[LintCode] Find the Missing Number

Given an array contains N numbers of 0 .. N, find which number doesn't exist in the array.

Example

Given N = 3 and the array [0, 1, 3], return 2.

Challenge

Do it in-place with O(1) extra memory and O(n) time.

Solution 1. O(n*logn) runtime using sorting

Sort the input array;

Then iterate through the array and find i that nums[i] != i;

If found one, return it; If not, return nums.length.

 1 public class Solution {
 2     public int findMissing(int[] nums) {
 3         if(nums == null || nums.length == 0){
 4             return -1;
 5         }
 6         Arrays.sort(nums);
 7         int i = 0;
 8         for(; i < nums.length; i++){
 9             if(nums[i] != i){
10                 return i;
11             }
12         }
13         return i;
14     }
15 }

Solution 2. O(n) runtime, O(1) space, by swapping elements to make nums[i] = i.

In solution 1, the key idea is to change the input array into an ascending order then find the index that nums[index] != index.

The bottleneck here is the O(n*logn) sorting. We can leverage the same idea from First Missing Positive and achieve

the re-ordering in O(n) runtime by only swapping elements.

Algorithm: provide the input array the property of nums[i] = i by swapping elements in O(n) time.

Since there is one number missing from 0 to N, a linear scan of the changed array finds the index i that nums[i] != i, this

index is the missing number.

Each while loop moves the current number nums[i] to its right location that satisfies nums[i] = i. The only exception is that

if the array has the number N. Because the largest index of nums is N - 1, nums[N] causes array index out of bound error.

So if nums[i] == N, we skip swapping it.

 1 public class Solution{
 2     public int findMissing(int[] nums){
 3         if(nums == null || nums.length == 0){
 4             return - 1;
 5         }
 6         for(int i = 0; i < nums.length; i++){
 7             while(nums[i] != i && nums[i] < nums.length){
 8                 int temp = nums[nums[i]];
 9                 nums[nums[i]] = nums[i];
10                 nums[i] = temp;
11             } 
12         }
13         int missing = 0;
14         for(; missing < nums.length; missing++){
15             if(nums[missing] != missing){
16                 break;    
17             }    
18         }
19         return missing;
20     }
21 }

Solution 3. O(n) runtime, O(1) space, find summation difference

This solution is not good as solution 2 or 4 because the sum here can be very big and causes overflow.

The following code uses type long to address this issue. But in the extreme cases even long type may

overflow.

 1 public class Solution {
 2     public int findMissing(int[] nums) {
 3         if(nums == null || nums.length == 0){
 4             return -1;
 5         }
 6         long sum1 = 0, sum2 = 0;
 7         for(int i = 0; i <= nums.length; i++){
 8             sum1 += i;
 9         }
10         for(int i = 0; i < nums.length; i++){
11             sum2 += nums[i];
12         }
13         return (int)(sum1 - sum2);
14     }
15 }

Solution 4. O(n) time, O(1) space, Bitwise operation

Algorithm:

1. xor all the array elements, let the result be x1;

2. xor all numbers from 0 to n, let the result be x2;

3. x1 ^ x2 gives the missing number.

Proof of correctness:

Say we are give A0, A1, A2, A4; A3 is missing.

X1 = A0 ^ A1 ^ A2 ^ A4;

X2 = A0 ^ A1 ^ A2 ^ A3 ^ A4;

X1 ^ X2 = (A0 ^ A1 ^ A2 ^ A4) ^ (A0 ^ A1 ^ A2 ^ A3 ^ A4)

　　 = (A0 ^ A0) ^ (A1 ^ A1) ^ (A2 ^ A2) ^ A3 ^ (A4 ^ A4)

= 0 ^ 0 ^ 0 ^ A3 ^ 0

　　　　= A3

 1 public class Solution {
 2     public int findMissing(int[] nums) {
 3         if(nums == null || nums.length == 0){
 4             return -1;
 5         }
 6         int xor1 = 0, xor2 = nums[0];
 7         for(int i = 1; i <= nums.length; i++){
 8             xor1 ^= i;
 9         }
10         for(int i = 1; i < nums.length; i++){
11             xor2 ^= nums[i];
12         }
13         return xor1 ^ xor2;
14     }
15 }