移位操作搞定两数之商

五一漫长的假期，外面的世界是人山人海，反而在家刷题算得上一个好的休闲方式。刚好我开始写这道题：

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Given two integers `dividend` and `divisor`, divide two integers **without** using multiplication, division, and mod operator.

The integer division should truncate toward zero, which means losing its fractional part. For example, `8.345` would be truncated to `8`, and `-2.7335` would be truncated to `-2`.

Return *the **quotient** after dividing* `dividend` *by* `divisor`.

**Note:** Assume we are dealing with an environment that could only store integers within the **32-bit** signed integer range: `[−231, 231 − 1]`. For this problem, if the quotient is **strictly greater than** `231 - 1`, then return `231 - 1`, and if the quotient is **strictly less than** `-231`, then return `-231`.

 

**Example 1:**

Input: dividend = 10, divisor = 3

Output: 3

Explanation: 10/3 = 3.33333.. which is truncated to 3.

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**Example 2:**

Input: dividend = 7, divisor = -3

Output: -2

Explanation: 7/-3 = -2.33333.. which is truncated to -2.

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**Constraints:**

-   `-231 <= dividend, divisor <= 231 - 1`
-   `divisor != 0`

看懂题目上说的，就是不能用乘法、除法以及取余操作来算出两个给定整数的商。这个时候我想到利用移位操作来实现。

虽然工作多年，但是真正在实际项目中用到移位操作的时候是很少的。

逻辑移位：

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-   逻辑移位将位向左或向右移动，并在空位填充零。
-   在左逻辑移位（<<）中，位向左移动，从右侧填充零。
-   在右逻辑移位（>>）中，位向右移动，从左侧填充零。

简单来说，如果1<<2, 就是1乘以2的2次方，以此类推。

所以我的解法就很明确了, 处理好被除数和除数的符号，然后再通过循环里面使用移位操作计算出商的：

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func divide(dividend int, divisor int) int {
	quotient := 0
	maxInt := math.MaxInt32
	minInt := math.MinInt32
	if divisor == 0 || (dividend == minInt && divisor == -1) {
		return maxInt
	}

	// determine the sign of the quotient
	sign := -1
	if (dividend > 0 && divisor > 0) || (dividend < 0 && divisor < 0) {
		sign = 1
	}

	// Convert the dividend and divisor to be positive number
	positiveDividend, positiveDivisor := int(math.Abs(float64(dividend))), int(math.Abs(float64(divisor)))

	for positiveDividend >= positiveDivisor {
		shift := 0
		for positiveDividend >= (positiveDivisor << shift) {
			shift += 1
		}

		quotient += (1<< (shift - 1))
		positiveDividend -= (positiveDivisor << (shift - 1))
	}
	
	return int(math.Min(float64(maxInt), math.Max(float64(sign) * float64(quotient), float64(minInt))))
}

总结

移位操作虽然好，但也不是唯一解，回忆一下小时候还没学乘法的时候，我们也可以用加法去模拟乘法，所以利用累加来模拟出两数之商更直观。

经过我的尝试，我发现完全减法会导致超时问题，不得不结合shifting操作，代码如下所示：

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func divide(dividend int, divisor int) int {
	quotient := 0
	maxInt := math.MaxInt32
	minInt := math.MinInt32
	if divisor == 0 || (dividend == minInt && divisor == -1) {
		return maxInt
	}
	// determine the sign of the quotient
	sign := -1
	if (dividend ^ divisor) >= 0 {
		sign = 1
	}

	// Convert the dividend and divisor to be positive number
	positiveDividend, positiveDivisor := int(math.Abs(float64(dividend))), int(math.Abs(float64(divisor)))

	// Every time postiveDividend subtract with the value of the divisor
	for positiveDividend >= positiveDivisor {
		// Initialize variables for binary search
        tempDivisor := positiveDivisor
        multiple := 1
        
        // Perform binary search-like division
        for positiveDividend >= (tempDivisor << 1) {
            tempDivisor <<= 1
            multiple <<= 1
        }
        
        // Subtract the multiple of divisor from dividend
        positiveDividend -= tempDivisor
        // Add the multiple to quotient
        quotient += multiple
	}
	return int(math.Min(float64(maxInt), math.Max(float64(sign) * float64(quotient), float64(minInt))))
 }