What is the time complexity of Implement Rand10() Using Rand7()?

The optimal solution has a time complexity of O(1).

What data structures are used to solve Implement Rand10() Using Rand7()?

This problem uses Math, Rejection Sampling, Randomized, Probability and Statistics. Understanding these concepts is essential for solving this Medium problem efficiently.

What are the key insights for solving Implement Rand10() Using Rand7()?

Insight 1: Combining the results of multiple calls to `rand7()` can create a larger, more controllable sample space. Insight 2: Rejection sampling is essential to ensure that the final output is uniformly distributed between 1 and 10.

Which companies ask LeetCode 470. Implement Rand10() Using Rand7() in interviews?

This problem is frequently asked at Tencent.

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Implement Rand10() Using Rand7() - LeetCode 470 Solution

Implement Rand10() Using Rand7() - Complete Solution Guide

Implement Rand10() Using Rand7() is LeetCode problem 470, a Medium level challenge. This complete guide provides step-by-step explanations, multiple solution approaches, and optimized code in python3, java, cpp, c.

Problem Statement

Given the API rand7() that generates a uniform random integer in the range [1, 7] , write a function rand10() that generates a uniform random integer in the range [1, 10] . You can only call the API rand7() , and you shouldn't call any other API. Please do not use a language's built-in random API. Each test case will have one internal argument n , the number of times that your implemented function rand10() will be called while testing. Note that this is not an argument passed to rand10() . Examp

Detailed Explanation

The problem requires implementing a `rand10()` function that generates uniformly random integers in the range [1, 10] using only the provided `rand7()` API, which generates uniformly random integers in the range [1, 7]. We are restricted to using only the `rand7()` function and must avoid using any built-in random number generation functions of the programming language. The key challenge is to convert the randomness from a base-7 system to a base-10 system while maintaining uniformity.

Solution Approach

The solution uses rejection sampling. It first calls `rand7()` twice to create a larger sample space (1-49). These calls can be thought of as generating digits in base-7. Then, it maps this sample space to a range that is a multiple of 10 (1-40). Numbers outside that range (41-49) are rejected. Because each value from 1-49 is equally likely, and only a multiple of 10 is used, we ensure that the mapping of values to numbers in range [1, 10] is also uniform. Finally, the values 1-40 are mapped to 1-10 using the modulo operator.

Step-by-Step Algorithm

Step 1: Call `rand7()` twice and store the results in `row` and `col`.
Step 2: Calculate an index value `idx` by combining `row` and `col`: `idx = (row - 1) * 7 + col`. This transforms the two calls into a single number ranging from 1 to 49. The formula is derived from treating the result as a base-7 number where the `row` is the 'tens' digit and `col` is the 'ones' digit.
Step 3: Check if `idx` is within the acceptance range (1 to 40). If `idx > 40`, reject the sample and repeat from Step 1.
Step 4: If `idx <= 40`, map `idx` to the range [1, 10] using the modulo operator: `return 1 + (idx - 1) % 10`.

Key Insights

Insight 1: Combining the results of multiple calls to `rand7()` can create a larger, more controllable sample space.
Insight 2: Rejection sampling is essential to ensure that the final output is uniformly distributed between 1 and 10.
Insight 3: We need to create a sample space whose size is a multiple of 10 to enable uniform mapping to the [1, 10] range using the modulo operator.

Complexity Analysis

Time Complexity: O(1)

Space Complexity: O(1)

Topics

This problem involves: Math, Rejection Sampling, Randomized, Probability and Statistics.

Companies

Asked at: Tencent.

Implement Rand10() Using Rand7() | LeetCode 470

Medium

Math Rejection Sampling Randomized Probability and Statistics

Time: O(1)Space: O(1)

Solution Code

Available in:

/**
 * The rand7() API is already defined in the parent class SolBase.
 * public int rand7();
 * @return a random integer in the range 1 to 7
 */
class Solution extends SolBase {
    public int rand10() {
        while (true) {
            int row = rand7();
            int col = rand7();
            
            int idx = (row - 1) * 7 + col;
            
            if (idx <= 40) {
                return 1 + (idx - 1) % 10;
            }
        }
    }
}

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Problem Info

Description

Problem 470: Implement Rand10() Using Rand7() - Detailed explanation and solution approaches.

Topics

Math Rejection Sampling Randomized Probability and Statistics

Companies

Tencent

Stats

Difficulty:Medium

Languages:4

Companies:1

Problem #:470

Problem Explanation

💡

Think Before You Code

🤔 Ask Yourself

→What is the input? What output do I need?
→Can I solve a smaller version of this problem first?
→What patterns have I seen in similar problems?

✏️ Try This First

Before looking at the solution, try to solve it on paper with a simple example!

🚀 Try on LeetCode

Solution Approach

📋

Step-by-Step Solution

4 steps

1
Step 1: Call `rand7()` twice and store the results in `row` and `col`.
2
Step 2: Calculate an index value `idx` by combining `row` and `col`: `idx = (row - 1) * 7 + col`. This transforms the two calls into a single number ranging from 1 to 49. The formula is derived from treating the result as a base-7 number where the `row` is the 'tens' digit and `col` is the 'ones' digit.
3
Step 3: Check if `idx` is within the acceptance range (1 to 40). If `idx > 40`, reject the sample and repeat from Step 1.
4
Step 4: If `idx <= 40`, map `idx` to the range [1, 10] using the modulo operator: `return 1 + (idx - 1) % 10`.

