AMAZON Coding Question – Solved

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To solve the problem of making all product sizes distinct at minimal total cost, you need to carefully analyze the interplay between the size values, their duplicates, and the cost of increasing those sizes. The key challenge is that increasing the size of a product has a linear cost dependent on how much you increase it, and you want to avoid paying unnecessary increments while ensuring no two sizes are the same.

1. **Understanding the Problem:**
You are given two arrays of length n:
- `size` where each element represents the initial size of a product.
- `cost` where each element represents the cost per unit increase to make that product’s size bigger.

Your goal is to adjust sizes by incrementing some sizes so that all values in the `size` array become distinct. The total cost you pay is the sum over all products of `(cost[i] * (new_size[i] - size[i]))`. The task is to minimize this total cost.

2. **Identifying the Core Challenge:**
When sizes are unique, no increments are needed, and cost is zero. But when sizes repeat, you must increase the duplicates to some unique sizes that are not already taken, potentially incurring cost. Since increments can be any positive integer number, you have flexibility on how far you increase duplicates.

The key observation is that the order of increments and the target new sizes can greatly affect total cost. Simply increasing duplicates in a greedy way (like always increasing by 1) may not be optimal because it might cause later products to have to increase by more, resulting in higher total cost.

3. **Insights into the Problem:**
- You want to find a set of distinct sizes, each at least equal to the original size for the corresponding product, because sizes can only increase.
- Since cost per increment varies by product, it might be cheaper to increase a product with lower cost more than one with higher cost.
- The increments have to be chosen carefully so that sizes do not clash, yet total increment cost remains minimal.

4. **Thinking About Sorting and Ordering:**
To systematically assign new sizes and avoid confusion, consider sorting the products based on their initial sizes. Sorting helps because when you look at sizes in ascending order, you can try to assign sizes that are at least the current product’s size and greater than any size previously assigned.
This approach ensures you do not assign a smaller number after a bigger number, maintaining a logical incremental progression of sizes.

5. **Choosing the Target Sizes:**
After sorting, the problem reduces to selecting a distinct number for each product starting from its original size or more, such that these numbers form a strictly increasing sequence and the cost is minimized.

You can think of the problem as a variant of "assigning increasing numbers" under constraints:
- The assigned size must be at least the original size.
- The cost for increasing from original size to assigned size varies per product.

6. **Exploring Dynamic Programming or Greedy Strategies:**
One way to think about this is to consider a sequence of candidate sizes you can assign (for example, all integers starting from the smallest size in the input up to some maximum possible size).

For each product, you might consider the minimal cost to assign it a certain size from these candidates, given the assignments of previous products.

This naturally suggests a dynamic programming approach where:
- The state could represent the minimal cost to assign sizes to the first i products with the last assigned size being some value.
- Transition involves choosing a valid next size (greater than the previous assigned size and at least the original size of the current product) and adding the cost of increasing the current product’s size to that chosen value.

This can be computationally expensive if done naïvely, so optimizations or pruning would be necessary.

7. **Key Considerations for Optimization:**
- Since increasing sizes arbitrarily large can be costly and unnecessary, you can limit the range of candidate sizes considered, for example by bounding them by the maximum initial size plus n.
- The problem’s cost structure and ordering might allow for a greedy-like approach with careful consideration: For example, assign the smallest possible size to each product that is greater than the previous assigned size and at least its own original size. This greedy approach works in many cases, but you must account for the varying costs.

8. **Weighing Products by Cost:**
Because products have different costs per increment, you might want to prioritize incrementing sizes for products with lower costs, while products with higher costs should preferably be assigned sizes closer to their original sizes.

Hence, it might be beneficial to:
- Sort products by their original sizes to impose order.
- Within groups of duplicates, prioritize assigning increments to products with lower cost to minimize overall expense.

9. **Summary of Thought Process:**
- Sort products by original size.
- Start assigning sizes from smallest upwards, making sure the size assigned to the i-th product is at least as big as its original size and strictly greater than the size assigned to the (i-1)-th product.
- When there are duplicates or multiple products with the same size, assign increments to those with lower cost first.
- Calculate increment costs accordingly.
- Keep track of the total cost and aim for minimal sum.

10. **Verifying and Testing:**
Once you decide on an approach, test it on examples with duplicates and varying costs. Adjust your method if the cost isn’t minimal.

11. **Final Thoughts:**
The key to this problem lies in understanding the relationship between size increments and their costs, and carefully choosing increments to minimize total cost. It blends sorting, greedy heuristics, and potentially dynamic programming, depending on problem size and complexity.

A structured approach to assign the minimal valid size for each product in order, considering their costs, will lead you towards the minimal total cost solution.
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