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Solution
def solution (N, K, cost, sell):
diff = [(j-i, i) for i,j in zip(cost, sell) if j-i > 0]
diff.sort(key = lambda x: (x[0], -x[1]), reverse=True)
// ... rest of solution available after purchaseExplanation
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To solve the problem of maximizing John's profit given constraints on capital, item availability, and storage, it's important to first understand the structure and nature of the challenge. At its core, this is a resource-constrained selection problem, similar to a dynamic investment strategy, where at each step John decides which item to buy and sell based on profitability and affordability.
You're given N items, each with a cost and a selling price. John starts with an initial capital K and zero items in the shop. One key constraint is that his shop can only hold one item at a time. This limitation means he must sell the current item before buying the next. Also, each item can only be bought and sold once, and items can be dealt with in any order. This opens up the possibility of reordering transactions for maximum profit.
Now, consider what profit means in this context. For each item i, if John buys it at cost[i] and sells it at sell[i], the profit from that transaction is sell[i] - cost[i]. However, John can only buy an item if he currently has enough money (i.e., K or whatever amount he's grown to through previous transactions). Therefore, every decision is gated not just by profitability but also by affordability.
To determine an approach, consider viewing each item as a potential transaction: it requires some capital to invest (cost[i]) and gives a return (sell[i]). The difference is the net profit. John's task is to sequence these transactions such that he always has enough money to invest in the next item, while maximizing the total profit across all possible transactions.
This suggests a greedy-style strategy might be helpful, but not a naive one. A truly greedy strategy might sort the items by descending profit and try them one by one. However, this would fail when the most profitable items are too expensive early on. Instead, John may need to start with less profitable, cheaper items that he can afford early, use those to build up his capital, and then tackle more expensive, higher-profit items later.
So how do you reason through the best order of transactions? One way is to simulate the process. At each step, John should look at all the items he hasn’t yet used and check which ones he can afford with his current capital. Among these, he should choose the one that gives the most profit. After selling it, he adds the profit to his capital and marks the item as used. He continues this process until there are no more items he can afford to buy. The total profit is the sum of all individual profits earned through these transactions.
This simulation-style reasoning aligns with a priority-based selection mechanism. At each decision point, the system evaluates the current state (capital) and decides on the best affordable next step. The challenge lies in tracking which items have already been used, managing the limited capacity (only one item in the shop at a time), and ensuring transactions are only carried out when they are financially feasible.
It's also helpful to notice that any item with cost[i] greater than sell[i] yields a negative profit, and should never be chosen. So you can filter out such items at the very beginning. From there, you can maintain a list of all remaining viable transactions.
One edge case to think about is when the initial capital K is too low to buy any item. In that case, no transactions can happen, and the profit is zero. Similarly, if all items yield zero or negative profit, John should avoid buying anything.
You can also think about the sequence of steps as a loop:
1. At each iteration, filter the list of unused items to those that John can currently afford.
2. Among those, select the item with the maximum profit.
3. Execute the transaction: subtract the cost from the capital, add the selling price, and mark the item as used.
4. Repeat until no more items can be bought.
This logic ensures that John builds his capital gradually, using early opportunities wisely to unlock bigger profits later. It mirrors a real-world investment strategy: start small, reinvest, and aim for increasingly larger gains. The main focus of your thought process should be how to simulate or prioritize transactions given changing capital, one-slot storage, and one-time-use items.
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