本文概述
- C ++
- Java
- Python3
- C#
例子:
输入:X = 21, arr [] = {2, 3, 4, 5}简单的方法:最简单的方法是尝试给定面额的所有可能的组合,在每个组合中,硬币的总和等于x。从这些组合中,选择一个有最少硬币数量的并打印它。如果任意组合的和不等于X,则输出-1。
输出:2 4 5 5 5
说明:一种可能的解决方案是{2, 4, 5, 5, 5}, 其中2 + 4 + 5 + 5 + 5 =21。另一个可能的解决方案是{3, 3, 5, 5, 5}。
输入:X = 1, arr [] = {2, 4, 6, 9}
输出:-1
说明:所有硬币都大于1。因此, 不存在解。
时间复杂度:O(X^N)
辅助空间:O(N)
有效方法:上述方法可以通过动态规划进行优化,找到最小硬币数量。在找到最小硬币数量的同时,回溯可以用来追踪使其总和等于x所需的硬币,按照以下步骤解决问题:
- 初始化辅助数组dp [], 在哪里dp [i]将存储求和所需的最小硬币数量等于一世.
- 找到使它们的总和等于的最小硬币数量X使用中讨论的方法这个文章。
- 找到最小数量的硬币后, 使用回溯技术追踪使用的硬币, 使总和等于X.
- 在回溯中遍历数组并选择一个比当前总和还小的硬币dp [current_sum]等于dp [current_sum – selected_coin] +1。将所选硬币存储在阵列中。
- 完成上述步骤后, 通过传递当前金额as(当前金额–选择的硬币价值).
- 找到解决方案后, 打印选定硬币的阵列。
C ++
//C++ program for the above approach
#include <
bits/stdc++.h>
using namespace std;
#define MAX 100000
//dp array to memoize the results
int dp[MAX + 1];
//List to store the result
list<
int>
denomination;
//Function to find the minimum number of
//coins to make the sum equals to X
int countMinCoins( int n, int C[], int m)
{
//Base case
if (n == 0) {
dp[0] = 0;
return 0;
}
//If previously computed
//subproblem occurred
if (dp[n] != -1)
return dp[n];
//Initialize result
int ret = INT_MAX;
//Try every coin that has smaller
//value than n
for ( int i = 0;
i <
m;
i++) {
if (C[i] <
= n) {
int x
= countMinCoins(n - C[i], C, m);
//Check for INT_MAX to avoid
//overflow and see if result
//can be minimized
if (x != INT_MAX)
ret = min(ret, 1 + x);
}
}
//Memoizing value of current state
dp[n] = ret;
return ret;
}
//Function to find the possible
//combination of coins to make
//the sum equal to X
void findSolution( int n, int C[], int m)
{
//Base Case
if (n == 0) {
//Print Solutions
for ( auto it : denomination) {
cout <
<
it <
<
' ' ;
}
return ;
}
for ( int i = 0;
i <
m;
i++) {
//Try every coin that has
//value smaller than n
if (n - C[i]>
= 0
and dp[n - C[i]] + 1
== dp[n]) {
//Add current denominations
denomination.push_back(C[i]);
//Backtrack
findSolution(n - C[i], C, m);
break ;
}
}
}
//Function to find the minimum
//combinations of coins for value X
void countMinCoinsUtil( int X, int C[], int N)
{
//Initialize dp with -1
memset (dp, -1, sizeof (dp));
//Min coins
int isPossible
= countMinCoins(X, C, N);
//If no solution exists
if (isPossible == INT_MAX) {
cout <
<
"-1" ;
}
//Backtrack to find the solution
else {
findSolution(X, C, N);
}
}
//Driver code
int main()
{
int X = 21;
//Set of possible denominations
int arr[] = { 2, 3, 4, 5 };
int N = sizeof (arr) /sizeof (arr[0]);
//Function Call
countMinCoinsUtil(X, arr, N);
return 0;
}
Java
//Java program for
//the above approach
import java.util.*;
class GFG{static final int MAX = 100000 ;
//dp array to memoize the results
static int []dp = new int [MAX + 1 ];
//List to store the result
static List<
Integer>
denomination =
new LinkedList<
Integer>
();
//Function to find the minimum
//number of coins to make the
//sum equals to X
static int countMinCoins( int n, int C[], int m)
{
//Base case
if (n == 0 )
{
dp[ 0 ] = 0 ;
return 0 ;
}
//If previously computed
//subproblem occurred
if (dp[n] != - 1 )
return dp[n];
//Initialize result
int ret = Integer.MAX_VALUE;
//Try every coin that has smaller
//value than n
for ( int i = 0 ;
i <
m;
i++)
{
if (C[i] <
= n)
{
int x = countMinCoins(n - C[i], C, m);
//Check for Integer.MAX_VALUE to avoid
//overflow and see if result
//can be minimized
if (x != Integer.MAX_VALUE)
ret = Math.min(ret, 1 + x);
}
}
//Memoizing value of current state
dp[n] = ret;
return ret;
}
//Function to find the possible
//combination of coins to make
//the sum equal to X
static void findSolution( int n, int C[], int m)
{
//Base Case
if (n == 0 )
{
//Print Solutions
for ( int it : denomination)
{
System.out.print(it + " " );
}
return ;
}
for ( int i = 0 ;
i <
m;
i++)
{
//Try every coin that has
//value smaller than n
if (n - C[i]>
= 0 &
&
dp[n - C[i]] + 1 ==
dp[n])
{
//Add current denominations
denomination.add(C[i]);
//Backtrack
findSolution(n - C[i], C, m);
break ;
}
}
}
//Function to find the minimum
//combinations of coins for value X
static void countMinCoinsUtil( int X, int C[], int N)
{
//Initialize dp with -1
for ( int i = 0 ;
i <
dp.length;
i++)
dp[i] = - 1 ;
//Min coins
int isPossible = countMinCoins(X, C, N);
//If no solution exists
if (isPossible == Integer.MAX_VALUE)
