Problem Description

You are scheduled to attend a meeting and have a fixed number of hours, hoursBefore, to cover a certain distance through n roads. The lengths of these roads are represented as an integer array dist, where dist[i] is the length of the ith road in kilometers. You are given a consistent speed in km/h, denoted by the variable speed.

There is a constraint imposed on your travel -- after completing each road, you're required to rest until the start of the next whole hour before you commence the next leg of your journey. However, this mandatory rest period is waived for the final road as you've already arrived at your destination by then.

Interestingly, to help you arrive on time, you are permitted to skip some of these rest periods. Skipping a rest means you don't have to wait for the next whole hour, which might help you align your travel more closely with the hours available.

The core challenge of the problem is determining the minimum number of rest periods you must skip to ensure you arrive at your meeting on time, if at all possible. If there's no way to make it on time, you should return -1.

Intuition

To solve this problem, dynamic programming is an appropriate approach because we can break down the larger problem into smaller subproblems. Each subproblem involves evaluating the minimal time required to reach the end of a certain road with a specific number of skips.

The intuition behind this approach begins with examining every possible scenario where you choose to either wait for the rest period or skip it after each road. We aim to find the strategy that takes the least amount of time to arrive at the meeting on time while skipping the fewest rest periods.

We maintain an array f, where f[i][j] represents the minimum time to reach the end of the ith road with j skips. The algorithm iterates through each road and each possible number of skips, updating the time calculation to reflect whether we wait or skip after each road.

By initializing f[i][j] with an infinitely large number, we ensure that any actual calculated time will be considered in our comparisons. The iterative process compares the current minimum time f[i][j] with the time achieved by either waiting or skipping the rest period, updating as needed to hold the least time.

With all the subproblems solved, we look for the minimum number of skips needed that still allows us to arrive within hoursBefore. We iterate through the possible number of skips and return the smallest number for which the corresponding time does not exceed hoursBefore. If all scenarios exceed the time limit, we return -1, signaling it's impossible to attend the meeting on time.

The provided solution code represents this approach and calculates the minimum skips required accurately using dynamic programming.

Learn more about Dynamic Programming patterns.

Solution Approach

The solution utilizes dynamic programming, which involves breaking down the problem into smaller, manageable subproblems and then building up to the solution.

Data Structures:

• A 2D list f of size (n + 1) x (n + 1) is initialized with inf (representing infinity). This array will store the minimum time needed to reach each road with a given number of skips.

Algorithms and Patterns:

1. Initialization: Since Python lists start at index 0 but roads start at 1, we set f[0][0] to 0 to denote that at the beginning (before traveling any road), no time has passed.

2. Dynamic Programming Iteration:

   • The outer loop iterates through each road (denoted by i), starting from 1, since we consider 0 as the starting point before traveling any road.
   • The inner loop iterates through every possible number of skips (j) that could be applied up to the current road i.

3. State Transition:

   • Inside the inner loop, the algorithm considers two scenarios to find the minimum time to complete road i with j skips:
     • Waiting: The algorithm calculates the time if we wait for the next integer hour mark after road i-1. This is done by adding x / speed (time to travel the current road) to f[i - 1][j] (minimum time to reach the previous road with the same number of skips) and applying ceil to round up to the next hour. An epsilon eps is subtracted before applying the ceil function to handle floating point precision issues.
     • Skipping: The time is directly added without the ceiling for the case where we skip the rest period after road i-1. This is only applicable if j is non-zero, i.e., we have skipped at least once.

4. Comparing and Updating States:

   • The algorithm compares the outcomes of waiting and skipping, updating f[i][j] with the minimum value.

5. Finding the Result:

   • After we've filled in the f array with the minimum traveling times for all possible skips for each road, the algorithm looks for the smallest number of skips that would still allow us to arrive within hoursBefore.
   • This is done by iterating over the possible number of skips for the final road (f[n][j] where j ranges from 0 to n) and checking if that time is less than or equal to hoursBefore. A small epsilon eps is added to the time limit to handle floating point precision issues.
   • If such a j is found, it's returned as the minimum number of skips required. If all possible skips result in times greater than hoursBefore, the function returns -1.

