Minimum Number of Refueling Stops

To solve this problem, we can use a max heap to keep track of the gas stations we have passed along the way, with the amount of fuel as the key. We use a greedy approach, where we refuel at the gas station that gives us the most fuel when needed.

Here’s the step-by-step algorithm to reach the solution:

  1. Initialize the Variables: Set the current position and current fuel according to the startFuel. Initialize a max heap for gas stations, and a variable to keep track of the number of refueling stops.

  2. Iterate through Stations: Loop through each station and do the following: a. Check Distance: If the position of the current station is further than what the current fuel can reach, try to refuel at previous stations until we can reach the current station or the heap is empty. b. Add to Heap: If we can reach the current station, add the fuel from this station to the max heap. c. Check Target: If we can reach the target at any point, return the number of refueling stops.

  3. Final Check: After iterating through the stations, check if we can reach the target by refueling at any remaining stations in the max heap.

  4. Return Result: If we can reach the target, return the number of refueling stops. Otherwise, return -1.

Here’s the code:

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import heapq

class Solution:
    def minRefuelStops(self, target: int, startFuel: int, stations: List[List[int]]) -> int:
        max_heap = []
        refuel_stops = 0
        curr_fuel = startFuel
        curr_position = 0

        for position, fuel in stations:
            # Check if we can reach the current station
            while curr_position + curr_fuel < position and max_heap:
                curr_fuel += -heapq.heappop(max_heap)
                refuel_stops += 1

            # If we still can't reach the current station
            if curr_position + curr_fuel < position:
                return -1

            # Add current station's fuel to the max heap
            heapq.heappush(max_heap, -fuel)

            # Update current fuel after reaching the current station
            curr_fuel -= position - curr_position
            curr_position = position

        # Check if we can reach the target with the remaining stations
        while curr_position + curr_fuel < target and max_heap:
            curr_fuel += -heapq.heappop(max_heap)
            refuel_stops += 1

        return refuel_stops if curr_position + curr_fuel >= target else -1

The time complexity of this solution is O(N * log(N)), where N is the number of stations, and the space complexity is O(N).

10 Prerequisite LeetCode Problems

“871. Minimum Number of Refueling Stops” involves dynamic programming and priority queue. Here are some simpler problems to prepare for this:

  1. LeetCode 322. Coin Change

    • This problem introduces the concept of dynamic programming. The idea is to find the minimum number of coins needed to make a certain amount.
  2. LeetCode 518. Coin Change 2

    • This problem involves finding the number of combinations that make up a particular amount. It’s similar to the “Coin Change” problem, but focuses on the count instead of the minimum number.
  3. LeetCode 215. Kth Largest Element in an Array

    • This problem requires understanding of priority queues. The idea is to find the kth largest element in an unsorted array.
  4. LeetCode 787. Cheapest Flights Within K Stops

    • This problem is about finding the cheapest price for a journey with a given number of stops. It also uses a priority queue.
  5. LeetCode 743. Network Delay Time

    • This problem involves finding the time it will take for all nodes to receive a signal. It’s a bit more advanced and involves the use of Dijkstra’s algorithm with a priority queue.
  6. LeetCode 406. Queue Reconstruction by Height

    • This problem involves understanding of sorting and queue manipulation.
  7. LeetCode 1046. Last Stone Weight

    • This problem is about repeatedly choosing the two heaviest stones and smashing them together until there is only one stone left, or none at all. It introduces you to the use of a priority queue to solve problems.
  8. LeetCode 198. House Robber

    • This problem is a classic dynamic programming problem where you need to find the maximum amount of money you can rob tonight without alerting the police.
  9. LeetCode 120. Triangle

    • This problem involves finding the minimum path sum from top to bottom in a triangle. It helps in understanding the concept of dynamic programming.
  10. LeetCode 64. Minimum Path Sum

    • This problem is a grid-based dynamic programming problem where you need to find a path from top left to bottom right which minimizes the sum of all numbers along its path.

Problem Boundary

How to establish the boundary of this problem?

Problem Classification

Problem Statement:A car travels from a starting position to a destination which is target miles east of the starting position.

