I found a paper by Grover titled "How significant are the known collision and element distinctness quantum algorithms?", in which he expressed criticism to several famous algorithms, including Ambainis's algorithm for element distinctness. More precisely, he argues that all those algorithms use too much space, and there is a trivial way to obtain the same speedup(using same amount of memory) by separating the search space into several parts and search them on independent processors simultaneously. It seems Ambainis admitted this in another article.
My questions are:
1.Is Grover correct about those algorithms? That is, are they "trivial" in Grover's sense?
2.Have there been new algorithms obtaining the same speedup and using less space since then?