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Maximum Profit in Stock Buy and sell with at most K Transaction

Here, we are going to learn about Maximum Profit in Stock Buy and sell with at most K Transaction using dynamic programming.
Submitted by Radib Kar, on January 05, 2020

This is a very popular interview problem to find maximum profit in stock buying and selling with at most K transactions. This problem has been featured in the interview rounds of Amazon.

Problem statement:

In stock market, a person buys a stock and sells it on some future date. Given the stock prices of N days in form of an array Amount and a positive integer K, find out the maximum profit a person can make in at most K transactions. A transaction is buying the stock on some day and selling the stock at some other future day and new transaction can start only when the previous transaction has been completed.

    Input:
    K=3
    N=7

    Stock prices on N days:
    10 25 38 40 45 5 58

    Output:
    88

Example:

    Number of maximum transactions: 3
    Total number of days, N: 7

    To achieve maximum profit:
    Stock bought at day1    (-10)
    Stock sold at day5      (+45)
    Stock bought at day6    (-5)
    Stock sold at day7      (+58)

    Total profit = 88
    Total transactions made = 2

Explanation:

Let there are N number of days for transactions, say x₁, x₂, ..., x_n

Now for any day x_i

1. Don't do any transaction on the day x_i
2. Transact on day x_i, i.e., buy stock on some day x_j and sell on day x_i where j<i and i,j Є N

Total number of transactions can be made at most = K

Now we can formulate the maximum profit case using above two condition.

Let,

    f(t,i)=maximum profit upto ith day and t transactions

Considering the above two facts about x_i

    f(t,i)=f(t,i-1)  if there is no transaction made on day xi ...(1)
        Max(f(t-1,j)+amount[i]-amount[j])       
        where j<i and i,j Є N if there is transaction  on day xi ...(2)
    Obviously, to maximize the profit we would take the maximum of (1)  and (2)

    Number of maximum transactions: 3
    Total number of days, N: 7
    Stock prices on the days are:
    10 25 38 40 45 5 58

Below is a part of recursion tree which can show that how many overlapping sub problems there will be.

Figure 1: Partial recursion tree to show overlapping sub-problems

So we need dynamic programming...

Problem Solution

Recursive Algorithm:

    Function(t, i): //f(t, i)
        If i=0
            f(t, i)=0
        If t=0
            f(t, i)=0

        f(t, i)= f(t, i-1); //no transaction on day xi
        For j =0: i
            Find max(f(t-1,j) + amount(i)- amount(j)
        End For
        If the maximum found > f(t, i)
            update f(t, i)
        End IF
    Return f(t, i)

Conversion to DP

    For tabulation we need a 2D array, DP[k+1][n] to store f(t,i)

    Base case,
    for i 0 to k,   DP[i][0]=0 //no profit on 0th day
    for i 0 to n-1,   DP[0][j]=0 //no profit on 0 transaction


    To fill the higher values,
    for t=1 to k
        for i =1 to n-1
            DP[t][i]=DP[t][i-1]
            for j= 0 to i-1 //buying on jth day and selling on ith day   
                DP[t][i]=max(DP[t][i],DP[t-1][j]+ amount[i] –amount[j])
            End for
        End for
    End for
    Result would be f(k,n) that is value of DP[k][n-1]

Initial DP table

    DP[4][7] //K=3, N=7

Try yourself to compute the DP table manually following the above algorithm and find out the result. Take some small example if necessary.

C++ implementation:

#include <bits/stdc++.h>
using namespace std;

int Max_profit(vector<int> amount, int n, int k)
{
    int DP[k + 1][n];
    memset(DP, 0, sizeof(DP));

    //on 0th day
    for (int i = 0; i <= k; i++)
        DP[i][0] = 0;
    //on 0 transaction made
    for (int i = 0; i <= n; i++)
        DP[0][i] = 0;

    for (int t = 1; t <= k; t++) {
        for (int i = 1; i < n; i++) {
            DP[t][i] = DP[t][i - 1];
            int maxV = INT_MIN;
            //buying on jth day and selling on ith day
            for (int j = 0; j < i; j++) {

                if (maxV < DP[t - 1][j] + amount[i] - amount[j])
                    maxV = DP[t - 1][j] + amount[i] - amount[j];
            }
            if (DP[t][i] < maxV)
                DP[t][i] = maxV;
        }
    }
    return DP[k][n - 1];
}

int main()
{
    int n, item, k;

    cout << "Enter maximum no of transactions:\n";
    cin >> k;
    cout << "Enter number of days\n";
    cin >> n;
    vector<int> amount;
    cout << "Enter stock values on corresponding days\n";
    for (int j = 0; j < n; j++) {
        scanf("%d", &item);
        amount.push_back(item);
    }

    cout << "Maximum profit that can be achieved: " << Max_profit(amount, n, k) << endl;

    return 0;
}

Output

Enter maximum no of transactions:
3
Enter number of days
7
Enter stock values on corresponding days
10 25 38 40 45 5 58
Maximum profit that can be achieved: 88