There are gas stations along the way. The gas stations are represented as an array stations where stations[i] = [positioni, fueli] indicates that the ith gas station is positioni miles east of the starting position and has fueli liters of gas.

The car starts with an infinite tank of gas, which initially has startFuel liters of fuel in it. It uses one liter of gas per one mile that it drives. When the car reaches a gas station, it may stop and refuel, transferring all the gas from the station into the car.

Return the minimum number of refueling stops the car must make in order to reach its destination. If it cannot reach the destination, return -1.

Note that if the car reaches a gas station with 0 fuel left, the car can still refuel there. If the car reaches the destination with 0 fuel left, it is still considered to have arrived.

Example 1:

Input: target = 1, startFuel = 1, stations = [] Output: 0 Explanation: We can reach the target without refueling. Example 2:

Input: target = 100, startFuel = 1, stations = [[10,100]] Output: -1 Explanation: We can not reach the target (or even the first gas station). Example 3:

Input: target = 100, startFuel = 10, stations = [[10,60],[20,30],[30,30],[60,40]] Output: 2 Explanation: We start with 10 liters of fuel. We drive to position 10, expending 10 liters of fuel. We refuel from 0 liters to 60 liters of gas. Then, we drive from position 10 to position 60 (expending 50 liters of fuel), and refuel from 10 liters to 50 liters of gas. We then drive to and reach the target. We made 2 refueling stops along the way, so we return 2.

Constraints:

1 <= target, startFuel <= 109 0 <= stations.length <= 500 1 <= positioni < positioni+1 < target 1 <= fueli < 109

Analyze the provided problem statement. Categorize it based on its domain, ignoring ‘How’ it might be solved. Identify and list out the ‘What’ components. Based on these, further classify the problem. Explain your categorizations.

Distilling the Problem to Its Core Elements

In order to have me distill a problem to its core, you could ask questions that prompt for a deeper analysis of the problem, understanding of the underlying concepts, and simplification of the problem’s essence. Here are some examples of such prompts:

  1. Can you identify the fundamental concept or principle this problem is based upon? Please explain.
  2. What is the simplest way you would describe this problem to someone unfamiliar with the subject?
  3. What is the core problem we are trying to solve? Can we simplify the problem statement?
  4. Can you break down the problem into its key components?
  5. What is the minimal set of operations we need to perform to solve this problem?

These prompts guide the discussion towards simplifying the problem, stripping it down to its essential elements, and understanding the core problem to be solved.

Visual Model of the Problem

How to visualize the problem statement for this problem?

Problem Restatement

Could you start by paraphrasing the problem statement in your own words? Try to distill the problem into its essential elements and make sure to clarify the requirements and constraints. This exercise should aid in understanding the problem better and aligning our thought process before jumping into solving it.

Abstract Representation of the Problem

Could you help me formulate an abstract representation of this problem?

Given this problem, how can we describe it in an abstract way that emphasizes the structure and key elements, without the specific real-world details?

Terminology

Are there any specialized terms, jargon, or technical concepts that are crucial to understanding this problem or solution? Could you define them and explain their role within the context of this problem?

Problem Simplification and Explanation

Could you please break down this problem into simpler terms? What are the key concepts involved and how do they interact? Can you also provide a metaphor or analogy to help me understand the problem better?

Constraints

Given the problem statement and the constraints provided, identify specific characteristics or conditions that can be exploited to our advantage in finding an efficient solution. Look for patterns or specific numerical ranges that could be useful in manipulating or interpreting the data.

What are the key insights from analyzing the constraints?

Case Analysis

Could you please provide additional examples or test cases that cover a wider range of the input space, including edge and boundary conditions? In doing so, could you also analyze each example to highlight different aspects of the problem, key constraints and potential pitfalls, as well as the reasoning behind the expected output for each case? This should help in generating key insights about the problem and ensuring the solution is robust and handles all possible scenarios.

Provide names by categorizing these cases

What are the key insights from analyzing the different cases?

Identification of Applicable Theoretical Concepts

Can you identify any mathematical or algorithmic concepts or properties that can be applied to simplify the problem or make it more manageable? Think about the nature of the operations or manipulations required by the problem statement. Are there existing theories, metrics, or methodologies in mathematics, computer science, or related fields that can be applied to calculate, measure, or perform these operations more effectively or efficiently?

