Minimum Number of Taps to Open to Water a Garden

To find the minimum number of taps to water the whole garden, we can use a greedy approach. The idea is to find the farthest reach at each step, considering the area that needs to be watered and the range of each tap. Here’s how you can do it:

  1. Transform the ranges into intervals for each tap.
  2. Iterate through the intervals and keep track of the current reach, the next reach, and the number of taps needed.
  3. Return the minimum number of taps or -1 if the garden cannot be watered.

Here’s the code:

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from typing import List

class Solution:
    def minTaps(self, n: int, ranges: List[int]) -> int:
        # Initialize intervals for each tap
        intervals = []
        for i, r in enumerate(ranges):
            intervals.append((i - r, i + r))

        # Sort intervals by the starting point
        intervals.sort()

        # Initialize variables to keep track of the current reach, next reach, and taps needed
        current_reach, next_reach, taps = 0, 0, 0
        i = 0

        # Iterate through the intervals
        while current_reach < n:
            # Check each interval to find the next reach
            while i < len(intervals) and intervals[i][0] <= current_reach:
                next_reach = max(next_reach, intervals[i][1])
                i += 1

            # If no next reach is found, the garden cannot be watered
            if current_reach == next_reach:
                return -1

            # Update the current reach and increase the number of taps needed
            current_reach = next_reach
            taps += 1

        return taps

This code takes the input n and ranges and returns the minimum number of taps needed to water the whole garden. If the garden cannot be watered, it returns -1, as required by the problem statement.

10 Prerequisite LeetCode Problems

This is about finding the minimum number of taps to open to water the entire garden, where the garden is represented as an integer array ranges of length n+1, where each ranges[i] (0-indexed) means the tap can water the area from [i-ranges[i]] to [i+ranges[i]]. The problem falls into the category of interval scheduling, which is a classic problem in computer science.

Here are some simpler problems to prepare for this:

  1. 435. Non-overlapping Intervals: This problem asks to find the minimum number of intervals you need to remove to make the rest of the intervals non-overlapping. It also involves sorting and greedy strategy.

  2. 452. Minimum Number of Arrows to Burst Balloons: This problem requires you to find the minimum number of arrows to burst all balloons. The intervals represent the balloons, which is quite similar to watering the garden.

  3. 406. Queue Reconstruction by Height: This problem can help you understand how to sort and manage intervals or ranges.

  4. 1024. Video Stitching: In this problem, you’re asked to stitch videos to reach up to the target time, which is very similar to watering the whole garden.

  5. 763. Partition Labels: This problem is about partitioning a string such that each letter appears in at most one partition. It involves managing and tracking ranges.

  6. 56. Merge Intervals: In this problem, you need to merge overlapping intervals. It can help you get used to handling intervals and ranges.

  7. 57. Insert Interval: This problem asks you to insert a new interval into a list of non-overlapping intervals. It requires the handling of ranges and intervals.

  8. 253. Meeting Rooms II: This problem asks for the minimum number of meeting rooms given an array of meeting time intervals.

  9. 986. Interval List Intersections: This problem requires you to find all the intersections of two interval lists.

  10. 1288. Remove Covered Intervals: In this problem, you’re asked to remove all intervals that are covered by another interval.

Problem Classification

The problem statement falls under the category “Greedy Algorithms” or “Interval Scheduling.”

‘What’ Components

  • Input: An integer n representing the length of the garden and an integer array ranges of length n+1 indicating the area each tap can water.
  • Output: A single integer representing the minimum number of taps needed to water the whole garden, or -1 if it’s impossible.
  • Constraints:
    • 1 <= n <= 10^4
    • ranges.length == n + 1
    • 0 <= ranges[i] <= 100

The problem can be further classified as an “Optimization Problem” where the objective is to find the minimum number of taps required. This problem requires both a decision (which taps to open) and an optimization (minimize the number of taps).

Given the constraints and the need for optimization, this problem aligns well with Greedy Algorithmic techniques, where we make a sequence of choices that look the best at each step.

This is an Optimization Problem within the domain of Algorithmic Problem Solving, well-suited for a Greedy Algorithmic approach.

