Given n wines in a row, with integers denoting the cost of each wine respectively. Each year you can sale the first or the last wine in a row. However, the price of wines increases over time. Let the initial profits from the wines be P1, P2, P3…Pn. In the Yth year, the profit from the ith wine will be Y*Pi. For each year, your task is to print “beg” or “end” denoting whether first or last wine should be sold. Also, calculate the maximum profit from all the wines.
Examples :
Input: Price of wines: 2 4 6 2 5
Output: beg end end beg beg
64
Explanation :
Approach: It is a standard Dynamic Programming problem. It initially looks like a greedy problem in which we should sell cheaper of wines each year but the example case (year 2) clearly proves the approach is wrong. Sometimes we need to sell an expensive wine earlier to save relatively costly wines for later years (Here, if 4 were sold in the 2nd year, in the 4th year we had to sell 2 which would be a waste of a heavy coefficient).
The second problem is to “store the strategy” to obtain the calculated price which has a fairly standard method that can be used in other problems as well. The idea is to store the optimal action for each state and use that to navigate through the optimal states starting from the initial state.
// Program to calculate maximum price of wines #include <bits/stdc++.h> using namespace std;
#define N 1000 int dp[N][N];
// This array stores the "optimal action" // for each state i, j int sell[N][N];
// Function to maximize profit int maxProfitUtil( int price[], int begin,
int end, int n) {
if (dp[begin][end] != -1)
return dp[begin][end];
int year = n - (end - begin);
if (begin == end)
return year * price[begin];
// x = maximum profit on selling the
// wine from the front this year
int x = price[begin] * year +
maxProfitUtil(price, begin + 1, end, n);
// y = maximum profit on selling the
// wine from the end this year
int y = price[end] * year +
maxProfitUtil(price, begin, end - 1, n);
int ans = max(x, y);
dp[begin][end] = ans;
if (x >= y)
sell[begin][end] = 0;
else
sell[begin][end] = 1;
return ans;
} // Util Function to calculate maxProfit int maxProfit( int price[], int n) {
// resetting the dp table
for ( int i = 0; i < N; i++)
for ( int j = 0; j < N; j++)
dp[i][j] = -1;
int ans = maxProfitUtil(price, 0, n - 1, n);
int i = 0, j = n - 1;
while (i <= j) {
// sell[i][j]=0 implies selling the
// wine from beginning will be more
// profitable in the long run
if (sell[i][j] == 0) {
cout << "beg " ;
i++;
} else {
cout << "end " ;
j--;
}
}
cout << endl;
return ans;
} // Driver code int main() {
// Price array
int price[] = { 2, 4, 6, 2, 5 };
int n = sizeof (price) / sizeof (price[0]);
int ans = maxProfit(price, n);
cout << ans << endl;
return 0;
} |
// Program to calculate maximum price of wines import java.io.*;
class GFG {
static int N = 1000 ;
static int [][]dp = new int [N][N];
// This array stores the "optimal action"
// for each state i, j
static int [][]sell = new int [N][N];
// Function to maximize profit
static int maxProfitUtil( int price[],
int begin, int end, int n)
{
if (dp[begin][end] != - 1 )
return dp[begin][end];
int year = n - (end - begin);
if (begin == end)
return year * price[begin];
// x = maximum profit on selling the
// wine from the front this year
int x = price[begin] * year +
