Bitcoin secures its network by forcing miners to solve a cryptographic puzzle who's difficulty increases based on the total hashing power of the network. If someone were to develop an algorithm that enables them to solve this puzzle say 1,000 times more efficiently (faster) than anyone else on the network they would be able to mine most (or a good portion) of new Bitcoin for pennies on the dollar. Assuming the person only uses his knowledge to mine Bitcoin at a much greater speed/efficiency compared to other miners (and does not hack anything or attempt to double spend etc) would the network be able to detect that something like this is happening?
I suppose not, because the hashing power of the network is not something that is known explicitly. It is something that is implicitly calculated based on the assumption that there is no possible way for a single individual to solve the cryptographic puzzle more efficiently than anyone else.
Market laws suggest that if said person would use his 1000x more efficient miner in any non-trivial scale, he would push other miners out of the market. The difficulty would steadily increase and other miners would be operating at a loss.
So I guess the answer is yes, other people would notice if such thing happened.
Average interblock times is 10 minutes. Every 2016 blocks (2 weeks if the hashrate is constant) difficulty is readjusted. If you start mining much quicker, the interblock time of these two weeks will be less than 10 minutes and the difficulty will increase. The opposite is also true.
The total hashrate is estimated using the average interblock times in a time interval and the current difficulty.
The units of hashrate is hashes per second. Whether you found a more efficient mining method or added more computers to the network doesn't matter to the rest of the network.