By the end of the iteration, the solution will have considered every possible way to allocate the skips to minimize the travel time and determined whether it is possible to arrive on time.

Mathematical Formulas:

• The ceil function is used to account for the need to round up to the next integer hour when waiting after each road, and subtraction of an epsilon is used to correct for any floating point errors that could incorrectly round down when right on the hour mark.

The algorithm's complexity is bounded by the number of roads n and the number of possible skips, which can also be at most n. Hence, we have a time complexity of O(n^2), which is generally acceptable for the road lengths and number of hours before the meeting typically encountered in this type of problem.

Example Walkthrough

To illustrate the solution approach, let's take a small example. Assume we need to cover a distance through 3 roads with lengths given by the array dist = [5, 3, 6], where hoursBefore = 6 hours and our consistent speed is speed = 1 km/h.

Let's create an array f of size (n + 1) x (n + 1) filled with inf to represent the minimum time taken with a corresponding number of skips. The f array would be a 4x4 array in this case, considering n = 3.

We first initialize f[0][0] = 0. This denotes that we're at the starting point and haven't spent any time or skips yet.

The possible states as we iterate through our array f:

f[0][0] = 0

Moving on to the first road of 5 km:

• Without skipping, we'll spend ceil(5 / 1) = 5 hours, so f[1][0] = 5.
• Skipping is not an option since it's the first road.

f[0][0] = 0
f[1][0] = 5

Next, the second road of 3 km:

• Without skipping, we'll spend ceil((5 + 3) / 1) = 8 hours in total (5 from the first road, and 3 for this road), so f[2][0] = 8.
• With one skip (assuming we skipped after finishing the first road), we would spend 5 (from the first road without waiting) + 3 (second road) hours, so f[2][1] = 8.

f[0][0] = 0
f[1][0] = 5
f[2][0] = 8
f[2][1] = 8

Finally, the third road of 6 km:

• Without skipping, we're now at ceil((8 + 6) / 1) = 14 hours, but since we can't reach our destination on time without skipping, this attempt is infeasible, so we skip considering f[3][0].
• With one skip carried over and skipping after the second road, we would spend 5 + 3 + 6 = 14 hours, which is again infeasible, so f[3][1] doesn't need to be considered.
• With two skips, assuming we skipped after the first and second roads, we would spend 5 + 3 + 6 = 14 hours. Since we can skip the rest on the last road, f[3][2] = 14.

f[0][0] = 0
f[1][0] = 5
f[2][0] = 8 (infeasible as exceeds `hoursBefore`)
f[2][1] = 8 (infeasible as exceeds `hoursBefore`)
f[3][0] = x
f[3][1] = x
f[3][2] = 14

Now, we must consider all options for the number of skips (up to the number of roads n=3). Ideally, we would look for f[n][j] where j ranges from 0 to n (3 in this case) such that f[n][j] <= hoursBefore (6).

When evaluating f[3][2], we can see the value is 14 which exceeds hoursBefore. Thus, none of our options meet the requirement, and we conclude that it is impossible to arrive on time. We return -1.

Solution Implementation

Time and Space Complexity

The time complexity of the provided code is O(n^3), where n is the length of the dist array. This is because there are three nested loops: the outermost loop iterating through each element in dist, and two inner loops, where one loop iterates through j from 0 to i+1, and the other considers two possible scenarios for each j (either skipping or not skipping). Each operation inside the innermost loop is O(1), but since the loops are nested, the total time complexity scales cubically.

The space complexity of the code is O(n^2) because of the 2D list f that has dimensions n+1 by n+1. This list stores the minimum time required to reach each point with a given number of skips. Since it needs to hold a value for every combination of distances and skips, the space required grows quadratically with the number of distances.

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