When we say that an algorithm XXX is asymptotically more efficient than another algorithm YYY, it means that for sufficiently large input sizes, XXX requires fewer computational steps or resources (such as time or space) than YYY. More formally, there exists some input size n0n_0n0​ such that for all input sizes n>n0n>n_0n>n0​, the performance (e.g., running time) of XXX is better (smaller) than that of YYY

. Key points to understand this concept:

• Focus on large inputs: Asymptotic efficiency concerns the behavior of algorithms as the input size grows very large, ignoring constant factors and lower-order terms that might affect performance on smaller inputs

• Growth rate comparison: It compares the growth rates of resource usage (time or space) as functions of input size nnn, typically expressed using Big O notation or related asymptotic notations

• Not necessarily better for small inputs: An algorithm that is asymptotically more efficient might not be faster for small input sizes due to higher constant factors or overhead, but it will outperform the other algorithm beyond some threshold input size

In summary, saying XXX is asymptotically more efficient than YYY means XXX scales better and will be more efficient than YYY for large enough inputs, making it the preferable choice when dealing with large datasets or problem sizes