Visual Model of the Problem

Visualizing this problem can help clarify its constraints and requirements. Here’s how you can visualize it:

1. Number Line for the Garden

Imagine a number line that represents the positions along the garden, from 0 to n.

2. Taps as Points

Place points or markers at every integer position from 0 to n on this number line. These points represent the locations of the taps.

3. Watering Ranges as Intervals

Draw intervals around each tap point to represent the area that the tap can water. The interval for a tap at position i would extend from (i - ranges[i]) to (i + ranges[i]).

For example, for ranges = [3, 4, 1, 1, 0, 0] and n = 5:

  • The tap at point 0 can water from -3 to 3. So draw an interval from -3 to 3.
  • The tap at point 1 can water from -3 to 5. Draw an interval from -3 to 5.
  • Continue this for each tap.

4. Overlapping Intervals

Pay attention to where the intervals overlap. Overlaps may represent areas where you have a choice of taps to use.

5. Gaps

Look for any gaps between intervals. Gaps mean that you can’t water the whole garden, so the answer would be -1.

Here’s a simple illustration based on the first example:

Number Line:  -3 -2 -1  0  1  2  3  4  5
                  |----------------|
                      |------------------|
                            |----|
                                  |----|
                                        ||  
                                              ||

Taps:                  0    1    2    3    4    5

In this case, you can see that just turning on the tap at point 1 can cover the entire garden from 0 to 5.

By visualizing the problem in this manner, you get a clearer idea of which taps are critical for covering the whole garden and which taps overlap, providing you with options. This visualization sets the stage for a greedy algorithmic approach to solve the problem.

Problem Restatement

You have a straight-line garden that starts at point 0 and ends at point n. There are n + 1 water taps placed at every point from 0 to n along this line. Each tap has a certain range it can water, given by an array called ranges.

The aim is to find the smallest number of these taps you need to turn on so that every point in the garden gets watered. If you can’t water the entire garden with the given taps, return -1.

Requirements:

  • Each tap can water an area defined as [i - ranges[i], i + ranges[i]], where i is the tap’s position.
  • You need to water the garden completely, from point 0 to point n.

Constraints:

  • The garden length n will be at least 1 and at most 10,000.
  • The ranges array will have n + 1 elements.
  • Each element in ranges will be between 0 and 100.

The goal is to find the least number of taps to turn on for full garden coverage, or conclude that it’s not possible.

Abstract Representation of the Problem

An abstract representation often strips away the context to focus on the core mechanics and relationships. Let’s do that for this problem.

Elements

  1. Set of Points: A set P representing discrete positions from 0 to n.
  2. Set of Ranges: A set R where each element r_i is a tuple (start, end) indicating the interval that a specific “resource” (in our context, a tap) can cover.
  3. Objective Function: A function f() that minimizes the count of resources required to cover all points in P.

Constraints

  1. |P| = n + 1 (Cardinality of set P)
  2. Each range r_i is derived from an array element ranges[i], such that start = i - ranges[i] and end = i + ranges[i].
  3. start and end must be integers and fall within the global range [0, n].

Goal

Find the minimal subset of R such that the union of its intervals covers the entire set P. If impossible, return a special flag (in our case, -1).

By abstracting the problem in this manner, we separate the ‘what’ from the ‘how,’ making it easier to focus on the core algorithmic challenge. This also helps in identifying the mathematical or algorithmic techniques that could be applied for solving it.

Terminology

1. Interval

Definition: A range of numbers between a starting and an ending point.
Role: Each tap’s reach is represented as an interval on the number line. Understanding how intervals overlap, merge, or leave gaps is essential for solving this problem.

2. Greedy Algorithm

Definition: An algorithmic paradigm that follows the problem-solving heuristic of making the locally optimal choice at each stage.
Role: A greedy approach can be used to solve this problem by always choosing the tap that waters the furthest point in the garden that hasn’t been watered yet.

3. Optimization Problem

Definition: A type of problem where the goal is to find the best solution from a set of possible solutions, often subject to certain constraints.
Role: This problem is an optimization problem where the aim is to minimize the number of taps needed to cover the entire garden.

4. Union of Intervals

Definition: The combined coverage of multiple intervals.
Role: The union of intervals defines the areas that get watered when multiple taps are opened. To solve the problem, the union of the selected intervals must cover the entire garden.