maxProfitUtil(price, begin + 1 ,
end, n);
// y = maximum profit on selling the
// wine from the end this year
int y = price[end] * year +
maxProfitUtil(price, begin,
end - 1 , n);
int ans = Math.max(x, y);
dp[begin][end] = ans;
if (x >= y)
sell[begin][end] = 0 ;
else
sell[begin][end] = 1 ;
return ans;
}
// Util Function to calculate maxProfit
static int maxProfit( int price[], int n)
{
// resetting the dp table
for ( int i = 0 ; i < N; i++)
for ( int j = 0 ; j < N; j++)
dp[i][j] = - 1 ;
int ans = maxProfitUtil(price, 0 ,
n - 1 , n);
int i = 0 , j = n - 1 ;
while (i <= j) {
// sell[i][j]=0 implies selling
// the wine from beginning will
// be more profitable in the
// long run
if (sell[i][j] == 0 ){
System.out.print( "beg " );
i++;
}
else
{
System.out.print( "end " );
j--;
}
}
System.out.println();
return ans;
}
// Driver code
public static void main (String[] args)
{
// Price array
int price[] = { 2 , 4 , 6 , 2 , 5 };
int n = price.length;
int ans = maxProfit(price, n);
System.out.println( ans );
}
} // This code is contributed by anuj_67. |
# Python3 Program to calculate # maximum price of wines N = 1000
dp = [ [ - 1 for col in range (N)]
for row in range (N)]
# This array stores the "optimal action" # for each state i, j sell = [ [ 0 for col in range (N)]
for row in range (N)]
# Function to maximize profit def maxProfitUtil(price, begin, end, n):
if (dp[begin][end] ! = - 1 ):
return dp[begin][end]
year = n - (end - begin)
if (begin = = end):
return year * price[begin]
# x = maximum profit on selling the
# wine from the front this year
x = price[begin] * year + \
maxProfitUtil(price, begin + 1 , end, n)
# y = maximum profit on selling the
# wine from the end this year
y = price[end] * year + \
maxProfitUtil(price, begin, end - 1 , n)
ans = max (x, y)
dp[begin][end] = ans
if (x > = y):
sell[begin][end] = 0
else :
sell[begin][end] = 1
return ans
# Util Function to calculate maxProfit def maxProfit(price, n):
ans = maxProfitUtil(price, 0 , n - 1 , n)
i = 0
j = n - 1
while (i < = j):
# sell[i][j]=0 implies selling the
# wine from beginning will be more
# profitable in the long run
if (sell[i][j] = = 0 ):
print ( "beg" , end = " " )
i = i + 1
else :
print ( "end" , end = " " )
j = j - 1
print ( " " )
return ans
# Driver code # Price array price = [ 2 , 4 , 6 , 2 , 5 ]
size = 5
ans = maxProfit(price, size);
print (ans)
# This code is contributed by ashutosh450 |
// C# Program to calculate maximum // price of wines using System;
class GFG {
static int N = 1000;
static int [,] dp = new int [N, N];
// This array stores the "optimal action"
// for each state i, j
static int [,] sell = new int [N, N];
// Function to maximize profit
static int maxProfitUtil( int [] price, int begin, int end, int n)
{
// Check if the result for this state is already calculated
if (dp[begin, end] != -1)
return dp[begin, end];
// Calculate the current year
int year = n - (end - begin);
if (begin == end)
return year * price[begin];
// Calculate maximum profit by selling from the front
int x = price[begin] * year +
maxProfitUtil(price, begin + 1, end, n);
// Calculate maximum profit by selling from the end
int y = price[end] * year +
maxProfitUtil(price, begin, end - 1, n);
// Choose the maximum of the two options
int ans = Math.Max(x, y);
dp[begin, end] = ans;
