#### Approach #1: Adjacent Symbols [Accepted]

Intuition

Between every group of vertical dominoes ('.'), we have up to two non-vertical dominoes bordering this group. Since additional dominoes outside this group do not affect the outcome, we can analyze these situations individually: there are 9 of them (as the border could be empty). Actually, if we border the dominoes by 'L' and 'R', there are only 4 cases. We'll write new letters between these symbols depending on each case.

Algorithm

Continuing our explanation, we analyze cases:

• If we have say "A....B", where A = B, then we should write "AAAAAA".

• If we have "R....L", then we will write "RRRLLL", or "RRR.LLL" if we have an odd number of dots. If the initial symbols are at positions i and j, we can check our distance k-i and j-k to decide at position k whether to write 'L', 'R', or '.'.

• (If we have "L....R" we don't do anything. We can skip this case.)

Complexity Analysis

• Time and Space Complexity: , where is the length of dominoes.

#### Approach #2: Calculate Force [Accepted]

Intuition

We can calculate the net force applied on every domino. The forces we care about are how close a domino is to a leftward 'R', and to a rightward 'L': the closer we are, the stronger the force.

Algorithm

Scanning from left to right, our force decays by 1 every iteration, and resets to N if we meet an 'R', so that force[i] is higher (than force[j]) if and only if dominoes[i] is closer (looking leftward) to 'R' (than dominoes[j]).

Similarly, scanning from right to left, we can find the force going rightward (closeness to 'L').

For some domino answer[i], if the forces are equal, then the answer is '.'. Otherwise, the answer is implied by whichever force is stronger.

Example

Here is a worked example on the string S = 'R.R...L': We find the force going from left to right is [7, 6, 7, 6, 5, 4, 0]. The force going from right to left is [0, 0, 0, -4, -5, -6, -7]. Combining them (taking their vector addition), the combined force is [7, 6, 7, 2, 0, -2, -7], for a final answer of RRRR.LL.

Complexity Analysis

• Time and Space Complexity: .

Analysis written by: @awice.