5. Subset

Definition: A set that contains only elements that are also in another set.
Role: We’re interested in finding the smallest subset of taps (intervals) whose union covers the entire garden.

Understanding these specialized terms and concepts will enable a clearer comprehension of both the problem and any proposed solution. They serve as the building blocks for formulating the algorithm to solve the problem.

Problem Simplification and Explanation

Key Concepts

  1. Garden: Think of it as a long, straight road that needs to be wet from start to finish.
  2. Taps: These are like sprinklers placed along this road at every step.
  3. Water Range: Each sprinkler can wet the road only up to a certain distance on both sides.

Your goal is to wet the entire road using the fewest sprinklers possible.

Interactions

  1. Starting Point: You begin at the start of the road (point 0).
  2. Choices: At each step, you can decide to turn on a sprinkler.
  3. Coverage: Each sprinkler you turn on will wet a section of the road, both ahead and behind it.
  4. Overlap: Sometimes, the area one sprinkler wets will overlap with another. That’s okay; it just gives you choices.
  5. Gaps: If there’s a part of the road that no sprinkler can reach, you can’t wet the entire road.

Metaphor

Imagine you are painting a long fence, and you have a bunch of friends to help you. Each friend has a paint roller with a different width—some can paint a large area in one go, while others can only paint a small patch.

  1. Choosing Friends: You want to finish painting with the least number of friends. So, you start by picking the friend who can paint the largest area first.
  2. Filling Gaps: If there are unpainted sections, you then select another friend who can paint the farthest unpainted area.
  3. Overlap: It’s okay if your friends paint over some of the same sections; you’re more concerned about missing spots.
  4. Impossible Task: If you find an area that none of your friends’ paint rollers can reach, you realize you can’t complete the job.

Your goal is to finish painting the fence using the fewest friends possible, just like you want to water the entire garden with the fewest taps.

Constraints

Understanding the characteristics and constraints can lead to a more efficient algorithm. Here are some specifics to consider:

1. Sorted Intervals

The tap locations are already sorted from 0 to n. This is beneficial when considering greedy or dynamic programming approaches, as you don’t need to sort the taps yourself.

2. Small and Fixed Maximum Range

The maximum range of any tap is 100, which is relatively small. This can be exploited for optimizing the search for the next best tap, especially if you use data structures like arrays to quickly look up ranges.

3. Discrete Points

The garden points and tap ranges are all integers. This eliminates the need for any complex floating-point arithmetic or approximations. It simplifies the task of finding exact coverages.

4. Limited Garden Length

The maximum garden length n is 10,000, which is not excessively large. Algorithms with time complexity like O(n log n) or even O(n^2) could be acceptable, though of course, more efficient is better.

5. Goal is Minimization

You are trying to minimize the number of taps, not maximize coverage. This insight lends itself well to a greedy approach where you pick the “best” option available at each step.

6. Possible Overlaps

The taps’ ranges can overlap, providing you choices. In a greedy approach, you would pick the tap that extends furthest into the yet-to-be-watered region.

7. No Negative Ranges

All ranges are zero or positive. This means a tap will always at least cover the point where it’s located. You don’t need to consider cases where the tap range could be a negative number.

8. Starting and Ending Points

The problem clearly states that taps are also located at the starting point 0 and the ending point n. This means that the coverage should begin and end with these points in mind.

By identifying these patterns and numerical ranges, you can tailor your algorithm to be as efficient as possible, making good use of the constraints and conditions provided.

The key insights from analyzing the constraints are as follows:

1. Bounded Complexity

The garden length n is capped at 10,000 and the maximum range for any tap is 100. This suggests that solutions with a time complexity like O(n log n) or even O(n^2) may be acceptable. However, the bounded constraints make it possible to aim for an even more efficient solution.

2. Integer-Based Calculations

All the points and ranges are integers, simplifying the calculations and data manipulation. This removes the complexity of dealing with floating-point arithmetic.

3. Sorted Positions

Taps are sorted in their positions from 0 to n, aiding in potential greedy or dynamic programming approaches. You don’t need extra time to sort them.

4. Non-Negative Ranges

All tap ranges are zero or positive. This eliminates the need to check for negative ranges, which simplifies the algorithm further.