Simple Explanation

To explain a problem in simple, non-technical language:

  1. Can you explain [problem] in simple terms or like you would explain to a non-technical person?
  2. Imagine you’re explaining [problem] to someone without a background in programming. How would you describe it?
  3. If you had to explain [problem] to a child or someone who doesn’t know anything about coding, how would you do it?
  4. In layman’s terms, how would you explain the concept of [problem]?
  5. Could you provide a metaphor or everyday example to explain the idea of [problem]?

These prompts encourage an explanation that avoids jargon or assumes minimal technical knowledge, making the concept more accessible to a broader audience.

Problem Breakdown and Solution Methodology

Given the problem statement, can you explain in detail how you would approach solving it? Please break down the process into smaller steps, illustrating how each step contributes to the overall solution. If applicable, consider using metaphors, analogies, or visual representations to make your explanation more intuitive. After explaining the process, can you also discuss how specific operations or changes in the problem’s parameters would affect the solution? Lastly, demonstrate the workings of your approach using one or more example cases.

Inference of Problem-Solving Approach from the Problem Statement

Can you identify the key terms or concepts in this problem and explain how they inform your approach to solving it? Please list each keyword and how it guides you towards using a specific strategy or method.

How did you infer from the problem statement that this problem can be solved using ?

Stepwise Refinement

  1. Could you please provide a stepwise refinement of our approach to solving this problem?

  2. How can we take the high-level solution approach and distill it into more granular, actionable steps?

  3. Could you identify any parts of the problem that can be solved independently?

  4. Are there any repeatable patterns within our solution?

Solution Approach and Analysis

Given the problem statement, can you explain in detail how you would approach solving it? Please break down the process into smaller steps, illustrating how each step contributes to the overall solution. If applicable, consider using metaphors, analogies, or visual representations to make your explanation more intuitive. After explaining the process, can you also discuss how specific operations or changes in the problem’s parameters would affect the solution? Lastly, demonstrate the workings of your approach using one or more example cases.

Identify Invariant

What is the invariant in this problem?

Identify Loop Invariant

What is the loop invariant in this problem?

Thought Process

Can you explain the basic thought process and steps involved in solving this type of problem?

Explain the thought process by thinking step by step to solve this problem from the problem statement and code the final solution. Write code in Python3. What are the cues in the problem statement? What direction does it suggest in the approach to the problem? Generate insights about the problem statement.

Establishing Preconditions and Postconditions

  1. Problem Name:

    • What is the problem that you are trying to solve?
  2. Method Name:

    • What is the name of the method/function that you are using to solve this problem?
  3. Parameters:

    • What are the inputs to the method?
    • What types are these parameters?
    • What do these parameters represent in the context of the problem?
  4. Preconditions:

    • Before this method is called, what must be true about the state of the program or the values of the parameters?
    • Are there any constraints on the input parameters?
    • Is there a specific state that the program or some part of it must be in?
  5. Method Functionality:

    • What is this method expected to do?
    • How does it interact with the inputs and the current state of the program?
  6. Postconditions:

    • After the method has been called and has returned, what is now true about the state of the program or the values of the parameters?
    • What does the return value represent or indicate?
    • What side effects, if any, does the method have?
  7. Error Handling:

    • How does the method respond if the preconditions are not met?
    • Does it throw an exception, return a special value, or do something else?

By answering these questions for each method in your program, you can ensure that you have a clear understanding of what each part of your code is doing and how it should behave. This will help prevent bugs and make your code easier to read and maintain.

Problem Decomposition

  1. Problem Name:

    • What is the complex problem that you are trying to solve?
  2. Problem Understanding:

    • Can you explain the problem in your own words? What are the key components and requirements?
  3. Initial Breakdown:

    • Start by identifying the major parts or stages of the problem. How can you break the problem into several broad subproblems?
  4. Subproblem Refinement:

    • For each subproblem identified, ask yourself if it can be further broken down. What are the smaller tasks that need to be done to solve each subproblem?
  5. Task Identification:

    • Within these smaller tasks, are there any that are repeated or very similar? Could these be generalized into a single, reusable task?
  6. Task Abstraction:

    • For each task you’ve identified, is it abstracted enough to be clear and reusable, but still makes sense in the context of the problem?
  7. Method Naming:

    • Can you give each task a simple, descriptive name that makes its purpose clear?
  8. Subproblem Interactions:

    • How do these subproblems or tasks interact with each other? In what order do they need to be performed? Are there any dependencies?

