O(nlogn) Clean and easy Java DP + Binary Search solution with detailed explanation

• I use an array dp[] to store the largest elements in the increasing subsequences and the array's index is the length of the subsequence (since this, we can make sure that the array dp[] is in-order). The idea is keep checking if nums[i] is bigger than dp[len] or not: nums[i] > dp[len] means we can retrieve a longer subsequence by adding the current element nums[i] into the result, so we just increase the pointer 'len' by one and put the current element into the new index; otherwise we need to do a binary search to find out the index of the largest element st. dp[index] < nums[i] && dp[index + 1] > nums[i] and update the dp[index] with value nums[i].

Let's see an example:
intput: 2 4 9 3 7 8

The array 'dp' looks like these in first two iterations:

2 4 0 0 0 (len = 1)

2 4 9 0 0 (len = 2)

In the third iteration, we notice that dp[len] ('9' here) > nums[i] ('3' here), so we update the dp[index] with value '3', then we got:

2 3 9 0 0 (len = 2) <---- Attention here, this array doesn't mean we can retrieve subsequence [2,3,9], it only means the largest element in a length 1 subsequence is '2' ([2]), the largest element in a length 2 subsequence is '3' ([2,3]) and the largest element in a length 3 subsequence is '9' ([2,4,9]) by far.

In next iteration, we update the dp[index] with value '7' again and we got:
2 3 7 0 0 (len = 2)

In the final round, obviously dp[len] (7) < nums[i] (8) so we increase the 'len' by one and put '8' into dp[3] then we got:
2 3 7 8 0 (len = 3)

Clearly len + 1 = 4 is our result : )

PS: Keep in mind that 'len' always indicate the length of the longest increasing subsequence in each iteration.

public int lengthOfLIS(int[] nums) {
if(nums == null || nums.length == 0) {
return 0;
}
int[] dp = new int[nums.length];
dp[0] = nums[0];
int len = 0;
for(int i = 1; i < nums.length; i++) {
if(nums[i] > dp[len]) {
dp[++len] = nums[i];
}
else {
int index = search(dp, len, nums[i]);
dp[index] = nums[i];
}
}
return len + 1;
}

private int search(int[] dp, int len, int val) {
int start = 0;
while(start <= len) {
int mid = start + (len - start) / 2;
if(dp[mid] == val) {
return mid;
}
else if(dp[mid] < val) {
start = mid + 1;
}
else {
len = mid - 1;
}
}
return start;
}

• You have said the wrong explanation.
You said: otherwise we need to do a binary search to find out the index of the largest element st. dp[index] < nums[i] && dp[index + 1] > nums[i] and update the dp[index] with value nums[i].
But in your example, in the step:2 3 9 0 0 (len = 2) <---- Attention here, i=3, nums[i]=3, index = 0, dp[0]<3<dp[1], so we should update dp[0] rather than dp[1].
It would be better if you can update this conflict in your explanation.

• I don't think it's the largest under given length, but the smallest at that position. Consider the sequence 2 4 9 3 1 10, you will get 1 3 9 10 at last.

In this case, 3 is the smallest number that can have increasing sequence with length 2. That's why the problem can be considered into category of DP.

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