5. Minimization Goal

The goal is to minimize the number of taps needed to cover the entire garden. This is a strong hint towards using a greedy approach, where you aim for the ’local maximum’ coverage at each step.

6. Defined Edge Points

Taps are available at the starting point (0) and ending point (n). These fixed points can be exploited to reduce edge cases in your algorithm.

By considering these insights, you can design an algorithm that specifically takes advantage of these constraints, thereby improving its efficiency.

Case Analysis

Test Case 1: Smallest Garden Size (Edge Case)

  • Input: n = 1, ranges = [0, 0]
  • Output: -1
  • Analysis: In this case, neither tap can water any part of the garden other than their own position. The garden cannot be watered entirely, so the output is -1.

Test Case 2: All Taps Have Zero Range (Boundary Condition)

  • Input: n = 4, ranges = [0, 0, 0, 0, 0]
  • Output: -1
  • Analysis: Again, each tap only waters its own position. Since none have a range that extends beyond themselves, the garden cannot be completely watered.

Test Case 3: Single Tap Can Cover Entire Garden (Best Case)

  • Input: n = 5, ranges = [0, 0, 0, 3, 0, 0]
  • Output: 1
  • Analysis: Here, the tap at position 3 can cover the whole garden from [0, 5]. Thus, only one tap is needed.

Test Case 4: Multiple Taps Needed, No Overlaps

  • Input: n = 5, ranges = [1, 1, 1, 1, 1, 1]
  • Output: 3
  • Analysis: Each tap covers two adjacent points (itself and one neighbor). You will need three taps at positions 1, 3, and 5 to cover the entire garden.

Test Case 5: Multiple Taps Needed, With Overlaps

  • Input: n = 4, ranges = [2, 1, 2, 1, 0]
  • Output: 1
  • Analysis: Here, the tap at position 0 can cover from [-2, 2], and the tap at position 2 can cover [0, 4]. Just opening the tap at position 2 can water the whole garden from [0, 4].

Test Case 6: Multiple Taps Needed, Partial Overlap (Pitfall)

  • Input: n = 7, ranges = [1, 2, 1, 0, 2, 1, 0, 0]
  • Output: 2
  • Analysis: Here, two taps at positions 1 and 4 can water the garden completely. While the tap at position 1 waters [-1, 3], the tap at 4 waters [2, 6]. It may look tempting to use the tap at 2, but it would require a third tap to cover the end.

These test cases explore the minimum and maximum boundaries of the input space, cases with and without overlaps, and scenarios where not all taps need to be open. They help us understand the range of conditions any proposed algorithm needs to handle effectively.

Identification of Applicable Theoretical Concepts

1. Greedy Algorithm

Given that the problem aims for a minimization, a Greedy Algorithm is suitable here. At each step, choose the tap that provides the furthest coverage not yet achieved.

2. Dynamic Programming

Dynamic programming could also be a good fit, especially for finding the minimum taps required to cover each point in the garden, although it may not be as efficient as a greedy approach.

3. Intervals and Segment Coverage

The problem is fundamentally about covering a segment (the garden) with other smaller segments (ranges of taps). This relates to the Interval Covering problem in computational geometry and scheduling.

4. Sliding Window

A sliding window approach can help you track the area that a particular set of taps can cover as you move along the garden.

5. Range Query Data Structures

Data structures like Segment Trees or Fenwick Trees could be used for range queries, but in this problem, they may be overkill given the constraints. However, knowing these exist can be helpful for similar but more complex problems.

6. Sorting Algorithms

While the taps are already sorted by their location, additional sorting based on their range can provide a performance benefit in certain algorithmic approaches.

7. Array Manipulation

Because all values are integers and the ranges are relatively small, array manipulation could be used to mark the covered points efficiently.

8. Graph Theory

You could theoretically model the problem as a flow network where taps are nodes and edges indicate possible water flow between segments, though this may be a more complicated way to approach this particular problem.

In some cases, especially if the tap ranges were not sorted, a binary search could be employed to find the next best tap to open for a specific garden segment.

By employing these mathematical and algorithmic concepts, you could create a more efficient and simplified solution to the problem.

