
Array becomes {2, 4, 5, 3, 7, 8}
then, swap element at index 2 with 3 – cost(arr[2]*arr[3]) = (5*3) = 15,
Array becomes {2, 4, 3, 5, 7, 8}
then, swap element at index 1 with 2 – cost(arr[1]*arr[2]) = (4*3) = 12,
Array becomes {2, 3, 4, 5, 7, 8}
Array is now sorted and total cost = 14+16+15+12 = 57.
Input: arr[] = {1, 8, 9, 7, 6}
Output: 36
方法：这个想法是，为了对一个循环进行排序，我们有两种选择，要么只使用循环的局部最小值，要么使用数组的局部和整体最小值。选择一个交换成本较低的元素。步骤如下：
- 计算作为当前循环中最小元素的局部最小值(例如local_minimum )和作为整个数组中最小元素的整体最小值(例如total_minimum)。
- 仅使用局部最小值来计算和存储对周期进行排序的成本(例如cost1 )。
- 同样，通过使用局部最小值和整体最小值来计算和存储对循环进行排序的成本(例如cost2 )。
- 现在，分类此循环的最低成本将是cost1和cost2中的最低成本。将此费用加到总费用中。
下面是数组arr [] = {1、8、9、7、6}的插图：
- 在上图中，可以使用局部最小元素6或整体最小元素1对循环{8,9,7,6}进行分类。通过仅使用局部最小值元素，即交换6和9，交换6和7，交换6和8。因此，总成本为6 * 9 + 6 * 7 + 6 * 8 = 144 。
- 通过同时使用整体最小值和局部最小值元素，即交换1和6，交换1和9，交换1和7，交换1和8，交换1和6。因此，总成本为1 * 6 + 1 * 9 + 1 * 7 + 1 * 8 + 1 * 6 = 36。
- 上述费用的最小值为36。
下面是上述方法的实现：
C++
// C++ program for the above approach
#include
using namespace std;
// Function returns the minimum cost
// to sort the given array
int minCost(int arr[], int n)
{
// Create array of pairs in which
// 1st element is the array element
// and 2nd element is index of first
pair sorted[n];
// Initialize the total cost
int total_cost = 0;
for (int i = 0; i < n; i++) {
sorted[i].first = arr[i];
sorted[i].second = i;
}
// Sort the array with respect to
// array value
sort(sorted, sorted + n);
// Initialize the overall minimum
// which is the 1st element
int overall_minimum = sorted[0].first;
// To keep track of visited elements
// create a visited array & initialize
// all elements as not visited
bool vis[n] = { false };
// Iterate over every element
// of the array
for (int i = 0; i < n; i++) {
// If the element is visited or
// in the sorted position, and
// check for next element
if (vis[i] && sorted[i].second == i)
continue;
// Create a vector which stores
// all elements of a cycle
vector v;
int j = i;
// It covers all the elements
// of a cycle
while (!vis[j]) {
vis[j] = true;
v.push_back(sorted[j].first);
j = sorted[j].second;
}
// If cycle is found then the
// swapping is required
if (v.size() > 0) {
// Initialize local minimum with
// 1st element of the vector as
// it contains the smallest
// element in the beginning
int local_minimum = v[0], result1 = 0,
result2 = 0;
// Stores the cost with using only
// local minimum value.
