Maximum Value of K Coins From Piles

Maximum Value of K Coins From Piles

What is this problem about?

The "Maximum Value of K Coins From Piles" coding problem presents you with several piles of coins, where each pile contains a varying number of coins, and each coin has a specific value. Your goal is to select exactly k coins in total from these piles such that the sum of the values of the chosen coins is maximized. The catch is that you can only take coins from the top of each pile. This means if you decide to take coins from a particular pile, you must take them in sequential order from the top. This problem often requires careful decision-making and is a classic application of dynamic programming, sometimes enhanced with prefix sums for efficient calculation.

Why is this asked in interviews?

This Maximum Value of K Coins From Piles interview question is a popular choice for assessing a candidate's understanding and application of dynamic programming, especially in scenarios involving multiple choices and constraints. Companies like Microsoft and Google use it to evaluate your ability to define a clear DP state that captures the essential information (e.g., dp[pile_index][remaining_k_coins]), formulate recurrence relations, and optimize calculations. It tests your problem-solving skills in breaking down a complex problem into smaller, overlapping subproblems. The use of prefix sums for optimizing the inner loops also highlights your attention to efficiency and algorithmic refinement, making it a robust dynamic programming interview question.

Algorithmic pattern used

The "Maximum Value of K Coins From Piles" problem is best solved using Dynamic Programming, often optimized with Prefix Sums. The DP state can be defined as dp[i][j], representing the maximum value obtainable by picking j coins from the first i piles. The recurrence relation involves iterating through the number of coins we can take from the i-th pile. For each i-th pile, if we decide to take x coins (where 0 <= x <= number_of_coins_in_pile_i and x <= j): dp[i][j] = max(dp[i][j], dp[i-1][j-x] + value_of_first_x_coins_from_pile_i) The value_of_first_x_coins_from_pile_i can be pre-calculated efficiently using prefix sums for each pile. This transforms the calculation of the sum of x coins from O(x) to O(1). The base case is dp[0][0] = 0. The final answer would be dp[num_piles][k]. This array, dynamic programming, and prefix sum interview pattern allows for an efficient solution by avoiding redundant computations and speeding up inner loop calculations.

Example explanation

Suppose piles = [[10, 20, 30], [5, 10]] and k = 3. We want to pick 3 coins for maximum value.

Precompute Prefix Sums for each pile:

• Pile 0: [0, 10, 30, 60] (sum of 0, 1, 2, 3 coins)
• Pile 1: [0, 5, 15] (sum of 0, 1, 2 coins)

Initialize dp table: dp[num_piles + 1][k + 1] with 0s.

Iterate through piles and number of coins j: Consider Pile 0: [10, 20, 30]

• dp[1][0] = dp[0][0] + 0 = 0 (take 0 coins from pile 0)
• dp[1][1] = max(dp[0][1] + 10, dp[0][0] + 10) = 10
• dp[1][2] = max(dp[0][2] + 30, dp[0][1] + 20, dp[0][0] + 30) = 30
• dp[1][3] = max(dp[0][3] + 60, dp[0][2] + 50, dp[0][1] + 40, dp[0][0] + 60) = 60

Consider Pile 1: [5, 10]

• When calculating dp[2][j], we consider taking x coins from Pile 1 and j-x coins from Pile 0.
• For dp[2][3] (final answer with 3 coins):
  • Take 0 coins from Pile 1: dp[1][3] + 0 = 60 + 0 = 60
  • Take 1 coin from Pile 1 (value 5): dp[1][2] + 5 = 30 + 5 = 35
  • Take 2 coins from Pile 1 (value 5+10=15): dp[1][1] + 15 = 10 + 15 = 25
  • dp[2][3] = max(60, 35, 25) = 60.

The maximum value of K coins is 60. This illustrates the dynamic programming approach to the Maximum Value of K Coins From Piles coding problem.

Common mistakes candidates make

A common mistake in the Maximum Value of K Coins From Piles coding problem is a brute-force approach trying all combinations, which leads to exponential time complexity. Another frequent error is not using prefix sums, resulting in an O(K * Piles * MaxCoinsInPile) complexity, which might be too slow if MaxCoinsInPile is large. Incorrectly defining the DP state or formulating the recurrence relation can also lead to issues, especially regarding how to track the number of coins already picked and which piles have been considered. Overlapping subproblems are central to DP, and neglecting to store and reuse these results will lead to inefficiency.

Interview preparation tip

To excel in the Maximum Value of K Coins From Piles interview question, solidify your understanding of both dynamic programming and prefix sums. Practice identifying problems where these two patterns can be combined for optimal efficiency. Focus on clearly defining your DP state and transitions, and always consider how to pre-process data (like using prefix sums) to speed up calculations within the DP loop. Work through small examples by hand to ensure your recurrence relation and base cases are correct. Regular practice with array and dynamic programming interview pattern questions will build the necessary intuition.