The thread: The model has parameters — page 2
The number that would choose a cap
A depth histogram is one linear pass — 3.16 operations a character over thirty-two thousand of them — and it says the whole text sits at a mean depth of 3.98 with a worst of ten. Nobody prints it, and every choice of cap in this collection was made without it.
When it does not fitWhat insurance against an estimate costs
A planner that trusts its row estimate expects to pay 1.057 times the better plan and risks 7.76. One that insures itself by halving its estimate before it decides expects 1.057 and risks 4.10 — the insurance is free. At a read ratio of sixteen the same insurance costs six per cent in expectation and makes the worst case worse. Whether a conservative planner is paying a sensible premium depends on two numbers the planner can measure and usually does not — its device's read ratio and the direction its own errors run.
When the algorithm is a tableThe lattice that decides the ties
Rounding a fitted substitution matrix to whole bits puts 63 of 200 alignments on a tie where the exact fit puts 15. Rounding to half bits — a finer grain, and the obvious repair — puts 79. What tracks the ties is not how fine the lattice is but how many of the six fitted costs it keeps apart: whole and half bits both leave three, an eighth of a bit leaves all six, and matches the exact fit exactly.
One pass, and no roomThe period that is not a promise
Greenwald–Khanna's ε appears twice — once as the rank tolerance the structure promises, and once as ⌊1/2ε⌋, the number of updates between compressions. Unhook the second from the first and sweep it across a thousand-fold range. The tuples held move by 10%, the worst rank error by 21%, the peak by ten times and the housekeeping by seventy.
What is taught wronglyThe histogram that cannot see the order
A prediction accurate to one per cent on three streams is seven times out on the fourth, and the input that fails is the one every capacity plan is built from. A statistic computed from the same input says in advance which case is in hand — and misses one of the two ways it can go wrong.
Two parametersOne Bellman–Ford buys every Dijkstra
A directed graph of 256 vertices with a third of its arcs negative needs shortest paths between every pair. Running Bellman–Ford from every source costs 25.8 million counted operations on the densest graph drawn; running it once, repricing every arc by what it found, and then running Dijkstra from every source costs 13.1 million, and the one Bellman–Ford is under one per cent of that. Floyd–Warshall's 16.8 million is never the cheapest count on the plate. On the sparsest graphs the repeated Bellman–Ford wins, because its early exit makes nine passes rather than 255.
What is taught wronglyThe boundary that hides the burst
A window whose blocks hold five hundred and twelve arrivals reports a perfectly even stream — every block the same duration to the tick — while the arrivals it is retiring have an index of dispersion of 0.57. Move the block to a hundred and twenty-five and the same stream varies by 1.8 times.
Two parametersWhere the landmarks stand
Four tables of exact distances, each from a chosen cell, turn a straight-line estimate that barely helps on rough terrain into one that cuts a search by a factor of six. Averaged over 1,200 queries on eight maps, the same four tables expand 126 cells a query when their cells are the map's corners and 423 when they are near its centre. The standard choice, each landmark as far as possible from the ones before, expands 141 and loses to the corners on all eight maps. Moving four landmarks to the right places buys more than doubling their number.
What is taught wronglyThe sampler that cannot alias
A block window's boundary fires on an arrival count, and a stream whose burst repeats every sixty-four arrivals is reported as perfectly even by a block of five hundred and twelve. A quantile summary compresses on an update count, which is the same arrangement. Swept against three periodic value processes and their shuffles, it does not alias — and the reason is one line of arithmetic rather than a lucky sweep.
Two parametersWhat the queries know that the map does not
A greedy rule that chooses landmark cells by rerunning a sample of past queries needs two hundred of them to draw level with a rule that reads only the map — and what it finally chooses, on map after map, is the four corners. Give the queries a destination instead of scattering them, and twenty are enough to beat the corners by 29% on eight maps out of eight. A query log is worth reading exactly to the extent that it is not uniform.
When it does not fitThe read a filter has no key for
A Bloom filter on every run of a log-structured store turns a lookup for a missing key from a read of every level into a fraction of one — 0.72 transfers across eight levels at five bits a key, and 0.00027 at twenty. A range query over the same store reads nine transfers at five bits and nine at twenty, because a filter answers whether one named key is in a run and a range has no key to name.
