Approach #1: Adjacent Symbols [Accepted]
Intuition
Between every group of vertical dominoes ('.'
), we have up to two nonvertical 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 positionsi
andj
, we can check our distanceki
andjk
to decide at positionk
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.