Problem Description

The problem involves a game played with a collection of stones, each with a certain weight. During the game, two stones are picked in every turn and smashed against each other. If the weights of the stones are equal (x == y), both stones are destroyed. If they are not (x != y), the lighter stone is destroyed and the heavier stone's weight is reduced by the weight of the lighter stone. The aim of the game is to minimize the weight of the last remaining stone after all turns have been played. If no stones remain, the weight is considered to be 0.

Intuition

To solve this problem, we can think of it in terms of a 0/1 Knapsack problem, a common strategy used in dynamic programming. In the 0/1 Knapsack problem, we aim to maximize the value of items that can be fit into a knapsack of limited capacity without exceeding the weight limit.

For this game, we can imagine having a "knapsack" with a capacity that is half the sum of all stone weights. The reason we use half the sum is that we seek to partition the stones into two groups that are as equal in total weight as possible. By doing so, when the stones from these groups battle each other, the resulting stone (if any) will have the minimal possible weight.

So, we construct a dynamic programming array dp where dp[j] keeps the maximum achievable sum of stones that does not exceed the capacity j. We iterate over each stone and update the dp array, considering the current stone to either be included or excluded from our "knapsack". Eventually, dp[-1] will contain the largest weight that is less than or equal to the sum of all stones divided by two.

The smallest possible weight of the final stone can then be calculated as the total weight of all stones minus twice the weight indicated by dp[-1], because for every weight we put in one group, we are effectively removing it from what would be the opposing group of stones in the "game".

Learn more about Dynamic Programming patterns.

Solution Approach

The solution uses dynamic programming to solve this modified knapsack problem. Here's a step-by-step walkthrough of the implementation:

1. Calculate the sum s of all stones. The objective is to partition the stones into two groups with a total weight that is as close to s / 2 as possible.

2. Define n as s / 2 because this represents the largest possible weight our "knapsack" may store. Since we only need an integer knapsack weight, we use a bitwise right shift s >> 1, equivalent to s / 2 but faster computationally.

3. Initialize a dynamic programming array dp of size n + 1 with zeros. This array will help us to keep track of the maximum weight that can be achieved for each capacity up to n. dp[j] will represent the maximum weight that can be accumulated with a knapsack capacity of j.

4. The outer loop goes through each stone's value v in the stones array.

5. The inner loop is a reversed loop from n to v. This loop updates the dp array. For each capacity j from n down to the weight of the current stone v, we check whether including the stone would lead to a better result compared to not including it.

6. Within the inner loop, the expression dp[j] = max(dp[j], dp[j - v] + v) updates the current dp[j] value. It evaluates two situations: not taking the current stone (leaving dp[j] as it is) or taking the stone (which would be dp[j - v] + v). If the latter offers a higher weight without exceeding the capacity j, we update dp[j].

7. After filling the dp array, the maximum sum we can get from one group of stones is dp[-1]. The smallest possible weight of the last remaining stone is then s - dp[-1] * 2. We subtract double dp[-1] from the total sum s because, effectively, for every stone placed in one group, we remove that weight from the potential remaining stone.

By using dynamic programming, this solution efficiently determines the best way to "pack" the stones into two groups with the goal of minimizing the weight of the last remaining stone after all possible smashes.

Example Walkthrough

Let's go through a small example to illustrate the solution approach with a stones array stones = [2,7,4,1,8,1].

1. We first calculate the sum s of all stones, which is 2 + 7 + 4 + 1 + 8 + 1 = 23. The goal is to partition the stones into two groups with a total weight that is as close to s / 2 = 23 / 2 = 11.5 as possible for minimizing the last stone's weight.

2. We define n as the integer part of s / 2, which gives us n = 11 since we're dealing with integer weights.

3. We initialize a dynamic programming array dp of size n + 1 with zeros: dp = [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], corresponding to knapsack capacities from 0 to 11.

4. We start iterating through the stones array:

   • For stone value v = 2, we update dp from end to start where j >= v.
   • For stone value v = 7, we repeat the updating process.
   • We continue until all stones are considered.

5. For the first iteration with v = 2, the inner loop will check capacities from 11 down to 2. The update looks like:

   • For j = 11, we consider including the stone or not: dp[11] = max(dp[11], dp[11 - 2] + 2).
   • For j = 10, decide between dp[10] and dp[10 - 2] + 2.
   • And so on, down to j = 2.

6. After we process v = 2, the dp array changes indicating the best way to pack stones up to now: dp = [0, 0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]

7. We process all stones this way, and the dp array continues to update with the maximum weights for each capacity.

8. At the end, assuming after all updates our dp array looks like: dp = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]

9. The maximum sum we can get from one group of stones is dp[-1] which is 11. This means that one group of stones has a total weight of 11, and so does the second group, as we wanted two equal groups.

10. The smallest possible weight of the last remaining stone is then s - dp[-1] * 2 = 23 - 11 * 2 = 1.

So, the algorithm leads to the partitioning of the stones into two groups whose weights differ by at most 1. The weight of the last stone—if any—is minimized, which in this case is 1, as this is the least weight we can achieve by using dynamic programming and the knapsack analogy.

Solution Implementation

Time and Space Complexity

The given code is an implementation of a dynamic programming solution to solve a variation of the classic knapsack problem. To analyze the time complexity and space complexity, let us explore the code:

Time Complexity

The time complexity of the code can be determined by looking at the nested loops. There is an outer loop that iterates over each stone, and an inner loop that runs in reverse from "n" (where "n" is half of the sum of stones, "s >> 1") to the value of the current stone "v" (i.e., "range(n, v - 1, -1)"). The inner loop ensures that each sub-sum only considers each stone once.

• The outer loop runs once for each stone, so it runs "m" times where "m" is the number of stones ("len(stones)").
• The inner loop runs (at most) "n" times for each outer iteration, where "n" is half the sum of all stone weights, rounded down ("s >> 1").

Therefore, the overall time complexity is O(mn), where "m" is the number of stones and "n" is half the sum of the stones' weights.

Space Complexity

The space complexity of the code is determined by the storage requirements. The array "dp" of size "n + 1" is created to store the maximum achievable weight for each sub-sum. No other data structures are used that grow with the size of the input. Thus, the space complexity is O(n), where "n" is half the sum of the stones' weights.

Learn more about how to find time and space complexity quickly using problem constraints.