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:
 Transform the ranges into intervals for each tap.
 Iterate through the intervals and keep track of the current reach, the next reach, and the number of taps needed.
 Return the minimum number of taps or 1 if the garden cannot be watered.
Here’s the code:


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] (0indexed) means the tap can water the area from [iranges[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:
435. Nonoverlapping Intervals: This problem asks to find the minimum number of intervals you need to remove to make the rest of the intervals nonoverlapping. It also involves sorting and greedy strategy.
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.
406. Queue Reconstruction by Height: This problem can help you understand how to sort and manage intervals or ranges.
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.
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.
56. Merge Intervals: In this problem, you need to merge overlapping intervals. It can help you get used to handling intervals and ranges.
57. Insert Interval: This problem asks you to insert a new interval into a list of nonoverlapping intervals. It requires the handling of ranges and intervals.
253. Meeting Rooms II: This problem asks for the minimum number of meeting rooms given an array of meeting time intervals.
986. Interval List Intersections: This problem requires you to find all the intersections of two interval lists.
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 arrayranges
of lengthn+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, wellsuited 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 straightline 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]]
, wherei
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 haven + 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
 Set of Points: A set
P
representing discrete positions from 0 ton
.  Set of Ranges: A set
R
where each elementr_i
is a tuple(start, end)
indicating the interval that a specific “resource” (in our context, a tap) can cover.  Objective Function: A function
f()
that minimizes the count of resources required to cover all points inP
.
Constraints
P = n + 1
(Cardinality of setP
) Each range
r_i
is derived from an array elementranges[i]
, such thatstart = i  ranges[i]
andend = i + ranges[i]
. start
andend
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 problemsolving 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
 Garden: Think of it as a long, straight road that needs to be wet from start to finish.
 Taps: These are like sprinklers placed along this road at every step.
 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
 Starting Point: You begin at the start of the road (point 0).
 Choices: At each step, you can decide to turn on a sprinkler.
 Coverage: Each sprinkler you turn on will wet a section of the road, both ahead and behind it.
 Overlap: Sometimes, the area one sprinkler wets will overlap with another. That’s okay; it just gives you choices.
 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 widthâ€”some can paint a large area in one go, while others can only paint a small patch.
 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.
 Filling Gaps: If there are unpainted sections, you then select another friend who can paint the farthest unpainted area.
 Overlap: It’s okay if your friends paint over some of the same sections; you’re more concerned about missing spots.
 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 floatingpoint 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 yettobewatered 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. IntegerBased Calculations
All the points and ranges are integers, simplifying the calculations and data manipulation. This removes the complexity of dealing with floatingpoint 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. NonNegative 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.
9. Binary Search
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]
 Initialize:
furthest = 0
,taps = 0
 Identify Reachable Points:
reachable = [1, 3, 3, 3, 6, 6, 6, 7]
 Sort by Reachability: Sorted indices: [1, 4, 2, 3, 5, 6, 7, 0]
 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.
 Verify and Return:
furthest = 6
is not beyond the garden end. Hence return1
.
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 ProblemSolving 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
Read Input Data: Read the garden length
n
and tapranges
array.Initialize Variables:
furthest = 0
taps = 0
reachable = []
Preprocessing:
 Calculate the furthest reachable point for each tap and store in the
reachable
array.
 Calculate the furthest reachable point for each tap and store in the
Sort Taps:
 Sort tap indices based on their furthest reachable point in descending order.
Main Loop:
 Set a variable
current_point = 0
 Loop while
current_point <= n
: Initialize a variable
next_furthest = current_point
 While there are taps that can cover
current_point
, find the one that extends furthest and updatenext_furthest
.  If
next_furthest == current_point
, return1
.  Update
current_point = next_furthest + 1
 Increment
taps
 Initialize a variable
 Set a variable
Return Result:
 Return the value of
taps
.
 Return the value of
Granular, Actionable Steps
Read Input Data:
 Use standard input methods to read
n
andranges
.
 Use standard input methods to read
Initialize Variables:
furthest = 0
taps = 0
reachable = [0] * (n + 1)
Preprocessing:
 Loop through
ranges
to populatereachable
array:reachable[i] = i + ranges[i]
.
 Loop through
Sort Taps:
 Use any efficient sorting algorithm to sort the tap indices based on the values in
reachable
.
 Use any efficient sorting algorithm to sort the tap indices based on the values in
Main Loop:
 Start from
current_point = 0
.  Use a while loop to iterate until
current_point <= n
.
 Start from
Parts That Can Be Solved Independently
 Reading the Input: This is an independent task.
 Initializing Variables and Preprocessing: Can be done once input is read.
 Sorting Taps: This can also be done independently once
reachable
is computed.  Main Loop: Depends on previous steps but is itself a selfcontained task.
Repeatable Patterns
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.
Incrementing Counters: The operation of moving the
current_point
and incrementingtaps
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.