Problem Breakdown and Solution Methodology

To solve this problem, let’s use the Greedy Algorithm approach. The steps can be broken down as follows:

Step 1: Initialize Variables

Initialize variables to keep track of the furthest point watered (furthest) and the number of taps opened (taps). Both start at zero.

Step 2: Identify Reachable Points

Create a variable to store the furthest point that can be reached from each tap if opened. We’ll call this array reachable.

Step 3: Sort by Reachability

Sort the taps in descending order based on how far they can water. This will help us choose the most promising tap at each step.

Step 4: Cover the Garden

Iterate through the garden, greedily choosing the tap that extends the furthest beyond furthest. Update furthest and increment taps every time you open a tap.

Step 5: Verify and Return

Once the loop is finished, check if furthest is beyond the end of the garden (n). If it is, return taps. Otherwise, return -1 as it’s impossible to cover the whole garden.

Metaphor

Imagine you are trying to illuminate a long hallway with flashlights. You’d want to place a flashlight that covers the darkest portion of the hallway as well as extends its light the furthest down the hallway.

How Parameters Affect the Solution

  • Increase in n: As the length of the garden grows, the time complexity remains largely the same due to our sorting and greedy approach, but you might need more taps.
  • Change in ranges: If all taps have higher ranges, you’ll likely need fewer taps. If they have lower ranges, you’ll need more.

Example Case: n = 7, ranges = [1, 2, 1, 0, 2, 1, 0, 0]

  1. Initialize: furthest = 0, taps = 0
  2. Identify Reachable Points: reachable = [1, 3, 3, 3, 6, 6, 6, 7]
  3. Sort by Reachability: Sorted indices: [1, 4, 2, 3, 5, 6, 7, 0]
  4. Cover the Garden:
    • furthest = 0: Open tap at index 1 (covers up to 3), furthest = 3, taps = 1
    • furthest = 3: Open tap at index 4 (covers up to 6), furthest = 6, taps = 2
    • furthest = 6: No need to open more taps.
  5. Verify and Return: furthest = 6 is not beyond the garden end. Hence return -1.

By following this approach, we can solve the problem efficiently while keeping track of the minimum number of taps needed to cover the entire garden.

Inference of Problem-Solving Approach from the Problem Statement

The problem involves finding a minimum number of elements (taps) that collectively meet a certain condition (covering the entire garden). Greedy algorithms are often effective at solving optimization problems where you’re looking to minimize or maximize a particular parameter. Here, the goal is to minimize the number of taps.

The problem lends itself to a greedy approach because each step (selecting a tap to open) is both local and independent, in the sense that choosing the best tap at each point doesn’t prevent us from making the best choice at the next point. The ‘best’ tap at each step can be defined as the one that waters the furthest point that hasn’t been watered yet. By always picking the ‘best’ available choice, we’re aiming to minimize the total number of taps needed, which makes the greedy approach appropriate for solving this problem.

Stepwise Refinement

  1. Read Input Data: Read the garden length n and tap ranges array.

  2. Initialize Variables:

    • furthest = 0
    • taps = 0
    • reachable = []

  3. Preprocessing:

    • Calculate the furthest reachable point for each tap and store in the reachable array.

  4. Sort Taps:

    • Sort tap indices based on their furthest reachable point in descending order.

  5. Main Loop:

    • Set a variable current_point = 0
    • Loop while current_point <= n:
      1. Initialize a variable next_furthest = current_point
      2. While there are taps that can cover current_point, find the one that extends furthest and update next_furthest.
      3. If next_furthest == current_point, return -1.
      4. Update current_point = next_furthest + 1
      5. Increment taps

  6. Return Result:

    • Return the value of taps.

Granular, Actionable Steps

  1. Read Input Data:

    • Use standard input methods to read n and ranges.

  2. Initialize Variables:

    • furthest = 0
    • taps = 0
    • reachable = [0] * (n + 1)

  3. Preprocessing:

    • Loop through ranges to populate reachable array: reachable[i] = i + ranges[i].

  4. Sort Taps:

    • Use any efficient sorting algorithm to sort the tap indices based on the values in reachable.

  5. Main Loop:

    • Start from current_point = 0.
    • Use a while loop to iterate until current_point <= n.

