Maximum Value of K Coins From Piles

Problem Description

You have n piles of coins on a table, where each pile consists of a positive number of coins of assorted denominations. In one move, you can choose any coin on top of any pile, remove it, and add it to your wallet. Given a list piles, where piles[i] is a list of integers denoting the composition of the i-th pile from top to bottom, and a positive integer k, return the maximum total value of coins you can have in your wallet if you choose exactly k coins optimally.

Key Insights

• You can only take coins from the top of each pile.
• The optimal selection of coins involves balancing between the number of coins you take from each pile and their values.
• Dynamic programming can be used to track the maximum value obtainable by taking different numbers of coins from the piles.

Space and Time Complexity

Time Complexity: O(n * k)
Space Complexity: O(k)

Solution

The problem can be solved using a dynamic programming approach. We maintain a DP array where dp[j] represents the maximum value obtainable with exactly j coins. For each pile, we calculate the maximum possible values by iterating through the coins in that pile and updating the DP array accordingly.

1. Start with a DP array initialized to zero.
2. Iterate through each pile and for each pile, iterate through the number of coins you can take (from 0 to the minimum of k or the number of coins in the current pile).
3. Update the DP array in reverse to ensure that we do not use the same pile's coins multiple times in the same iteration.

This method ensures that we are considering each pile optimally while keeping track of the maximum values possible for all selections of coins.

Code Solutions