By going through these steps for each complex problem, you can break it down into manageable parts, making it much easier to devise an effective solution.

From Brute Force to Optimal Solution

Could you please begin by illustrating a brute force solution for this problem? After detailing and discussing the inefficiencies of the brute force approach, could you then guide us through the process of optimizing this solution? Please explain each step towards optimization, discussing the reasoning behind each decision made, and how it improves upon the previous solution. Also, could you show how these optimizations impact the time and space complexity of our solution?

Code Explanation and Design Decisions

  1. Identify the initial parameters and explain their significance in the context of the problem statement or the solution domain.

  2. Discuss the primary loop or iteration over the input data. What does each iteration represent in terms of the problem you’re trying to solve? How does the iteration advance or contribute to the solution?

  3. If there are conditions or branches within the loop, what do these conditions signify? Explain the logical reasoning behind the branching in the context of the problem’s constraints or requirements.

  4. If there are updates or modifications to parameters within the loop, clarify why these changes are necessary. How do these modifications reflect changes in the state of the solution or the constraints of the problem?

  5. Describe any invariant that’s maintained throughout the code, and explain how it helps meet the problem’s constraints or objectives.

  6. Discuss the significance of the final output in relation to the problem statement or solution domain. What does it represent and how does it satisfy the problem’s requirements?

Remember, the focus here is not to explain what the code does on a syntactic level, but to communicate the intent and rationale behind the code in the context of the problem being solved.

Coding Constructs

Consider the following piece of complex software code.

  1. What are the high-level problem-solving strategies or techniques being used by this code?

  2. If you had to explain the purpose of this code to a non-programmer, what would you say?

  3. Can you identify the logical elements or constructs used in this code, independent of any programming language?

  4. Could you describe the algorithmic approach used by this code in plain English?

  5. What are the key steps or operations this code is performing on the input data, and why?

  6. Can you identify the algorithmic patterns or strategies used by this code, irrespective of the specific programming language syntax?

Language Agnostic Coding Drills

Your mission is to deconstruct this code into the smallest possible learning units, each corresponding to a separate coding concept. Consider these concepts as unique coding drills that can be individually implemented and later assembled into the final solution.

  1. Dissect the code and identify each distinct concept it contains. Remember, this process should be language-agnostic and generally applicable to most modern programming languages.

  2. Once you’ve identified these coding concepts or drills, list them out in order of increasing difficulty. Provide a brief description of each concept and why it is classified at its particular difficulty level.

  3. Next, describe the problem-solving approach that would lead from the problem statement to the final solution. Think about how each of these coding drills contributes to the overall solution. Elucidate the step-by-step process involved in using these drills to solve the problem. Please refrain from writing any actual code; we’re focusing on understanding the process and strategy.

Targeted Drills in Python

Now that you’ve identified and ordered the coding concepts from a complex software code in the previous exercise, let’s focus on creating Python-based coding drills for each of those concepts.

  1. Begin by writing a separate piece of Python code that encapsulates each identified concept. These individual drills should illustrate how to implement each concept in Python. Please ensure that these are suitable even for those with a basic understanding of Python.

  2. In addition to the general concepts, identify and write coding drills for any problem-specific concepts that might be needed to create a solution. Describe why these drills are essential for our problem.

  3. Once all drills have been coded, describe how these pieces can be integrated together in the right order to solve the initial problem. Each drill should contribute to building up to the final solution.

Remember, the goal is to not only to write these drills but also to ensure that they can be cohesively assembled into one comprehensive solution.

Q&A

What are the reasons for making these mistakes in the given code?

Similar Problems

Can you suggest 10 problems from LeetCode that require similar problem-solving strategies or use similar underlying concepts as the problem we’ve just solved? These problems can be from any domain or topic, but they should involve similar steps or techniques in the solution process. Also, please briefly explain why you consider each of these problems to be related to our original problem.