If the case was dismissed with prejudice, it cannot be refiled. If it was dismissed without prejudice, the state (prosecuor) has up to 2 years from the date of the alleged incident to refile it. Generally, dismissals by the prosecutor are without prejudice unless the order specifically says it is with prejudice. Talk with the attorney who represented you to make sure this was a with prejudice dismissal. Your criminal record will show you were charged with the assault and that is was dismissed. The only way the charge would be completely removed from your record is if you are successful in having the record expunged. This happens rarely.
Dismissed with prejudice means the charge cannot be re-filed. You can petition the Washington State Patrol to remove this "non-conviction data" from your arrest record 3 years after the date of dismissal (assuming you stay of out of trouble between now and then).
I would only add that you may even be eligible to petition to have the record expunged two years after the record became "non-conviction data" as the result of a "disposition favorable to the defendant." Like a dismissal with prejudice. RCW 10.97.060.