// Store the optimal action: 0 for selling from the front, 1 for selling from the end
if (x >= y)
sell[begin, end] = 0;
else
sell[begin, end] = 1;
return ans;
}
// Util Function to calculate maxProfit
static int maxProfit( int [] price, int n)
{
int i, j;
// Resetting the dp table
for (i = 0; i < N; i++)
for (j = 0; j < N; j++)
dp[i, j] = -1;
// Calculate the maximum profit
int ans = maxProfitUtil(price, 0, n - 1, n);
i = 0; j = n - 1;
// Print the optimal actions (selling from the beginning or end)
while (i <= j) {
// sell[i][j]=0 implies selling
// the wine from beginning will
// be more profitable in the
// long run
if (sell[i, j] == 0) {
Console.Write( "beg " );
i++;
}
else {
Console.Write( "end " );
j--;
}
}
Console.WriteLine();
return ans;
}
// Driver Code
public static void Main()
{
// Price array
int [] price = { 2, 4, 6, 2, 5 };
int n = price.Length;
// Call the maxProfit function and print the result
int ans = maxProfit(price, n);
Console.WriteLine(ans);
}
} // This code is contributed by anuj_67. |
<script> // Program to calculate maximum price of wines
let N = 1000;
let dp = new Array(N);
// This array stores the "optimal action"
// for each state i, j
let sell = new Array(N);
for (let i = 0; i < N; i++)
{
dp[i] = new Array(N);
sell[i] = new Array(N);
for (let j = 0; j < N; j++)
{
dp[i][j] = 0;
sell[i][j] = 0;
}
}
// Function to maximize profit
function maxProfitUtil(price, begin, end, n)
{
if (dp[begin][end] != -1)
return dp[begin][end];
let year = n - (end - begin);
if (begin == end)
return year * price[begin];
// x = maximum profit on selling the
// wine from the front this year
let x = price[begin] * year +
maxProfitUtil(price, begin + 1,
end, n);
// y = maximum profit on selling the
// wine from the end this year
let y = price[end] * year +
maxProfitUtil(price, begin,
end - 1, n);
let ans = Math.max(x, y);
dp[begin][end] = ans;
if (x >= y)
sell[begin][end] = 0;
else
sell[begin][end] = 1;
return ans;
}
// Util Function to calculate maxProfit
function maxProfit(price, n)
{
// resetting the dp table
for (let i = 0; i < N; i++)
for (let j = 0; j < N; j++)
dp[i][j] = -1;
let ans = maxProfitUtil(price, 0, n - 1, n);
let i = 0, j = n - 1;
while (i <= j) {
// sell[i][j]=0 implies selling
// the wine from beginning will
// be more profitable in the
// long run
if (sell[i][j] == 0){
document.write( "beg " );
i++;
}
else
{
document.write( "end " );
j--;
}
}
document.write( "</br>" );
return ans;
}
// Price array
let price = [ 2, 4, 6, 2, 5 ];
let n = price.length;
let ans = maxProfit(price, n);
document.write( ans );
// This code is contributed by divyeshrabadiya07.
</script> |
beg end end beg beg 64
Time Complexity: O(n2)
Auxiliary Space: O(n2)
Efficient approach : Using DP Tabulation method ( Iterative approach )
The approach to solve this problem is same but DP tabulation(bottom-up) method is better then Dp + memoization(top-down) because memoization method needs extra stack space of recursion calls.
Steps to solve this problem :
- Create a dp to store the solution of the subproblems.
- Create another table sell to storing information when to sell.
- Initialize the dp and sell with base cases
- Now Iterate over subproblems to get the value of current problem form previous computation of subproblems stored in dp.
- Return the final solution stored in dp[0][n-1].