for (int k = 1; k < v.size(); k++)
result1 += (local_minimum * v[k]);
// Stores the cost of using both
// local minimum and overall minimum
for (int k = 0; k < v.size(); k++)
result2 += (overall_minimum * v[k]);
// Update the result2
result2 += (overall_minimum
* local_minimum);
// Store the minimum of the
// two result to total cost
total_cost += min(result1, result2);
}
}
// Return the minimum cost
return total_cost;
}
// Driver Code
int main()
{
// Given array arr[]
int arr[] = { 1, 8, 9, 7, 6 };
int n = (sizeof(arr) / sizeof(int));
// Function Call
cout << minCost(arr, n);
return 0;
} Java
// Java program to implement
// the above approach
import java.util.*;
class GFG{
// Function returns the minimum cost
// to sort the given array
static int minCost(int arr[], int n)
{
// Create array of pairs in which
// 1st element is the array element
// and 2nd element is index of first
int[][] sorted = new int[n][2];
// Initialize the total cost
int total_cost = 0;
for(int i = 0; i < n; i++)
{
sorted[i][0] = arr[i];
sorted[i][1] = i;
}
// Sort the array with respect to
// array value
Arrays.sort(sorted, (a, b) -> a[0] - b[0]);
// Initialize the overall minimum
// which is the 1st element
int overall_minimum = sorted[0][0];
// To keep track of visited elements
// create a visited array & initialize
// all elements as not visited
boolean[] vis = new boolean[n];
// Iterate over every element
// of the array
for(int i = 0; i < n; i++)
{
// If the element is visited or
// in the sorted position, and
// check for next element
if (vis[i] && sorted[i][1] == i)
continue;
// Create a vector which stores
// all elements of a cycle
ArrayList v = new ArrayList<>();
int j = i;
// It covers all the elements
// of a cycle
while (!vis[j])
{
vis[j] = true;
v.add(sorted[j][0]);
j = sorted[j][1];
}
// If cycle is found then the
// swapping is required
if (v.size() > 0)
{
// Initialize local minimum with
// 1st element of the vector as
// it contains the smallest
// element in the beginning
int local_minimum = v.get(0), result1 = 0,
result2 = 0;
// Stores the cost with using only
// local minimum value.
for(int k = 1; k < v.size(); k++)
result1 += (local_minimum * v.get(k));
// Stores the cost of using both
// local minimum and overall minimum
for(int k = 0; k < v.size(); k++)
result2 += (overall_minimum * v.get(k));
// Update the result2
result2 += (overall_minimum *
local_minimum);
// Store the minimum of the
// two result to total cost
total_cost += Math.min(result1, result2);
}
}
// Return the minimum cost
return total_cost;
}
// Driver code
public static void main (String[] args)
{
// Given array arr[]
int arr[] = { 1, 8, 9, 7, 6 };
int n = arr.length;
// Function call
System.out.print(minCost(arr, n));
}
}
// This code is contributed by offbeat Python3
# Python3 program for the above approach
# Function returns the minimum cost
# to sort the given array
def minCost(arr, n):
# Create array of pairs in which
# 1st element is the array element
# and 2nd element is index of first
sortedarr = []
# Initialize the total cost
total_cost = 0
for i in range(n):
sortedarr.append([arr[i], i])
# Sort the array with respect to
# array value
sortedarr.sort()
# Initialize the overall minimum
# which is the 1st element
overall_minimum = sortedarr[0][0]
# To keep track of visited elements
# create a visited array & initialize
# all elements as not visited
vis = [False] * n
# Iterate over every element
# of the array
for i in range(n):
# If the element is visited or
# in the sorted position, and
# check for next element
if vis[i] and sortedarr[i][1] == i:
continue
# Create a vector which stores
# all elements of a cycle
v = []
j = i
size = 0
# It covers all the elements
# of a cycle
while vis[j] == False:
vis[j] = True
v.append(sortedarr[j][0])
j = sortedarr[j][1]
size += 1
# If cycle is found then the
# swapping is required
if size != 0:
# Initialize local minimum with
# 1st element of the vector as
# it contains the smallest
# element in the beginning
local_minimum = v[0]
result1 = 0
result2 = 0
# Stores the cost with using only
# local minimum value.
for k in range(1, size):
result1 += local_minimum * v[k]
# Stores the cost of using both
# local minimum and overall minimum
for k in range(size):
result2 += overall_minimum * v[k]
# Update the result2
result2 += (overall_minimum *
local_minimum)
# Store the minimum of the
# two result to total cost
total_cost += min(result1, result2)
# Return the minimum cost
return total_cost
# Driver code
# Given array arr[]
A = [ 1, 8, 9, 7, 6 ]
# Function call
ans = minCost(A, len(A))
print(ans)
# This code is contributed by kumarkashyap输出：
36
时间复杂度： O(N)
辅助空间： O(N)