It is important to understand that the "sub-sampling" mentioned is occurring in the frequency domain, not the time-domain-- so any aliasing that the OP is thinking of would all be time-domain aliasing, not aliasing in the frequency domain. To "sub-sample" the frequency spectrum, the time domain sampling rate is unchanged but the duration in time (number of samples in the DFT) is reduced. This results in each DFT bin being wider in frequency, which is a reduced frequency resolution.
Here is the general relationship in case this provides further insight:
Consider a sequence $x[n]$ of length $N$, whose DFT would be $X[k]$, and a shorter length $B$ which is equally divisible into $N$ ($N/B$ is an integer $D$).
To create the $B$-length DFT where the result is exactly every $D^\rm{th}$ sample of $X[k]$ you would need to do the following:
Split $x[n]$ into $D$ $B$-length blocks.
Take the $B$-length DFT of each of the $D$ blocks.
Sum the complex results in each bin.
This would result in an exact match of the original and down-sampled DFTs.
This does result in a computational savings of $\log_2(D)$: Comparing the number of operations in the DFT of the $N$-length sequence ($N\log_2(N)$ vs the DFT of $D$ $B$-length sequences $D(B\log_2(B))$, since $D = N/B$:
$$D\left(B\log_2(B)\right) = \frac NB\left(M\log_2(B)\right) = N\log_2(B)$$
There would be $N\log_2(B)$ operations (plus the $D$ summations) to compute a down-sampled DFT spectrum in comparison to the $N\log_2(N)$ operations for the original $N$-length sequence.
Doing the above is not the best solution!
The operations above describe true down-sampling (select every $D$th sample, throw away the rest.) However, by simply taking the DFT of one $B$ length sequence, which would further reduce the number of operations required by $B$, would result in a truly decimated result. With down-sampling alone as the above process describes and as described in the paper, we can lose critical information: consider a case where every $D^\rm{th}$ sample is zero but there is spectral content in every other bin. The process above would result in all zeros!
This is the result of the time-aliasing which is not a desired property.
Decimation is the combination of filtering and down-sampling. The result of a DFT over a single $B$-length sequence would not be the exact down-sampled value of the DFT as described above but would be the result of the average value over the adjacent $D$ bins. This would be equivalent to the combination of a moving average in frequency over $D$ samples of $X[k]$ followed by the down-sampling of every $D^\rm{th}$ sample--which would be then the decimation of the frequency spectrum. Further, the implied moving average in frequency eliminates the time-domain aliasing (to the same extent that doing a moving average in time before resampling would eliminate aliasing in the frequency domain).