Implementation :
#include <bits/stdc++.h> using namespace std;
#define N 1000 int dp[N][N]; // Dynamic programming table for storing maximum profits
int sell[N][N]; // Table for storing information about when to sell
int maxProfit( int price[], int n) {
// Initialize the diagonal elements of the tables
for ( int i = 0; i < n; i++) {
dp[i][i] = n * price[i]; // If only one year is left, sell at the current price
sell[i][i] = 0; // Mark as 'beg' (beginning) because only one year is left
}
// Calculate the maximum profit for different lengths of the subarray
for ( int len = 1; len < n; len++) {
for ( int i = 0; i < n - len; i++) {
int j = i + len;
int year = n - (j - i); // Calculate the current year
int x = price[i] * year + dp[i + 1][j]; // Sell at the current year price and calculate the profit for remaining subarray
int y = price[j] * year + dp[i][j - 1]; // Sell at the last year price and calculate the profit for remaining subarray
// Choose the option that gives maximum profit
if (x >= y) {
dp[i][j] = x; // Maximum profit if we sell at the current price
sell[i][j] = 0; // Mark as 'beg' (beginning) because we sell at the current price
} else {
dp[i][j] = y; // Maximum profit if we sell at the last year price
sell[i][j] = 1; // Mark as 'end' because we sell at the last year price
}
}
}
// Determine the sequence of selling 'beg' and 'end'
int i = 0, j = n - 1;
while (i <= j) {
if (sell[i][j] == 0) {
cout << "beg " ;
i++;
} else {
cout << "end " ;
j--;
}
}
cout << endl;
return dp[0][n - 1]; // Return the maximum profit for the entire array
} int main() {
int price[] = {2, 4, 6, 2, 5};
int n = sizeof (price) / sizeof (price[0]);
int ans = maxProfit(price, n);
cout << ans << endl;
return 0;
} |
class Main {
static final int N = 1000 ;
static int [][] dp = new int [N][N]; // Dynamic programming table for storing maximum profits
static int [][] sell = new int [N][N]; // Table for storing information about when to sell
static int maxProfit( int price[], int n) {
// Initialize the diagonal elements of the tables
for ( int i = 0 ; i < n; i++) {
dp[i][i] = n * price[i]; // If only one year is left, sell at the current price
sell[i][i] = 0 ; // Mark as 'beg' (beginning) because only one year is left
}
// Calculate the maximum profit for different lengths of the subarray
for ( int len = 1 ; len < n; len++) {
for ( int i = 0 ; i < n - len; i++) {
int j = i + len;
int year = n - (j - i); // Calculate the current year
int x = price[i] * year + dp[i + 1 ][j]; // Sell at the current year price and calculate the profit for remaining subarray
int y = price[j] * year + dp[i][j - 1 ]; // Sell at the last year price and calculate the profit for remaining subarray
// Choose the option that gives maximum profit
if (x >= y) {
dp[i][j] = x; // Maximum profit if we sell at the current price
sell[i][j] = 0 ; // Mark as 'beg' (beginning) because we sell at the current price
} else {
dp[i][j] = y; // Maximum profit if we sell at the last year price
sell[i][j] = 1 ; // Mark as 'end' because we sell at the last year price
}
}
}
// Determine the sequence of selling 'beg' and 'end'
int i = 0 , j = n - 1 ;
while (i <= j) {
if (sell[i][j] == 0 ) {
System.out.print( "beg " );
i++;
} else {
System.out.print( "end " );
j--;
}
}
System.out.println();
return dp[ 0 ][n - 1 ]; // Return the maximum profit for the entire array
}
public static void main(String args[]) {
int price[] = { 2 , 4 , 6 , 2 , 5 };
int n = price.length;
int ans = maxProfit(price, n);
System.out.println(ans);
}
} |
def maxProfit(price):
n = len (price)
dp = [[ 0 ] * n for _ in range (n)]
sell = [[ 0 ] * n for _ in range (n)]
# Initialize the diagonal elements of the tables
for i in range (n):
dp[i][i] = n * price[i] # If only one year is left, sell at the current price
sell[i][i] = 0 # Mark as 'beg' (beginning) because only one year is left
# Calculate the maximum profit for different lengths of the subarray
for length in range ( 1 , n):
for i in range (n - length):
j = i + length
year = n - (j - i) # Calculate the current year
x = price[i] * year + dp[i + 1 ][j] # Sell at the current year price and calculate the profit for the remaining subarray
