Leetcode- Single Number II

Given an array of integers, every element appears three times except for one. Find that single one.

Note: We generalize this problem to:

Given an array of integers, every element appears k times except for one which apprears l times. Find that single one whic appears l times.

Analysis:

The general solution is given in https://oj.leetcode.com/discuss/857/constant-space-solution

Now I try to explain the general solution.

First, we only consider one bit (either one in the bit sequence of each element in A). The idea is to record how many times 1 is appreaed for this bit. We use an array X with length k to indicate 0 - k-1 times. If 1 appreas j times, then X[j] is marked as 1. We scan the array A, and want to express such logic:

1. If A[i] on this bit is 0, then X array remains its original value.

2. If A[i] on this bit is 1, then

　　2.1 if 1 has appreared j-1 times, i.e., X[j-1]==1, then X[j] =1 now.

　　2.2 if 1 has appreared j times, i.e., X[j]==1, then X[j]=0 now.

　　2.3 if 1 has appreaed some other times, then X[j] remains the same, i.e., still 0.

We use logic table to express this logic:

 

X[j-1]	X[j]	A[i]	new X[j]
0	1	0	1
1	0	0	0
0	0	0	0
1	1	0	N/A
0	1	1	0
1	0	1	1
0	0	1	0
1	1	1	N/A

Note that X[j-1]==1 && X[j]==1 will not happen.

Then we change the table to logic expression formula:

new X[j] = (~X[j-1] & X[j] & ~A[i]) | (X[j-1] & ~X[j] & A[i]).

Since X[j] & X[j-1] will not happen, it does not matter what output it will be, so we assume the output will be 1 to simply the expression, then we have:

X[j] = (X[j] & ~A[i]) | (X[j-1] & A[i])

NOTE: This is the formula used in the general solution, but we have not reached there. CONTINUE.

Since if 1 apprears k times, then it equals to apprear 0 times, we have X[0] = (old X[k-1] & A[i]) | (X[0] & ~A[i]).

Now, with this formula, we can count how many times the 1 has appreared on some specific bit position for array A. It also means that we can know that what the value that apprears l times on some specific bit position. It is the value of X[l]. NOTE, it is either 1 or 0.

Second, since there is no relationship between each bit in a number, we can directly use the above formula to operate the number, the answer then will be the value that apprears exactly l times on each bit, which is the ANSWER we want.

Solution:

NOTE: This solution is posted by Ronmocy in LeetCode.

public class Solution {
    public int singleNumber(int[] A, int k, int l) {
        if (A == null) return 0;
        int t;
        int[] x = new int[k];
        x[0] = ~0;
        for (int i = 0; i < A.length; i++) {
            t = x[k-1];
            for (int j = k-1; j > 0; j--) {
                x[j] = (x[j-1] & A[i]) | (x[j] & ~A[i]);
            }
            x[0] = (t & A[i]) | (x[0] & ~A[i]);
        }
        return x[l];
    }
}

 Another solution, easier to figure out:

https://discuss.leetcode.com/topic/43166/java-o-n-easy-to-understand-solution-easily-extended-to-any-times-of-occurance

The usual bit manipulation code is bit hard to get and replicate. I like to think about the number in 32 bits and just count how many 1s are there in each bit, and sum %= 3 will clear it once it reaches 3. After running for all the numbers for each bit, if we have a 1, then that 1 belongs to the single number, we can simply move it back to its spot by doing ans |= sum << i;

This has complexity of O(32n), which is essentially O(n) and very easy to think and implement. Plus, you get a general solution for any times of occurrence. Say all the numbers have 5 times, just do sum %= 5.

public int singleNumber(int[] nums) {
    int ans = 0;
    for(int i = 0; i < 32; i++) {
        int sum = 0;
        for(int j = 0; j < nums.length; j++) {
            if(((nums[j] >> i) & 1) == 1) {
                sum++;
                sum %= 3;
            }
        }
        if(sum != 0) {
            ans |= sum << i;
        }
    }
    return ans;
}