# What is the meaning of the statement "a sequence of n PUSH, POP and MULTIPOP opreations"

I am reading CLRS 3rd Ed, chapter 17.1 (Aggregate analysis pg453) and I came across this statement.

Let us analyze a sequence of n PUSH, POP, and MULTIPOP operations on an initially empty stack.

I am confused as in:

1. Do the push, pop, and multipop cumulatively add up to n operations.
(there is a total of n operations which consists of x pushes, y pops, and z multipops where x+y+z = n)
or
2. Are they talking about n push, n pop, and n multipop operations?
or
3. 1st operation(push, pop, Multipop), 2nd operation(push, pop, Multipop), ..., nth operation(push, pop, Multipop)

The question is does the statement from the book imply 1 or 2 or 3 or something I did not mention above. Thanks

• I don't have the book with me so I am not posting this as an answer but in all likelihood $n$ refers to the number of operations and "pop, push and multipop" to the kind of operations, so number 1 of your suggestions. If the intended meaning was different the phrase would (probably) have a different structure. Commented Nov 3, 2021 at 16:32
• @phan801 thanks that makes sense. Commented Nov 3, 2021 at 19:12

As phan801 commented, the first interpretation, a sequence of $$n$$ operations, each of which is either push or pop or multipop, is correct.

Either one of the other two interpretations might stand a small chance without surrounding context or with a very different context. However, had it been the intended meaning, "the phrase would probably have a different structure", such as, "n push operations, n pop operations, and n multipop operations" or "n operations, each being a push operation followed by a pop operation followed by a multipop operation".

A strong indication of the actual meaning comes from the introductory sentence of this section, "17.1 Aggregate analysis".

In aggregate analysis, we show that for all $$n$$, a sequence of $$n$$ operations takes worst-case time $$T(n)$$ in total.

The interpretation 2 means three sequences of $$n$$ operations instead of "a sequence of $$n$$ operations".

The interpretation 3 could have been more possible if push-then-pop-then multipop is a reasonable combination. However, after push-then-pop, multipop operation will always be a non-operation, since the stack is assumed empty initially. There is nothing interesting to analyze for this interpretation. (Even if the stack is not empty initially, we would probably use "push-then-multipop" since we can combine a pop operation followed by a multipop operation into another multipop operation".) So this combination of three operations as one operation does not make much sense.

• By the way, the first interpretation fits the ensuing discussion in the textbook perfectly. Commented Nov 3, 2021 at 19:45