I want to write an algorithm that takes an $n \times n$ grid and a number $L$, generate a random walk of length $L$ on the grid that doesn't visit the same cell twice.
One simple solution would be keep randomly choosing a neighbour of the last cell to add to the path. I found that for small density (i.e. $L < 0.625n^2$), I can repeat this algorithm a large amount (at least $1000$) times and can almost find some complete path without deadlock, i.e. without a a state where the path is incomplete but there is no more move available (i.e. all neighbours of the current cell is already filled). A DFS can lead to a deadlock state, and from my experiments, most of the end states are deadlocks, hence it is very hard to get to a complete state with DFS. But for larger density, it is nearly impossible to avoid deadlock no matter how many times I repeat.
My second attempt is to backtrack one step if the current $i^{th}$ step results in a deadlock (i.e. delete the last action from the list of possible actions at $(i-1)^{th}$ step and randomly choose a remaining action to retry). However it apparently took forever to finish the path (it kept going forward and backward when $i$ reaches $0.625n^2$).
Is there any suggestion for a fast algorithm that have fairly high probability of avoiding deadlocks? I'm fine with pseudo-randomness.