{
System.out.print( "-1" );
}
//Backtrack to find the solution
else
{
findSolution(X, C, N);
}
}
//Driver code
public static void main(String[] args)
{
int X = 21 ;
//Set of possible denominations
int arr[] = { 2 , 3 , 4 , 5 };
int N = arr.length;
//Function Call
countMinCoinsUtil(X, arr, N);
}
}
//This code is contributed by Rajput-Ji
Python3
# Python3 program for the above approach
import sys
MAX = 100000
# dp array to memoize the results
dp = [ - 1 ] * ( MAX + 1 )
# List to store the result
denomination = []
# Function to find the minimum number of
# coins to make the sum equals to X
def countMinCoins(n, C, m):# Base case
if (n = = 0 ):
dp[ 0 ] = 0
return 0
# If previously computed
# subproblem occurred
if (dp[n] ! = - 1 ):
return dp[n]
# Initialize result
ret = sys.maxsize
# Try every coin that has smaller
# value than n
for i in range (m):
if (C[i] <
= n):
x = countMinCoins(n - C[i], C, m)
# Check for INT_MAX to avoid
# overflow and see if result
#. an be minimized
if (x ! = sys.maxsize):
ret = min (ret, 1 + x)
# Memoizing value of current state
dp[n] = ret
return ret
# Function to find the possible
# combination of coins to make
# the sum equal to X
def findSolution(n, C, m):# Base Case
if (n = = 0 ):
# Print Solutions
for it in denomination:
print (it, end = " " )
return
for i in range (m):
# Try every coin that has
# value smaller than n
if (n - C[i]>
= 0 and
dp[n - C[i]] + 1 = = dp[n]):
# Add current denominations
denomination.append(C[i])
# Backtrack
findSolution(n - C[i], C, m)
break
# Function to find the minimum
# combinations of coins for value X
def countMinCoinsUtil(X, C, N):
# Initialize dp with -1
# memset(dp, -1, sizeof(dp))
# Min coins
isPossible = countMinCoins(X, C, N)
# If no solution exists
if (isPossible = = sys.maxsize):
print ( "-1" )
# Backtrack to find the solution
else :
findSolution(X, C, N)
# Driver code
if __name__ = = '__main__' :X = 21
# Set of possible denominations
arr = [ 2 , 3 , 4 , 5 ]
N = len (arr)
# Function call
countMinCoinsUtil(X, arr, N)
# This code is contributed by mohit kumar 29
C#
//C# program for
//the above approach
using System;
using System.Collections.Generic;
class GFG{static readonly int MAX = 100000;
//dp array to memoize the results
static int []dp = new int [MAX + 1];
//List to store the result
static List<
int>
denomination =
new List<
int>
();
//Function to find the minimum
//number of coins to make the
//sum equals to X
static int countMinCoins( int n, int []C, int m)
{
//Base case
if (n == 0)
{
dp[0] = 0;
return 0;
}
//If previously computed
//subproblem occurred
if (dp[n] != -1)
return dp[n];
//Initialize result
int ret = int .MaxValue;
//Try every coin that has smaller
//value than n
for ( int i = 0;
i <
m;
i++)
{
if (C[i] <
= n)
{
int x = countMinCoins(n - C[i], C, m);
//Check for int.MaxValue to avoid
//overflow and see if result
//can be minimized
if (x != int .MaxValue)
ret = Math.Min(ret, 1 + x);
}
}
//Memoizing value of current state
dp[n] = ret;
return ret;
}
//Function to find the possible
//combination of coins to make
//the sum equal to X
static void findSolution( int n, int []C, int m)
{
//Base Case
if (n == 0)
{
//Print Solutions
foreach ( int it in denomination)
{
Console.Write(it + " " );
}
return ;
}
for ( int i = 0;
i <
m;
i++)
{
//Try every coin that has
//value smaller than n
if (n - C[i]>
= 0 &
&
dp[n - C[i]] + 1 ==
dp[n])
{
//Add current denominations
denomination.Add(C[i]);
//Backtrack
findSolution(n - C[i], C, m);
break ;
}
}
}
//Function to find the minimum
//combinations of coins for value X
static void countMinCoinsUtil( int X, int []C, int N)
{
//Initialize dp with -1
for ( int i = 0;
i <
dp.Length;
i++)
dp[i] = -1;
//Min coins
int isPossible = countMinCoins(X, C, N);
//If no solution exists
if (isPossible == int .MaxValue)
{
Console.Write( "-1" );
}
//Backtrack to find the solution
else
{
findSolution(X, C, N);
}
}
//Driver code
public static void Main(String[] args)
{
int X = 21;
//Set of possible denominations
int []arr = {2, 3, 4, 5};
int N = arr.Length;
//Function Call
countMinCoinsUtil(X, arr, N);
}
}
//This code is contributed by shikhasingrajput
输出如下:
2 4 5 5 5
时间复杂度:O(N * X), 其中N是给定数组的长度, X是给定整数。
【生成总和等于给定值的最小硬币组合】辅助空间:O(N)
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