Minimum Number of Taps to Open to Water a Garden

Minimum Number of Taps to Open to Water a Garden

1. What is this problem about?

Imagine a garden of length n that needs to be watered. You have n+1 taps located at positions 0, 1, ..., n. Each tap has a specific range r. If you open a tap at position i, it waters the area from i - r to i + r. The goal is to find the minimum number of taps you need to open to cover the entire garden from 0 to n.

2. Why is this asked in interviews?

This is a classic "interval coverage" problem asked by top-tier companies like Google, Amazon, and Uber. It tests a candidate's ability to transform a physical description into a mathematical model (intervals). It also evaluates proficiency in Greedy algorithms and Dynamic Programming, specifically how to select the best overlapping interval to maximize reach.

3. Algorithmic pattern used

The most efficient approach for the "Minimum Number of Taps to Open to Water a Garden coding problem" is the Greedy pattern (specifically the "Jump Game II" variant). You convert each tap's range into an interval [start, end]. Then, for each starting position, you track the furthest possible end position. By iterating through the garden and always "jumping" to the furthest reachable point, you can find the minimum taps in O(n) time.

4. Example explanation

Garden length n = 5. Tap ranges: [3, 4, 1, 1, 0, 0].

• Tap 0 (range 3) covers [-3, 3]. Effective: [0, 3].
• Tap 1 (range 4) covers [-3, 5]. Effective: [0, 5].
• Tap 2 (range 1) covers [1, 3]. If we open Tap 1, the entire garden [0, 5] is covered. Minimum taps = 1. The algorithm would compare Tap 0's reach (3) and Tap 1's reach (5) and choose Tap 1 because it's the furthest.

5. Common mistakes candidates make

A common error is not correctly handling the boundaries—forgetting that a tap at i with range r covers max(0, i-r) to min(n, i+r). Another mistake is attempting a sorting-based greedy approach (like the interval scheduling problem) without realizing that simple array pre-processing can achieve the same result in linear time. Failing to detect when the garden is impossible to water is also a frequent oversight.

6. Interview preparation tip

Whenever you see a problem about covering a range with intervals, think about how to represent the "furthest reach" for every point. If you can simplify the problem into a 1D "Jump Game," you'll often find a much more efficient solution than standard interval merging.