Parts That Can Be Solved Independently

  1. Reading the Input: This is an independent task.
  2. Initializing Variables and Preprocessing: Can be done once input is read.
  3. Sorting Taps: This can also be done independently once reachable is computed.
  4. Main Loop: Depends on previous steps but is itself a self-contained task.

Repeatable Patterns

  1. Finding the Next Furthest Point: In the main loop, we repeatedly look for the tap that can water the furthest point beyond the current furthest point. This is a repeatable pattern that can be isolated into a function.

  2. Incrementing Counters: The operation of moving the current_point and incrementing taps is a repeated action within the main loop.

Solution Approach and Analysis

Step 1: Read Input Data

Read the garden length n and the ranges array. This is the raw information we need to process.

Step 2: Initialize Variables

  • furthest = 0: Furthest point in the garden watered so far.
  • taps = 0: Count of taps used.

Step 3: Preprocessing

Calculate the furthest point each tap can water if opened. Store these in a reachable array.

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reachable = [0] * (n + 1)
for i in range(n + 1):
    left = max(0, i - ranges[i])
    right = min(n, i + ranges[i])
    reachable[left] = max(reachable[left], right)

Step 4: Sort Taps by Reachability

Sort tap indices based on their reachable values in descending order. We’ll use this sorted list to make the greedy choice at each step.

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sorted_indices = sorted(range(n + 1), key=lambda x: reachable[x], reverse=True)

Step 5: Main Loop to Cover Garden

Loop while furthest < n. In each loop, greedily choose the tap that extends the furthest beyond furthest.

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current_point = 0
while current_point <= n:
    next_furthest = current_point
    for i in sorted_indices:
        if i <= current_point and reachable[i] > next_furthest:
            next_furthest = reachable[i]
    if next_furthest == current_point:
        return -1  # Garden can't be fully covered
    current_point = next_furthest + 1
    taps += 1

Step 6: Return Result

If furthest >= n, the entire garden is covered. Return the number of taps opened, stored in taps. Else, return -1.

How Parameters Affect the Solution

  • Increase in n: Increasing the garden’s length will require a reassessment of the reachable array but won’t fundamentally change the approach.

  • Variation in ranges: More taps with higher ranges means fewer taps will be needed. Zeroes in ranges make certain points unreachable, affecting the result.

Example Case: n = 5, ranges = [3, 4, 1, 1, 0, 0]

  1. Read Input: n = 5, ranges = [3, 4, 1, 1, 0, 0]
  2. Initialize: furthest = 0, taps = 0
  3. Preprocessing: reachable = [3, 5, 3, 4, 4, 5]
  4. Sort Taps: Sorted indices based on reachability: [1, 5, 3, 4, 2, 0]
  5. Main Loop:
    • current_point = 0: Open tap at 1, furthest = 5, taps = 1
  6. Return: Since furthest = 5 is beyond n, we return taps = 1.

By breaking down the problem into these steps, we can solve it in a structured manner. Each step contributes to reaching the final solution, making sure the garden is fully watered with a minimal number of taps.

Thought Process

Cues in the Problem Statement

  1. One-dimensional garden: The garden exists on a line, suggesting a simpler problem than if it were 2D or 3D.
  2. Taps at [0, 1, ..., n]: The taps are equally spaced, a crucial simplification.
  3. Ranges of taps: The different watering ranges suggest some taps are more valuable than others.
  4. Minimum number of taps: The goal is to minimize something, often a hint that a greedy approach may be useful.

Direction from Cues

  1. We can think of each tap as covering a range on this line, effectively turning the problem into an interval covering problem.
  2. We need to cover the entire garden ([0, n]) using the fewest taps, which hints at a greedy approach where we always choose the tap that extends the farthest beyond our current position.

Insights

  • Because the garden is one-dimensional, each tap’s range is a simple interval on the number line.
  • The taps can be sorted based on the furthest point they can reach, aiding a greedy approach.

Code

Let’s translate these thoughts into Python3 code.

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def minTaps(n, ranges):
    # Step 2: Initialize Variables
    taps = 0

    # Step 3: Preprocessing - create reachable array
    reachable = [0] * (n + 1)
    for i, r in enumerate(ranges):
        left = max(0, i - r)
        right = min(n, i + r)
        reachable[left] = max(reachable[left], right)

    # Step 4: Sort Taps by Reachability
    # In this case, we don't have to sort but use reachable directly for a greedy choice.

