The LAX airport has an official long-term parking lot near the airport terminals, Economy Lot C, which is a much cheaper option compared to parking in a terminal garage, and cheaper than many nearby private parking lots (with rates on par to private lots further away).
Unfortunately, a large portion of the lot has been closed off due to construction work as part of a massive modernization program. Also, LAX doesn't accept parking reservations, so if one turns up there and the lot's full, one has to search for a parking space at a private lot.
Is it easy to find a parking space at Lot C given that much of the lot has been closed off, or should I pre-book a space at a private lot? Most private lots with similar rates are much further away from the airport, closer to the 405 freeway, which thus means a much longer shuttle ride to and from the airport than with Lot C.