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470. Implement Rand10() Using Rand7()

Description

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().

 

Example 1:

Input: n = 1
Output: [2]

Example 2:

Input: n = 2
Output: [2,8]

Example 3:

Input: n = 3
Output: [3,8,10]

 

Constraints:

  • 1 <= n <= 105

 

Follow up:

  • What is the expected value for the number of calls to rand7() function?
  • Could you minimize the number of calls to rand7()?

Solutions

Solution 1

Thinking

We must turn uniform \(rand7\) into uniform \(rand10\). \(\textit{rand7}\bmod 10\) is biased. Two calls give a uniform integer in \([1,49]\).

Rejection sampling keeps \([1,40]\) and returns \(x\bmod 10+1\); values \(41\)–\(49\) retry. \(40\) is a multiple of \(10\), so each residue appears four times.

The expected number of calls is constant. Rejecting the tail is what keeps every outcome equally likely.

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# The rand7() API is already defined for you.
# def rand7():
# @return a random integer in the range 1 to 7


class Solution:
    def rand10(self):
        """
        :rtype: int
        """
        while 1:
            i = rand7() - 1
            j = rand7()
            x = i * 7 + j
            if x <= 40:
                return x % 10 + 1

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/**
 * 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 i = rand7() - 1;
            int j = rand7();
            int x = i * 7 + j;
            if (x <= 40) {
                return x % 10 + 1;
            }
        }
    }
}

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// The rand7() API is already defined for you.
// int rand7();
// @return a random integer in the range 1 to 7

class Solution {
public:
    int rand10() {
        while (1) {
            int i = rand7() - 1;
            int j = rand7();
            int x = i * 7 + j;
            if (x <= 40) {
                return x % 10 + 1;
            }
        }
    }
};

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func rand10() int {
    for {
        i := rand7() - 1
        j := rand7()
        x := i*7 + j
        if x <= 40 {
            return x%10 + 1
        }
    }
}

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/**
 * The rand7() API is already defined for you.
 * function rand7(): number {}
 * @return a random integer in the range 1 to 7
 */

function rand10(): number {
    while (true) {
        const i = rand7() - 1;
        const j = rand7();
        const x = i * 7 + j;
        if (x <= 40) {
            return (x % 10) + 1;
        }
    }
}

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/**
 * The rand7() API is already defined for you.
 * @return a random integer in the range 1 to 7
 * fn rand7() -> i32;
 */

impl Solution {
    pub fn rand10() -> i32 {
        loop {
            let i = rand7() - 1;
            let j = rand7();
            let x = i * 7 + j;
            if x <= 40 {
                return (x % 10) + 1;
            }
        }
    }
}

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