Problem Description

You have a specific amount of time, hour, to commute to the office using a series of trains. Each segment of your journey has a fixed distance, and all segments must be completed sequentially. The key challenge is that trains can only leave at the start of each hour, meaning you might need to wait before boarding the next train. For instance, if a train ride takes 1.5 hours, a 0.5-hour wait is required before the next train departs. Your task is to determine the minimum train speed (in km/h) required to reach the office on time. This speed must be a positive integer, and if it's impossible to arrive on time, the function should return -1. The provided constraints ensure that the solution won't be higher than 10^7 and the hour won't have more than two decimals.

Intuition

The minimum speed to reach on time is not evidently clear and can range from 1 to 10^7. To find the minimum speed efficiently, we employ a binary search strategy. A binary search halves the potential search space by checking whether a particular condition (in this case, being able to make it to the office on time at a certain speed) is met or not.

Here's the thought process for arriving at the binary search solution:

1. If a particular speed allows us to reach the office on time (hour or less), then any speed higher than this would also suffice. Conversely, any speed lower than a speed that does not make it on time is also inadequate.

2. We define a check function to determine if the current speed is sufficient to reach on time, considering the waiting time between train rides. The last ride doesn't require any wait time.

3. Using a binary search, we find the minimum speed at which the check function returns true. As speeds lower than this minimum won't help us arrive on time, we restrict our search to the range where we might meet the condition.

4. We initiate a search range from 1 to 10^7 and repeatedly narrow it down using the check function until we pinpoint the minimum required speed.

5. If we're unable to find a speed that allows for a timely arrival within the maximum speed limit, we return -1, indicating that it's not possible to be punctual.

The binary search is implemented using the bisect_left function from the Python library. It operates by taking the range as an iterable, the condition to check, and a key function, which in this case is the check function. It then finds the leftmost value in the sorted range that satisfies the condition.

Learn more about Binary Search patterns.

Solution Approach

The solution to the problem leverages the binary search algorithm, which is ideal for the scenario because of its ability to reduce the search space in half with each iteration. The objective of the binary search in this context is to discover the minimum integer speed so that the total travel time does not exceed the given hour.

Here's how the binary search is used to implement the solution:

• Define a check function: This function accepts a speed and calculates the total time taken to travel all distances in the dist array at that speed. For all train segments except the last, the time is rounded up to the nearest integer to account for the rule that trains leave only at the beginning of every hour. For the last train segment, exact time can be used since you don't need to wait for another train after.

• Binary search algorithm: We perform a binary search to find the appropriate speed using two boundaries denoted as left and right, initially set to 1 and 10^7 + 1 respectively. The algorithm iteratively narrows these boundaries until the optimal speed is found.

  • Initialization: We set our initial search interval from 1 to 10^7 + 1, to cover the entire range of possible speeds as per the problem's constraints.

  • Binary search loop: At each step of the loop, the midpoint of the current interval is computed and passed to the check function. Based on the function's output:

    • If the check is true, it means the current speed or any higher could be the solution; hence we narrow the search to the lower half of the interval (right = mid).

    • If the check is false, it means the current speed is too slow, so we narrow the search to the upper half of the interval (left = mid + 1).

  • Checking for impossibility: If it's determined that even the highest possible speed doesn't allow us to be on time (at after searching the entire interval), we conclude that it's impossible to reach on time and return -1.

• Using bisect_left: The binary search is succinctly executed using the bisect_left function from the bisect module, which is applied to a range object representing our potential speeds. We use a lambda function as the key to the bisect_left, which internally applies the check function to the speeds to handle the binary search. The result is then adjusted by adding 1 to the speed since bisect_left finds the leftmost place where True could be inserted without changing the order.

• Mathematical and logical operations: The algorithm uses integer division (math.ceil) to account for the waiting time at stations and leverages boolean conditions to implement the binary search logic.

To sum it up, the solution efficiently navigates through a potentially massive search space to find the minimum speed necessary for timely arrival, using binary search principles encapsulated in Python's bisect_left function.

Example Walkthrough

Suppose we have the following input parameters:

• Distances array dist = [5, 7, 3], representing the distances of each train segment of the journey.
• Total time allowed hour = 2.5 hours to reach the office.

Now, let's walk through the binary search approach to find the minimum train speed required for the given journey:

1. Initialization: We start with a potential speed range of 1 to 10^7 + 1. This means our left boundary is 1 and our right boundary is 10^7 + 1.

2. First iteration of binary search:

   • The mid-speed is (1 + 10000001) / 2 = 5000001.
   • We check if a train at 5000001 km/h can reach within 2.5 hours.
   • We calculate the time for each segment at 5000001 km/h:
     • Segment 1: 5 / 5000001 hours - No need to round up because there is another segment after.
     • Segment 2: 7 / 5000001 hours - No need to round up because there is another segment after.
     • Segment 3: 3 / 5000001 hours - This is the last segment, we take the exact time.
   • We sum up these times and find that the total time taken is way less than 2.5 hours, so we set right to 5000001.

3. Second iteration of binary search:

   • The new mid-speed is (1 + 5000001) / 2 = 2500001.
   • We perform the same check function for the new mid-speed.
   • Again, the speed is high enough to get us there in under 2.5 hours, so we adjust right to 2500001.

4. Continuing the binary search:

   • We continue halving the interval and checking, bringing the left and right boundaries closer after each iteration.

5. Final step:

   • Eventually, we narrow down to a range where moving left by one would cause check to yield false, and we settle on the right boundary as the minimum sufficient speed.

   • Suppose after multiple iterations, we find that at 10 km/h, check is false, but at 11 km/h, check is true. This tells us that 11 km/h is the minimum speed required to reach the office on time.

6. Result:

   • The binary search terminates when left is equal to right, which means we've found the minimum train speed satisfying the conditions.
   • If no such speed allows arrival within 2.5 hours, our search boundaries would converge such that check always returns false, and we return -1.

In our example, by following the binary search algorithm steps, we would eventually find that the minimum speed required for the distances [5, 7, 3] to be covered within 2.5 hours is 11 km/h. This result is achieved without testing every speed from 1 to 10^7 individually.

Solution Implementation

Time and Space Complexity

The time complexity of the minSpeedOnTime function is determined by the binary search and the check function that is called at each step of the binary search.

The binary search runs in O(log R), where R is the range of possible speeds, which in this case is 10^7. The +1 correction does not affect the logarithmic complexity.

The check function runs in O(N), where N is the number of elements in the dist list because it needs to iterate over all the distances. Inside the check function, math.ceil function is called which has a constant time O(1). Therefore, for each call of the check function, the time complexity is linear with respect to the number of distances.

Therefore, the overall time complexity of the function is O(N log R) where N is the number of distances in the dist list and R is 10^7.

The space complexity of the function is O(1). No additional space is allocated that grows with the size of the input, except for the variable res, which uses a constant amount of space.

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