💡

Pro Tip for Students

Try tracing through each step with a small example on paper before coding. Understanding "why" each step happens is more important than memorizing the code!

⚡

Complexity Analysis

⏱️

Time Complexity

O(1)

⚡ Constant time - Super fast! Same speed regardless of input size.

💾

Space Complexity

O(1)

💪 Constant space - Uses same memory regardless of input!

📝

Why?

The time complexity is O(1) on average because the while loop terminates after a constant number of iterations on average. Each step inside the while loop takes constant time. The space complexity is O(1) because the solution only uses a fixed number of variables to store the results of `rand7()` and the calculated index.

📖New to Big O? Click to learn more

Big O Notation - Quick Guide

O(1)Best - Instant

O(log n)Excellent - Binary Search

O(n)Good - One loop

O(n log n)OK - Sorting

O(n²)Slow - Nested loops

O(2ⁿ)Very Slow - Recursion

Alternative Approaches

Using More Calls to Rand7

We can make more calls to rand7() to reduce the number of rejections. For instance we could call rand7() three times resulting in numbers from 1 to 343 (7*7*7). We then can accept results only less than or equal to 340 and use a modulo operation to map the number to the range [1, 10]. This approach reduces the number of rejections, but increases the computation cost inside the loop.

Time: O(1) on averageSpace: O(1)

Trade-offs: Tradeoff between number of calls to rand7() and rejections in the while loop. More calls to rand7() reduces rejection rate at the cost of increased calculation.

✨

Key Insights

1Insight 1: Combining the results of multiple calls to `rand7()` can create a larger, more controllable sample space.
2Insight 2: Rejection sampling is essential to ensure that the final output is uniformly distributed between 1 and 10.
3Insight 3: We need to create a sample space whose size is a multiple of 10 to enable uniform mapping to the [1, 10] range using the modulo operator.

📚

Key Terms Explained

For Students

Edge Cases

Edge case 1: The while loop might, in theory, run forever. However, the probability of this happening becomes increasingly small with each iteration, and the expected number of iterations is relatively low (approximately 49/40 = 1.225 calls to the (row - 1) * 7 + col operation).
Edge case 2: `rand7()` returns values from 1 to 7. Subtracting 1 and using modulo ensures that each number from 1 to 10 has equal chances (1/10).

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Not understanding the importance of rejection sampling to ensure uniformity. If the acceptance range isn't a multiple of 10, the resulting distribution will be biased.

✗Common mistake 2: Incorrectly mapping the results of `rand7()` to the range [1, 10]. Using a simple average or other non-uniform methods will violate the problem constraints.

✅

Learning Tip

Every mistake is a learning opportunity! When you make an error, understand why it happened before moving on.

🌍

Where is This Used in Real Life?

💼Real-world scenario 1: Simulations often require generating random numbers within a specific range using a limited set of random number generators. Rejection sampling is a common technique to adjust the distribution of the generated numbers to match the desired distribution.

🏦Real-world scenario 2: Cryptography sometimes requires deriving a random key or nonce of a particular size from a source that provides random bits of a different size. Techniques similar to this are used to ensure the derived key is uniform and unpredictable.

💡Understanding real-world applications helps you see the "why" behind algorithms. It's not just about passing interviews—these concepts power the apps and websites you use every day!

Author's Notes

💡 Key Insight

When dealing with arrays, always consider if sorting would simplify the problem. Many array problems have O(n) solutions using hash maps or two-pointer technique.

🎯 Interview Tip

Consider multiple approaches before coding. Mentioning alternatives shows breadth of knowledge.

⚠️ Watch Out

Always test your solution with edge cases: empty input, single element, and boundary values.

👨‍💻 From Experience

This problem pattern appears frequently in real codebases. I've used similar logic when building production systems.

Written by Saswata Mondal

Software Engineer • Algorithms Enthusiast • Interview Coach

👨‍💻

About the Author

👨‍💻

Saswata MondalAuthor400+ LC Problems

B.Tech CSE, IEM Kolkata | LeetCode Rating: 1672 (Top 15%)

Certified: IBM Java Developer, NPTEL Programming in Java

Last updated: September 19, 2024LeetCode Profile GitHub

This solution has been crafted based on my experience solving 400+ LeetCode problems. Each explanation is designed to help you understand the underlying concepts, not just memorize the code.

Community Tips & Insights

Don't forget to handle the edge case when the input is empty. Many candidates miss this in interviews.

@DevExpert92Jul 14

The key insight here is recognizing the pattern. Once you see it, the solution becomes straightforward.

@AlgoMasterJul 7

❓

Frequently Asked Questions

What techniques are commonly tested in medium-difficulty problems?

Medium problems often require combining multiple techniques: two-pointer with hash maps, BFS/DFS with state tracking, dynamic programming with optimization, or divide-and-conquer approaches. They test your ability to recognize patterns and choose the right combination of techniques.

How should I practice algorithm problems effectively?

Focus on understanding patterns rather than memorizing solutions. After solving a problem, review optimal solutions and understand the intuition. Group similar problems to recognize patterns. Practice explaining your approach out loud. Review problems after a few days to test retention.

What is the optimal approach for "Implement Rand10() Using Rand7()"?

For this medium-level Math problem, the key is to identify the core pattern. Analyze the input constraints to determine acceptable time complexity. Consider whether rejection sampling techniques can optimize the solution. Always handle edge cases and validate your approach with examples before coding.

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