    # Step 5: Main Loop to Cover Garden
    current_point = 0
    furthest = 0
    while current_point <= n:
        next_furthest = furthest
        while current_point <= n and current_point <= furthest:
            next_furthest = max(next_furthest, reachable[current_point])
            current_point += 1

        if next_furthest == furthest:
            return -1  # Garden can't be fully covered

        furthest = next_furthest
        taps += 1

    # Step 6: Return Result
    return taps

# Test the function
print(minTaps(5, [3, 4, 1, 1, 0, 0]))  # Output should be 1
print(minTaps(3, [0, 0, 0, 0]))  # Output should be -1

The function minTaps takes the garden length n and the ranges array as input and returns the minimum number of taps needed to water the entire garden. We followed the step-by-step approach we initially outlined to arrive at this solution.

From Brute Force to Optimal Solution

Brute-Force Solution

Generate all possible combinations of taps to check which combination can water the entire garden [0, n]. Pick the one with the smallest size.

Code

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from itertools import combinations

def minTaps_bruteforce(n, ranges):
    min_taps = float('inf')

    for r in range(1, n+2):
        for combo in combinations(range(n+1), r):
            garden = [0] * (n+1)
            for i in combo:
                left, right = max(0, i - ranges[i]), min(n, i + ranges[i])
                for j in range(left, right+1):
                    garden[j] = 1
            if all(garden):
                min_taps = min(min_taps, len(combo))

    return min_taps if min_taps != float('inf') else -1

Inefficiencies

  1. Time Complexity: Generating all combinations is (O(2^n)), where (n) is the garden length.
  2. Space Complexity: Storing the combinations requires significant memory, (O(2^n)).

Optimizing the Solution

Step 1: Observe Overlapping Intervals

In the brute-force approach, we checked every possible combination, many of which have overlapping intervals. We can reduce duplicate work by always selecting the tap that helps us cover the most ground.

Step 2: Sort by Reachability

Instead of sorting, preprocess the data to find the furthest point each tap can water. This will facilitate the greedy choice.

Step 3: Greedy Approach

Start at the beginning of the garden. Always pick the tap that waters the furthest point beyond the current position.

Code

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def minTaps(n, ranges):
    taps = 0
    reachable = [0] * (n + 1)
    for i, r in enumerate(ranges):
        left = max(0, i - r)
        right = min(n, i + r)
        reachable[left] = max(reachable[left], right)

    current_point = 0
    furthest = 0
    while current_point <= n:
        next_furthest = furthest
        while current_point <= n and current_point <= furthest:
            next_furthest = max(next_furthest, reachable[current_point])
            current_point += 1

        if next_furthest == furthest:
            return -1

        furthest = next_furthest
        taps += 1

    return taps

Time Complexity

The greedy approach works in (O(n)) time, where (n) is the garden length.

Space Complexity

The space complexity is (O(n)) for storing the reachable array.

By using the greedy approach, we’ve significantly optimized both the time and space complexity compared to the brute-force solution.

Coding Constructs

  1. High-Level Strategies:

    • Preprocessing: Pre-compute the furthest point each tap can water.
    • Greedy Algorithm: Always choose the tap that waters the most unwatered area next.

  2. Non-Programmer Explanation:

    • The code figures out the least number of taps needed to water a whole garden. It does so by always picking the tap that waters the most unwatered area next.

  3. Logical Elements:

    • Array: Storing the furthest point each tap can water.
    • Loop: Iterating over the garden.
    • Conditional Statements: Checking if the garden can be watered or not.

  4. Algorithm in Plain English:

    • First, make a list that shows how far each tap can water.
    • Start at the beginning of the garden.
    • Look at all the taps that can water the area from your current position to the furthest point you know a tap can water.
    • Pick the tap that waters the most unwatered area.
    • Move to the end of that watered area and repeat.
    • Keep track of how many taps you’ve used.

  5. Key Operations:

    • Precompute the furthest point each tap can reach.
    • Use two pointers to represent the current position in the garden and the furthest point that can be watered.
    • Use a loop to iterate over the garden, making greedy choices to minimize the number of taps.