y = price[j] * year + dp[i][j - 1 ] # Sell at the last year price and calculate the profit for the remaining subarray
# Choose the option that gives maximum profit
if x > = y:
dp[i][j] = x # Maximum profit if we sell at the current price
sell[i][j] = 0 # Mark as 'beg' (beginning) because we sell at the current price
else :
dp[i][j] = y # Maximum profit if we sell at the last year price
sell[i][j] = 1 # Mark as 'end' because we sell at the last year price
# Determine the sequence of selling 'beg' and 'end'
i, j = 0 , n - 1
while i < = j:
if sell[i][j] = = 0 :
print ( "beg" , end = " " )
i + = 1
else :
print ( "end" , end = " " )
j - = 1
print ()
return dp[ 0 ][n - 1 ] # Return the maximum profit for the entire array
price = [ 2 , 4 , 6 , 2 , 5 ]
ans = maxProfit(price)
print (ans)
|
using System;
class MaxProfit
{ // Function to calculate maximum profit and output the buying/selling actions
public static int MaxProfitCalc( int [] price)
{
int n = price.Length;
int [][] dp = new int [n][]; // Array to store maximum profit values
int [][] sell = new int [n][]; // Array to store buying/selling actions
// Initializing arrays
for ( int i = 0; i < n; i++)
{
dp[i] = new int [n];
sell[i] = new int [n];
}
// Initialize base cases for one-day transactions
for ( int i = 0; i < n; i++)
{
dp[i][i] = n * price[i];
sell[i][i] = 0; // No selling on the same day
}
// Dynamic Programming to fill in the dp and sell arrays
for ( int length = 1; length < n; length++)
{
for ( int i = 0; i < n - length; i++)
{
int j = i + length;
int year = n - (j - i);
int x = price[i] * year + dp[i + 1][j]; // If buying at the beginning is profitable
int y = price[j] * year + dp[i][j - 1]; // If buying at the end is profitable
// Choose the more profitable option and update the sell array accordingly
if (x >= y)
{
dp[i][j] = x;
sell[i][j] = 0; // 0 represents buying at the beginning
}
else
{
dp[i][j] = y;
sell[i][j] = 1; // 1 represents buying at the end
}
}
}
// Output the buying/selling actions
int iVal = 0;
int jVal = n - 1;
while (iVal <= jVal)
{
if (sell[iVal][jVal] == 0)
{
Console.Write( "beg " ); // Output "beg" for buying at the beginning
iVal++;
}
else
{
Console.Write( "end " ); // Output "end" for buying at the end
jVal--;
}
}
Console.WriteLine();
return dp[0][n - 1]; // Return the maximum profit
}
// Main function to demonstrate the MaxProfitCalc function
public static void Main( string [] args)
{
int [] price = { 2, 4, 6, 2, 5 };
int ans = MaxProfitCalc(price);
Console.WriteLine(ans);
}
} |
function maxProfit(price) {
const n = price.length;
const dp = Array.from({ length: n }, () => Array(n).fill(0));
const sell = Array.from({ length: n }, () => Array(n).fill(0));
// Initialize the diagonal elements of the tables
for (let i = 0; i < n; i++) {
dp[i][i] = n * price[i]; // If only one year is left, sell at the current price
sell[i][i] = 0; // Mark as 'beg' (beginning) because only one year is left
}
// Calculate the maximum profit for different lengths of the subarray
for (let len = 1; len < n; len++) {
for (let i = 0; i < n - len; i++) {
const j = i + len;
const year = n - (j - i); // Calculate the current year
const x = price[i] * year + dp[i + 1][j]; // Sell at the current year price and calculate the profit for remaining subarray
const y = price[j] * year + dp[i][j - 1]; // Sell at the last year price and calculate the profit for remaining subarray
// Choose the option that gives maximum profit
if (x >= y) {
dp[i][j] = x; // Maximum profit if we sell at the current price
sell[i][j] = 0; // Mark as 'beg' (beginning) because we sell at the current price
} else {
dp[i][j] = y; // Maximum profit if we sell at the last year price
sell[i][j] = 1; // Mark as 'end' because we sell at the last year price
}
}
}
// Determine the sequence of selling 'beg' and 'end'
let i = 0,
j = n - 1;
const sequence = [];
while (i <= j) {
if (sell[i][j] === 0) {
sequence.push( "beg" );
i++;
} else {
sequence.push( "end" );
j--;
}
}
console.log(sequence.join( " " ));
return dp[0][n - 1]; // Return the maximum profit for the entire array
} const price = [2, 4, 6, 2, 5]; const ans = maxProfit(price); console.log(ans); |
beg end end beg beg 64
Time Complexity: O(n^2)
Auxiliary Space: O(n^2)