Problem Description

Imagine you're playing a guessing game where the objective is to figure out a hidden number and you want to spend as little money as possible in the process. In this game, you will guess numbers between 1 and n, and each incorrect guess will cost you the value of the guess in dollars. If you get it right, you win! If you get it wrong, you'll be told if the hidden number is higher or lower than your guess. You want to know the least amount of money required to guarantee a win no matter what number is chosen by your opponent.

Intuition

The approach to solve this problem is to use dynamic programming, a method where we break the problem down into simpler subproblems and build up a solution to the larger problem based on the solutions to the smaller problems.

This problem requires a strategy to minimize the total cost while guaranteeing a win. To achieve this, we need to consider all possible outcomes and make the best worst-case decision at each guess. We think of the game in terms of ranges (from 1 to n) and then narrow down the range based on whether we need to guess higher or lower. This approach will help us figure out the minimum cost required to win the game.

We use a 2D array dp to record the cost to guarantee a win for every range [i, j]. The cost of the guess k is the maximum between dp[i][k-1] (if the number is lower than k) and dp[k+1][j] (if the number is higher than k) plus the cost k of making the guess. Since we want to minimize our cost, we take the minimum over all possible k's for the range [i, j]. The answer is then dp[1][n] since it's the minimum amount of money required to guarantee a win for the full range.

Learn more about Math and Dynamic Programming patterns.

Solution Approach

The implementation leverages the concept of dynamic programming to develop a strategy that ensures the minimum cost of a guaranteed win. Here's how the solution is built:

1. Dynamic Programming Table Initialization: Create a 2-dimensional array dp where dp[i][j] represents the minimum cost to guarantee a win for the range [i, j]. The size of the array is (n + 10) by (n + 10) to cover all the possible ranges between 1 and n, with a bit of extra space to avoid index out-of-bounds errors.

2. Filling the DP Table: Iterate over all the possible lengths l of the ranges from 2 to n+1. For each length, compute the minimum cost for all ranges of that length starting from 1.

3. Computing Costs: For each range [i, j] with the starting point i and the endpoint j, initialize the minimum cost dp[i][j] to infinity as a starting point. Then, try every possible guess k within the range [i, j]. For each guess, calculate the cost t, which is the maximum of either guessing too low (dp[i][k - 1]) or too high (dp[k + 1][j]), plus the cost k of making the current guess.

4. Minimizing Costs: The value of dp[i][j] is updated to the minimum cost out of all the guesses k.

5. Result Extraction: After populating the dp table with the costs for all ranges, the minimum cost to guarantee a win for the full range from 1 to n is found in dp[1][n].

The algorithm uses a bottom-up strategy, solving smaller range problems first and using their solutions to address larger ranges. This ensures that at every step, the optimal (minimum) cost for the current range is based on the best decisions made for the previous smaller ranges.

This approach ensures that we achieve a global optimum by making locally optimal choices, thanks to the overlapping subproblems and optimal substructure properties of dynamic programming.

Example Walkthrough

Let's illustrate the solution approach using a small example. Suppose n = 4, so the hidden number is between 1 and 4, and we want to find the minimum amount of money required to guarantee a win.

Step 1: Dynamic Programming Table Initialization
We initialize a 2D array dp that has 5x5 dimensions since n is 4 and we add 1 to have an inclusive range.

Initially, dp looks like this (with 0s where no cost is involved):

dp = [
  [0, 0, 0, 0, 0],
  [0, 0, 0, 0, 0],
  [0, 0, 0, 0, 0],
  [0, 0, 0, 0, 0],
  [0, 0, 0, 0, 0],
]

Step 2: Filling the DP Table
We consider ranges of lengths from 2 to 5.

Step 3: Computing Costs
When the length is 2, we consider ranges: [1, 2], [2, 3], [3, 4]. For example, for the range [1, 2], we can guess 1 or 2. If we guess 1 and the number is 2, we lose $1 and then pay $2 to guess correctly, which sums to $3. If we guess 2 and the number is 1, we lose $2 but then just pay $1 to guess it correctly, summing to $3. So, the minimum cost here is $3.

The table looks partially like this:

dp = [
  [0, 0, 0, 0, 0],
  [0, 0, 3, 0, 0],
  [0, 0, 0, 3, 0],
  [0, 0, 0, 0, 3],
  [0, 0, 0, 0, 0],
]

Step 4: Minimizing Costs
We move to ranges of length 3 and 4 and update dp accordingly. Take range [1, 3]. Our guesses are 1, 2, or 3:

• Guess 1: If it's wrong, we lose 1, and the minimum cost to guess between [2, 3] is \3 (from step 3). So, total cost = $1 + $3 = $4.
• Guess 2: If it's wrong, we can face a higher or lower number, resulting in a cost of max([1], [3]) = $2 (always correct on the second guess, 1 or 3). So, total cost = $2 + $2 = $4.
• Guess 3: Similar to guessing 1, total cost = $4.

Therefore, the cost to guarantee a win between [1, 3] is $4.

We proceed similarly for all lengths. Finally, for length 4, the range [1, 4] has multiple scenarios and we select the minimum cost of all.

Step 5: Result Extraction
After computing all possible ranges, our dp array for the range [1, n] (which is [1, 4] in our example) gives us the answer. For this example, dp[1][4] will end up being the minimum cost to guarantee a win for the game.

By building the solution from the smallest range to the largest and choosing the minimum cost at each step, we ensure the minimum amount of money required to guarantee a win for the entire range. This example uses smaller numbers for simplicity, but the actual implementation would handle larger values of n following the same process.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the provided code is O(n^3). This is because there are three nested loops: the outermost loop ranges over l, which goes from 2 to n; the second loop ranges over i, which goes from 1 to n - l + 1. For each combination of i and l, the innermost loop ranges over k, which goes from i to j (where j = i + l - 1). For each value of k, the code computes t which involves querying and updating values in the dp array, which takes constant time. The number of iterations is roughly n for the i loop, n for the l loop, and n again for the k loop, multiplying together for a cubic time complexity.

Space Complexity

The space complexity of the code is O(n^2). We allocate a dp array that has n + 10 rows and n + 10 columns to ensure that we do not index out of bounds when performing dynamic programming. Though not all of these entries are used, the overall space required is proportional to n^2 since the bulk of the space is occupied by the n * n subarray required to calculate the minimum cost for guessing the number.

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