#401

Easy

auto_awesomeBacktracking search with pruning

LeetCode Problem Workspace

Binary Watch

Binary Watch involves generating possible times on a binary watch given a number of turned-on LEDs.

Problem Statement

A binary watch consists of 4 LEDs for hours (0-11) and 6 LEDs for minutes (0-59). Each LED represents either a 0 or 1, with the least significant bit on the right. Given an integer turnedOn, which represents the number of LEDs that are currently on, the goal is to determine all possible times the watch could display.

The watch's hour component must not have a leading zero. The task is to return all valid times that could represent the watch, given the constraint on the number of turned-on LEDs. The output can be in any order, and the number of LEDs on should be between 0 and 10, inclusive.

Examples

Example 1

Input: turnedOn = 1

Output: ["0:01","0:02","0:04","0:08","0:16","0:32","1:00","2:00","4:00","8:00"]

Example details omitted.

Example 2

Input: turnedOn = 9

Output: []

Example details omitted.

Constraints

0 <= turnedOn <= 10

Solution Approach

Backtracking Search with Pruning

This problem can be solved through backtracking search, where we recursively explore the possibilities for hours and minutes. Pruning is essential to limit the exploration to valid times by comparing bit counts and avoiding invalid configurations, such as more than 4 LEDs for hours or more than 6 for minutes.

Bit Manipulation

To generate times efficiently, bit manipulation is useful. We can represent the watch's time using binary numbers, where each bit corresponds to an LED. By counting the number of set bits (turned-on LEDs), we can directly check for valid hour and minute combinations, avoiding unnecessary trials.

Valid Time Construction

To ensure the solution is valid, we need to construct the time by combining hours and minutes based on the turned-on LEDs. The hours range from 0-11, and minutes range from 0-59. Valid combinations should respect these ranges, and leading zeros in hours should be avoided.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity depends on the specific approach, but with backtracking and bit manipulation, it generally involves checking combinations of 10 LEDs, which leads to a reasonable solution space. The space complexity depends on storing the valid combinations of hours and minutes, but it remains manageable due to pruning and the constrained problem space.

What Interviewers Usually Probe

Look for an understanding of backtracking and how it can be applied to the problem.
Check for familiarity with bit manipulation and its role in efficiently solving problems with binary representations.
Ensure the candidate is aware of edge cases, such as when turnedOn is 9 (impossible configuration).

Common Pitfalls or Variants

Common pitfalls

Failing to handle edge cases like impossible configurations where turnedOn is too large to form a valid time.
Not properly handling leading zeros in the hour component.
Overcomplicating the solution by not utilizing bit manipulation and pruning effectively, leading to inefficiency.

Follow-up variants

Consider optimizing the solution for different input ranges, such as turnedOn being larger than 10.
Modify the problem to generate only times that are specific to a given hour or minute range.
Adapt the problem to include time intervals or add additional constraints, such as ensuring the time is within a certain range.

FAQ

What is the main approach to solving the Binary Watch problem?

The main approach involves using backtracking with pruning and bit manipulation to efficiently generate all valid times with the specified number of LEDs turned on.

How does bit manipulation help in solving this problem?

Bit manipulation allows you to efficiently represent the watch time, count the set bits (turned-on LEDs), and directly check for valid hour and minute combinations.

What are common mistakes when solving the Binary Watch problem?

Common mistakes include failing to handle edge cases, like impossible configurations, and not properly managing the leading zeros in the hour component.

How do we handle edge cases in the Binary Watch problem?

Edge cases, like when turnedOn is too large for a valid time, are handled by pruning the search space and skipping invalid combinations early in the backtracking process.

How can GhostInterview assist with the Binary Watch problem?

GhostInterview assists by providing solutions to optimize the backtracking process, using bit manipulation for efficient counting, and helping with common pitfalls like handling edge cases.

terminal

Solution

Solution 1: Enumerate Combinations

The problem can be converted to finding all possible combinations of $i \in [0, 12)$ and $j \in [0, 60)$.

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class Solution:
    def readBinaryWatch(self, turnedOn: int) -> List[str]:
        return [
            '{:d}:{:02d}'.format(i, j)
            for i in range(12)
            for j in range(60)
            if (bin(i) + bin(j)).count('1') == turnedOn
        ]

Solution 2: Binary Enumeration

We can use $10$ binary bits to represent the watch, where the first $4$ bits represent hours and the last $6$ bits represent minutes. Enumerate each number in $[0, 2^{10})$, check if the number of 1s in its binary representation equals $\textit{turnedOn}$, and if so, convert it to time format and add it to the answer.

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class Solution:
    def readBinaryWatch(self, turnedOn: int) -> List[str]:
        return [
            '{:d}:{:02d}'.format(i, j)
            for i in range(12)
            for j in range(60)
            if (bin(i) + bin(j)).count('1') == turnedOn
        ]

Continue Practicing

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