What is taught wronglyA parameter that waits for another
Four merge fold shapes over thirty-two evenly loaded shards leave errors of 665, 667, 665 and 667 — a fifth of a per cent apart. Give the same four shapes shards whose loads span seventeen-fold and they leave 148, 183, 323 and 403. The parameter did nothing until a second parameter moved, and every measurement that fixed the second one saw nothing.
What is taught wronglyThe floor a merge does not settle at
Compute a fold's level floors from the shard histograms and the prediction over-shoots by 1.73. A merged summary's floor is not the floor a summary settles at on the same arrivals — it is 0.90 of it at four shards and 0.70 at sixty-four, straight in log₂ m at a 3% residual, because merging preserves the heavy counters and never runs their eviction cascade.
Two parametersHow long a reweighting stays true
Johnson's one Bellman–Ford run costs under one per cent of an all-pairs computation because it is divided over every source. Asked one query at a time it is divided over nothing, and it still repays itself after 2.7 queries — because the preparation is one Bellman–Ford and every query saves a third of another. What decides the trade is not the query count but whether the graph holds still: at half a per cent of arcs redrawn between queries the stored potential is worth exactly nothing, and its life is geometric at a per-arc failure rate of 6.6%.
What is taught wronglyThe warning that is silent for the right reason
The statistic shipped to warn that a merge prediction is about to fail reads 0.160 on a stationary stream, 0.172 on a bursty one and 0.909 on a drifting one. It was asked to be looked at again because it does not catch a burst. It does not, and the reason is that on a burst there is nothing to catch.
What is taught wronglyThe threshold that reaches zero
At q = 5 and four errors on a twenty-four-character pattern the filter demands zero shared q-grams, proposes all 2,977 windows, and computes 986,266 table cells where filling the whole table would have cost 96,000. The failure is arithmetic and is knowable before a character is read.
What is taught wronglyThe cap that binds on one text and not another
A periodic text looks like the one made of chains and pays 1.11 times the phrases for a cap of four. A text that repeats itself pays 2.96. The guess is backwards, and the reason is that depth measures nesting rather than repetition.
What is taught wronglyA sixth of what, exactly
The saving from a counting-only reverse half is 30.5% at one sampled position in four and 14.0% at one in a hundred and twenty-eight. A number quoted without its sampling rate is a number about a setting somebody chose.
What is taught wronglyThe occurrences a join invents
Eight documents run together hold forty-nine eight-character windows that span a join, twenty-three of which occur in no document at all. Every index built over the concatenation reports them, and five essays of this collection paid that cost silently.
What a bound isTwo at binary, five at twenty-six
The saving is a factor in the alphabet, so a two-symbol alphabet gets two. Approximate matching in this field is mostly done on DNA, which sits near the bottom of the list at 2.7.
What is taught wronglyThe dial that has no setting
The generator has one parameter. The real version history's run count asks it for 2.3% and its phrase count asks for 3.2% — and the reason is not that the dial is badly calibrated. Real edits average 7.3 characters a block and generated ones average 1.04.
What a bound isThe flat bottom of a shallow curve
The low width is chosen as the floor of log of the universe over the count. Rounding it up instead costs one bit on five thousand, because the total is m·w plus n over two to the w and the minimum is where those two are equal.
What a bound isFlat in the budget, and not
One saving is eleven times at every error budget, because it is a property of the alphabet. The other moves between ninety-eight and a hundred and five, because it follows the share of extensions that find nothing. Two savings, two shapes, and neither line crosses the other.
What a bound isThe exponential is in the expression
The subset construction on one family reaches two to the k plus one states, exactly and not approximately. A literal of the same length gives eleven. Both are regular expressions and the difference is that one of them asks the machine to remember something.
What the libraries doA looser budget wastes a larger share
More errors permitted means more work, and the fraction of that work which was never going to help rises with it — from thirty-one per cent at no errors to seventy-five at two. The saving is worth most where the search is most expensive.
StructuresBits and steps on one frame
The size falls from ninety-nine per cent to eighty-six as the sampling thins, and the walk to a sampled position rises from two and a half steps to sixty-four. Neither line is the answer; the answer is a point on the pair.
What the libraries doWhere the table starts paying
Five thousand and sixty-five operations before the first character, then one per character. Against nothing before the first character and thirty-nine per character. They cross at two hundred and fifty-six characters, and that crossing is what an engine's compile decision actually is.