Problem Description

In this problem, we are tasked with calculating the minimum fatigue cost required to set a microwave to cook for a target duration in seconds. The microwave interface takes up to four digits input, interpreted as minutes and seconds, with the first two digits representing minutes (00 to 99) and the last two representing seconds (00 to 99). To set the microwave, we prepend zeroes to make sure we always have a four-digit number if fewer digits are entered.

The microwave setting process involves moving a finger from the starting digit (given by startAt) to other digits and pushing them to set the time. Each move incurs a moveCost and each push incurs a pushCost. There might be multiple ways to set the desired cooking time, but we are interested in finding the way that results in the least total fatigue cost.

Finally, the target cooking time is defined in seconds (targetSeconds). Since the microwave interprets inputs as minutes and seconds, we must convert the target time accordingly. One minute is equivalent to 60 seconds, so the target time could sometimes be set more efficiently by adjusting the balance between minutes and seconds.

Intuition

The intuition behind the solution is to break down the problem into simpler, logical steps and use a helper function to calculate the cost of setting the microwave to a given time.

1. Convert the targetSeconds to minutes and seconds. Since the maximum number of seconds the microwave can display is 99, we first divide the targetSeconds by 60 to find the number of whole minutes, and then we use modulo to find the remaining seconds.

2. We create a helper function, f(m, s), that calculates the cost of setting the microwave to m minutes and s seconds. It accounts for the cost of moving to and pushing the required digits. If the minutes or seconds exceed their respective bounds (e.g., more than 99), the function returns an infinite cost, indicating an invalid input.

3. The function first converts the minutes and seconds into an array of four digits, including leading zeros. It then ignores any leading zeros because they do not require any push.

4. It iterates through the array of digits, calculating the cost to move to and push each digit. If the current digit is the same as the previous one, we only consider the push cost; otherwise, we include both the move and push costs.

5. To find the minimum cost, we compare the cost of setting the exact minutes and seconds derived from targetSeconds and also consider the scenario where we decrement one minute and add 60 seconds (because sometimes it's cheaper to enter fewer minutes and more seconds, especially if it allows us to reduce the number of moves or pushes).

6. Finally, we return the minimum cost obtained from the two possibilities calculated using our helper function.

Using these steps, the solution efficiently examines the possible ways to enter the desired cooking time and selects the one with the lowest physical cost.

Learn more about Math patterns.

Solution Approach

The solution is implemented in Python and follows a step-by-step logical approach to minimize fatigue when setting the microwave timer. Let's walk through the implementation strategy:

• Defining The Helper Function (f): The heart of this solution is the helper function f(m, s), which calculates the cost for a proposed minute (m) and second (s) combination. It assumes that non-valid minute or second values (like more than 99 minutes or seconds) return an infinite cost denoted by inf. This prevents the use of invalid time formats.

• Converting Time Into Digits: Inside the helper function, we transform m and s into an array of individual digits, starting from the most significant to the least significant. This is done by integer division and modulo to split each one into tens and ones.

• Cost Calculation Logic: After getting the digit array, we iterate from the start (ignoring leading zeros) and calculate the cost as follows:

  • If the current digit is different from the previous one (prev), we have to move the finger, which incurs a moveCost.
  • Regardless of whether we moved or not, we always incur a pushCost when pressing a digit. The prev variable is then updated to the current digit for the next iteration.

• Finding The Cost For Two Scenarios: We call the helper function f twice with:

  • The exact minutes (m) and seconds (s) calculated from targetSeconds.
  • The adjusted values where we subtract one minute and add 60 seconds to s (only if the m is greater than 0 since we can't have negative minutes).

• Algorithm Pattern: The pattern employed by this solution is essentially a brute-force check of the possible valid combinations (there are only two), and simply choosing the one that returns the least cost. This is quick due to the low number of total possible and valid combinations.

• Data Structures: This solution relies on simple data structures: primarily an integer array to keep track of the digit-wise break down of the minute-second format required for setting the timer.