  6. Algorithmic Patterns:

    • Greedy Algorithm: Making the locally optimal choice at each stage.
    • Two Pointers: Keeping track of the current and furthest reachable points in the garden.
    • Preprocessing: Transforming the given data into a more easily used form (i.e., reachable array).

Language Agnostic Coding Drills

  1. Dissecting Code into Distinct Concepts:

    • Variable Initialization: Declaring and initializing variables to store intermediate and final results.

    • Array Manipulation: Reading and writing values to an array. Here, used for preprocessing the furthest points each tap can water.

    • Looping: Iterating through elements in an array. Required for preprocessing and the main algorithm.

    • Conditional Checks: Making decisions based on some conditions. Used to check if the garden can be watered from the current position.

    • Two Pointers: Keeping track of two different elements or indexes in an array simultaneously.

    • Greedy Choice: Making a locally optimal choice in hopes it leads to a globally optimal solution.

  2. List of Concepts by Increasing Difficulty:

    • Variable Initialization: Easiest, as it’s the basis of almost any program.

    • Array Manipulation: Still basic but requires understanding of indices and value assignment.

    • Looping: A bit more advanced because it requires understanding how to iterate through elements and when to stop.

    • Conditional Checks: This adds logic to the loops and requires understanding the conditions under which different blocks of code will execute.

    • Two Pointers: More difficult because it requires tracking two variables simultaneously and understanding how and when to move them.

    • Greedy Choice: Most complex as it involves a higher level algorithmic pattern that requires both intuition and proof that the greedy choice leads to an optimal solution.

  3. Problem-Solving Approach:

    • Variable Initialization: Start by initializing variables to keep track of the minimum taps needed, current position, and furthest reachable point.

    • Array Manipulation: Precompute an array that stores the furthest point each tap can water. This acts as the basis for making decisions later.

    • Looping: Loop through the precomputed array to find the furthest reachable point from the current position.

    • Conditional Checks: Inside the loop, use conditional checks to validate if it’s possible to water the whole garden from the current position. If not, return -1.

    • Two Pointers: Use two pointers to keep track of your current position and the furthest reachable point. The pointers will help you decide which tap to open next.

    • Greedy Choice: Within the loop, make a greedy choice by selecting the tap that can water the furthest point from your current position. Update the two pointers accordingly.

By sequentially applying these coding drills, you construct the full solution that solves the given problem efficiently.

Targeted Drills in Python

General Coding Drills

  1. Variable Initialization

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    # Initialize a variable to zero
min_taps = 0

  2. Array Manipulation

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    # Initialize an array with zeroes
arr = [0] * 5
# Change the value at index 2
arr[2] = 3

  3. Looping

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    # Loop through an array and print each element
for i in range(5):
    print(i)

  4. Conditional Checks

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    # Check if a condition is met
if min_taps > 0:
    print("We need some taps.")

  5. Two Pointers

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    # Initialize pointers
left = 0
right = 0
# Move the pointers
left += 1
right += 2

  6. Greedy Choice

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    # Choose the greater of two numbers
a = 3
b = 5
max_val = max(a, b)

Problem-Specific Drills

  1. Furthest Reachable Point

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    # Given a range, find the furthest point it can reach from its index
range_val = 3
index = 2
furthest_point = index + range_val

    This is crucial for our problem because we need to know how far each tap can water the garden.

Integration Steps

  1. Initialize Variables: Use the variable initialization drill to set up your initial variables like min_taps.

  2. Precompute Ranges: Use array manipulation and looping to go through the ranges list and populate an array that stores the furthest point each tap can reach.

  3. Main Loop: Begin the main loop to iterate over the precomputed array. Use conditional checks to validate your steps.

  4. Two Pointers: Initialize two pointers, left and right, to keep track of your current position and furthest reachable point, respectively.

  5. Greedy Choices: Inside the loop, apply the greedy choice to select the tap that waters the furthest point from your current position. Update min_taps.

  6. Conditional Checks: Before and after the loop, use conditional checks to return -1 if the garden cannot be watered.

  7. Return Result: At the end of the loop, return min_taps as the minimum number of taps needed.

Each of these drills and steps contributes to building the final solution. Once you’ve practiced these drills individually, combining them should give you a solid, well-structured solution to the problem.

Q&A

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