Leet Code OJ 338. Counting Bits [Difficulty: Medium]
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题目:
Given a non negative integer number num. For every numbers i in the range 0 ≤ i ≤ num calculate the number of 1’s in their binary representation and return them as an array.
Example:
For num = 5 you should return [0,1,1,2,1,2].
Follow up:
It is very easy to come up with a solution with run time O(n*sizeof(integer)). But can you do it in linear time O(n) /possibly in a single pass?
Space complexity should be O(n).
Can you do it like a boss? Do it without using any builtin function like __builtin_popcount in c++ or in any other language.
Hint:
You should make use of what you have produced already.
翻译:
给定一个非负整数num,对于每个0<=i<=num的整数i,计算i的二进制表示中1的个数,返回这些个数作为一个数组。
例如,输入num = 5 你应该返回 [0,1,1,2,1,2].
分析:
按照常规思路,很容易得出“Java代码2”的方案,但是这个方案的时间复杂度是O(nlogn)。
通过对数组的前64个元素进行分析(num=63),我们发现数组呈现一定的规律,不断重复,如下图所示:
0 1 1 2 1 2 2 3 1 2 2 3 2 3 3 4 1 2 2 3 2 3 3 4 2 3 3 4 3 4 4 5 1 2 2 3 2 3 3 4 2 3 3 4 3 4 4 5 2 3 3 4 3 4 4 5 3 4 4 5 4 5 5 6
由此我们发现0112是一个基础元素,不断循环反复,可以推论:如果已知第一个元素是result[0],那么第二第三个元素为result[0]+1,第四个元素为result[0]+2,由此获得前4个元素result[0]~result[3];以这4个元素为基础,我们可以得到
result[4]=result[0]+1,result[5]=result[1]+1…,
result[8]=result[0]+1,result[9]=result[1]+1… ,
result[12]=result[0]+2,result[13]=result[1]+2…;
以此类推可以获得全部的数组。
Java版代码1:
public class Solution { public int[] countBits(int num) { int[] result = new int[num + 1]; int range = 1; result[0] = 0; boolean stop = false; while (!stop) { stop = fillNum(result, range); range *= 4; } return result; } public boolean fillNum(int[] nums, int range) { for (int i = 0; i < range; i++) { if (range + i < nums.length) { nums[range + i] = nums[i] + 1; } else { return true; } if (2 * range + i < nums.length) { nums[2 * range + i] = nums[i] + 1; } if (3 * range + i < nums.length) { nums[3 * range + i] = nums[i] + 2; } } return false; }}
Java版代码2:
public class Solution { public int[] countBits(int num) { int[] result=new int[num+1]; result[0]=0; for(int i=1;i<=num;i++){ result[i]=getCount(i); } return result; } public int getCount(int num){ int count=0; while(num!=0){ if((num&1)==1){ count++; } num/=2; } return count; }}
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