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.


Step 5: Main Loop to Cover Garden
Loop while furthest < n
. In each loop, greedily choose the tap that extends the furthest beyond furthest
.


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 thereachable
array but won’t fundamentally change the approach.Variation in
ranges
: More taps with higher ranges means fewer taps will be needed. Zeroes inranges
make certain points unreachable, affecting the result.
Example Case: n = 5, ranges = [3, 4, 1, 1, 0, 0]
 Read Input:
n = 5, ranges = [3, 4, 1, 1, 0, 0]
 Initialize:
furthest = 0, taps = 0
 Preprocessing:
reachable = [3, 5, 3, 4, 4, 5]
 Sort Taps: Sorted indices based on reachability:
[1, 5, 3, 4, 2, 0]
 Main Loop:
current_point = 0
: Open tap at 1,furthest = 5, taps = 1
 Return: Since
furthest = 5
is beyondn
, we returntaps = 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
 Onedimensional garden: The garden exists on a line, suggesting a simpler problem than if it were 2D or 3D.
 Taps at
[0, 1, ..., n]
: The taps are equally spaced, a crucial simplification.  Ranges of taps: The different watering ranges suggest some taps are more valuable than others.
 Minimum number of taps: The goal is to minimize something, often a hint that a greedy approach may be useful.
Direction from Cues
 We can think of each tap as covering a range on this line, effectively turning the problem into an interval covering problem.
 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 onedimensional, 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.


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 stepbystep approach we initially outlined to arrive at this solution.
From Brute Force to Optimal Solution
BruteForce 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


Inefficiencies
 Time Complexity: Generating all combinations is (O(2^n)), where (n) is the garden length.
 Space Complexity: Storing the combinations requires significant memory, (O(2^n)).
Optimizing the Solution
Step 1: Observe Overlapping Intervals
In the bruteforce 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


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 bruteforce solution.
Coding Constructs
HighLevel Strategies:
 Preprocessing: Precompute the furthest point each tap can water.
 Greedy Algorithm: Always choose the tap that waters the most unwatered area next.
NonProgrammer 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.
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.
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.
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.
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
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.
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.
ProblemSolving 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
Variable Initialization
1 2
# Initialize a variable to zero min_taps = 0
Array Manipulation
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# Initialize an array with zeroes arr = [0] * 5 # Change the value at index 2 arr[2] = 3
Looping
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# Loop through an array and print each element for i in range(5): print(i)
Conditional Checks
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# Check if a condition is met if min_taps > 0: print("We need some taps.")
Two Pointers
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# Initialize pointers left = 0 right = 0 # Move the pointers left += 1 right += 2
Greedy Choice
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# Choose the greater of two numbers a = 3 b = 5 max_val = max(a, b)
ProblemSpecific Drills
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
Initialize Variables: Use the variable initialization drill to set up your initial variables like
min_taps
.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.Main Loop: Begin the main loop to iterate over the precomputed array. Use conditional checks to validate your steps.
Two Pointers: Initialize two pointers,
left
andright
, to keep track of your current position and furthest reachable point, respectively.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
.Conditional Checks: Before and after the loop, use conditional checks to return
1
if the garden cannot be watered.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, wellstructured solution to the problem.