<transcript><audio_15>we just need to extract the initial proof and applies apply every record update record in the list using the updated single proof algorithm</transcript><transcript><audio_16>which is o1 and to get the the newest proof uh the thing the the newest single proof pi j prime so if the record list is at the most</transcript><transcript><audio_17>square root of n record so the get one single proof algorithm needs at most so above all we need amortized</transcript><transcript><audio_18>n log n which is open or proof uh over square root of n which is square root of n log n time to do the update part and we need at most square relevant time to get any single proof since uh updated single proof</transcript> we just need to extract the initial proof and applies apply every record update record in the list using the updated single proof algorithm which is o1 and to get the the newest proof uh the thing the the newest single proof pi j prime so if the record list is at the most square root of n record so the get one single proof algorithm needs at most so above all we need amortized n log n which is open or proof uh over square root of n which is square root of n log n time to do the update part and we need at most square relevant time to get any single proof since uh updated single proof
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