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$ curl repovive.com/roadmaps/dynamic-programming/broken-profile-dp-plug-dp/domino-and-tromino-implementation

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$ curl repovive.com/roadmaps/dynamic-programming/broken-profile-dp-plug-dp/domino-and-tromino-implementation

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Domino and Tromino - Implementation - Broken Profile DP (Plug DP) | Dynamic Programming | Repovive

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Dynamic Programming21 sections · 918 units60h total

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Domino and Tromino - Implementation

DP or recurrence

Here is the formula-based solution. The formula comes from analyzing profile transitions for the $2$ -row board with dominoes and trominoes:

function numTilings(n):
    MOD := 1000000007
    if n = 1 then return 1
    if n = 2 then return 2
    p := array of size n+1
    p[1] := 1
    p[2] := 2
    for i from 3 to n:
        p[i] := (2 * p[i-1] + p[i-2]) mod MOD
    return p[n]

This runs in $O(n)$ time and $O(n)$ space. The formula $p[i] = 2p[i-1] + p[i-2]$ captures all tiling configurations by counting how the two tile types combine.

You could improve to $O(1)$ space by keeping only the last two values.