A distinct-customer figure must tie out exactly for an audit, and the job must finish inside a 20-minute window. Which escape does each requirement rule out?

## Requirements are constraints on the escape, not on the operation The operation itself does not change: an exact count of distinct customers is still one whose answer a later record can move, and it still needs reach over records it cannot hold. What the requirements decide is **which of the three currencies you are allowed to spend** — input and output volume, exactness, or one extra traversal. Reading them as constraints on the escape rather than as complaints about the data is what separates a diagnosis from a shrug. | Stated requirement | Escape it forbids | Why | |---|---|---| | The figure must tie out exactly for an audit | The bounded-error answer | A disagreement smaller than the bound cannot be adjudicated, so the figure stops being evidence | | A hard wall-clock budget | The spilled multi-pass run, and the cheap first traversal | Both pay the read cost again, and device throughput sets the clock | | The input can be read only once | Both multi-read escapes | The second read does not exist to be paid for | | Local storage is tight | The spilled multi-pass run | Intermediates need real headroom, often a multiple of the input | | The answer must be recomputable years later | The bounded-error answer, in practice | Reproducing an approximate figure means pinning the summary's behaviour as well as the data | ## Why exactness is a hard veto and not a preference It is tempting to argue that a bound of a fraction of a percent is smaller than the error already present in the source data, so an approximate figure ought to be acceptable. That argument is about the *data*; the requirement is about the *figure's role*. An audited number exists to be compared with another number produced independently. If the two differ, someone must be able to say which is wrong. With a stated error bound in play, every small difference has two explanations and no way to choose between them, so the reconciliation itself stops working. The veto is structural. This is also why the tolerance conversation, if there is to be one, happens **before** the work is built and with the people who consume the figure — not as a setting chosen by whoever writes the job. ## Why a wall-clock budget hits two escapes, not one Both multi-read escapes pay in the same currency even though they look different. The spilled run re-reads the input and writes intermediates alongside; the cheap first traversal writes nothing but reads the input a second time. In each case the extra work is **movement of bytes**, and on an input far larger than memory that movement, not the folding arithmetic, is what the clock is measuring. A budget that was comfortable for one read is often not comfortable for two, and "we will just read it twice" is exactly the assumption a deadline invalidates. Note the honest asymmetry: the second traversal of the narrowing technique is usually cheaper than the first, because it retains almost nothing. It is not free, and it is not a doubling either — say which, rather than quoting a multiplier. ## When the requirements leave nothing Stating "there is no escape that satisfies both" is a complete and correct answer, and the strongest candidates say it without embarrassment. What follows is a negotiation over the three things that are actually movable: 1. **The requirement.** Is the deadline a real downstream dependency or an inherited habit? Is the figure genuinely audited, or merely reported? One of these is often softer than it was presented. 2. **The hardware.** Renting more memory on one box, priced per hour, removes the need for any escape at all when the working set is within reach of a larger machine. It is an expense, not a defeat. 3. **What the operation has to see.** This is the most under-used lever. Fewer columns loaded, a filter applied at the read so that fewer records ever enter the computation, or a coarser reporting grain can move the working set inside memory, at which point the operation that could not be answered from pieces simply does not need to be. ## The failure mode this question is hunting The weak answer picks an escape and defends it against the requirement that forbids it — arguing the bound is small enough, or that the deadline can slip. The strong answer maps each requirement to the escape it vetoes, notices when the set is empty, and then negotiates on the movable axes instead of on the arithmetic.