Example Scenario Execution: If targetSeconds = 5500, this translates to 91 minutes and 40 seconds, but since the microwave cannot display more than 99 seconds, we have to subtract one minute and add those 60 seconds making the time 90 minutes and 100 seconds. Then, we sequentially apply the helper function to both time formats and choose the least costly one.

// Insert calculated cost calculation steps here

From the implementation given; it is clear that the approach taken to solve the problem is both efficient and effective. It cleverly utilizes the simplicity of brute force in a scenario where analyzing all combinations is trivial, combined with a well-defined helper function to abstract out the cost-calculation logic.

Example Walkthrough

Let's go through an example to illustrate the solution approach. Suppose that targetSeconds = 130, startAt = 1, moveCost = 2, and pushCost = 1. We want to find the minimum fatigue cost to set the microwave to cook for 130 seconds.

1. Convert targetSeconds to minutes and seconds.
   130 seconds is equal to 2 minutes and 10 seconds since ( 130 = 2 \times 60 + 10 ). So our inputs to the helper function will be m = 2 and s = 10.

2. Use Helper Function f(m, s).
   Implement f(2, 10):

   • Convert to array of digits: m = 2 becomes 02 and s = 10 becomes 10 for a complete array [0, 2, 1, 0].
   • Start at 1, move to 0 (start digit does not require push), move cost is 2.
   • Push 0, push cost is 1 (though first 0 is not pushed as it's leading).
   • Move to 2, move cost is 2.
   • Push 2, push cost is 1.
   • Remaining are 1 and 0. Since 1 is new, move and then push (2 move cost + 1 push cost).
   • Since we're already at 1, we only push 0 which incurs only 1 push cost.
   • The total minimum cost for f(2, 10) is ( 2 + 1 + 2 + 1 + 2 + 1 + 1 = 10 ).

3. Consider the second scenario where we subtract a minute and add 60 seconds.
   Since m = 2, it's possible to subtract one minute to get 1 minute. Add 60 seconds to 10 to get 70 seconds. Now we input f(1, 70):

   • Convert to array of digits: m = 1 becomes 01 and s = 70 becomes 70 for a complete array [0, 1, 7, 0].
   • Start at 1, move to 0 (start digit does not require push), move cost is 2.
   • Push 0, but since it's leading, we ignore this and move to 1.
   • Push 1, push cost is 1.
   • Move to 7, move cost is 2.
   • Push 7, push cost is 1.
   • Push 0, push cost is 1.
   • The total minimum cost for f(1, 70) is ( 2 + 1 + 2 + 1 + 1 = 7 ).

4. Select the minimum of the two.
   Now we simply select the minimum of f(2, 10) and f(1, 70), which in this case is f(1, 70) so, the minimum fatigue cost is 7.

With this example, we see the solution finds the most efficient manner to set the required time on the microwave by converting the time and comparing the fatigue cost of two close time settings, thus ensuring the user expends the least amount of physical effort.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the function is determined by the operations that it performs:

• The function f is called exactly two times, regardless of the input size, so we can consider it a constant-time operation in the context of analyzing the overall function. Within f, most operations are simple arithmetic operations, conditions, and a loop which, in the worst case, iterates at most 4 times, as it's dealing with time in minutes and seconds.

• The cost computation within the loop runs a constant number of times, as mentioned, at most 4, since the digits for minutes and seconds each can be in the range [0, 99]. The condition and the arithmetic inside the loop are constant-time operations.

Hence, the time complexity of the f function is O(1).

Since there are no other loops or recursive calls that depend on the size of the input in the main function and f is called only twice, the time complexity of the minCostSetTime function is also O(1).

Space Complexity

• The f function uses a fixed amount of additional space: an array arr of size 4 and a few integer variables to hold parameters and intermediate results.

• There is no dynamic data structure or additional memory usage that scales with the input size.

Therefore, the space complexity of the function is O(1).

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