The thread: Measured, not assumed — page 11
The room the lines lend
A blocked filter's threshold table gives each 512-bit line room for its closed blocks' values, and a group reserve sized for the spill had just put it under the ungrouped curve. A line under its room leaves slots empty that are already paid for, so its neighbours' spills could be carried forward into them, with no reserve at all: predicted at about 3.12 bits a block for 2.012 lines a lookup. The bits are right and the lines are not. At room for the mean a carried value passes 12.8 lines before it finds a slot, because a run of lines that is room-for-the-mean on average is a queue with nothing to spare. Given three quarters of a deviation of room, the walk falls under two lines and the carried table lands a hundredth of a bit under the reserve.
CountingA register dropped by what it already knew
Two registers beside a two-ended selection sort's extremes cut the reading of its record of adjacent order to a forty-fifth on short runs, and made it worse than reading the record plainly on long ones: 53.5 words a question on a sorted array of 1,024 keys against 22.1. The swaps were blamed, and the proposed repair kept the part of a broken run beyond the break as a register of its own. It changes the reading by under 1% anywhere. On the sorted array the survivors are kept 523,775 times and asked 1,022 times, and the sort makes no swap at all: every one of the 263,167 relearning drops there is a pair the record already held, rewritten at the same value by a neighbour's question that the previous scan's chains answered. The swap never broke the register. Learning what it already knew did.
When it does not fitA span the queries do not see
A rule for dividing a levelled store's range-filter memory by its levels' sizes, with a floor of half a bit a key, came within five points of the division fitted to the store's own queries at every budget from one bit up — but at half a bit the shuffled and the clustered store wanted opposite divisions of identically sized levels. A level's fences, its first and last keys, say how much of the key space it spans, and a clustered level spans little. Scaling each level's bits by that span was predicted to close the half-bit gap and change nothing above. It changes nothing at half a bit — where the floor is the whole budget, so there is nothing left to scale — and from one bit up it puts the clustered store 30 points behind the fitted division, where the floored rule was within one. A level holding one cluster spans 0.4% of the key space and meets 4.4% of the queries, because half of them start at a stored key: the fences measure the key space, and the queries are aimed at the keys.
What is taught wronglyThe level where compression stops paying
Choosing the best coding for every level of the grid separately, rather than one for all twelve, saves 26 bits out of 47,668 — five hundredths of one per cent. The apparatus for choosing costs more than that to describe.
What a bound isA factor of fourteen, for four per cent
The compound walk removes a factor of σ from every interval extension. It requires a tree whose leaves are in order, which costs between nothing and five per cent of the wavelet tree. Both numbers on one plate, and the trade is not close.
What the machine doesTwenty bits apart
Two representations of one sparse set, six thousand seven hundred and forty-five bits against six thousand seven hundred and sixty-five. One exploits sparsity and the other exploits runs, and on this set at this density they price identically.
CountingA register kept through a confirmation
Two registers beside a two-ended selection sort's extremes, holding where the run of known order from each one ends, read 53.5 words a question on a sorted array of 1,024 keys, more than reading the record plainly — because 263,167 times a neighbour's question, answered by the previous scan's chains, rewrote a pair at the value it already held, and each rewrite dropped a register. Kept through such a confirmation and dropped only when a pair is forgotten or turns round, the registers read 0.347 words a question on that array, a 154th, and 0.345 on 4,096 keys: the reading stops growing with n. They become the cheapest record at every run length, 49 to 73 times under the next on runs and 5.75 times under the run list's search on the sorted array, and the price of a saved comparison now falls as runs lengthen, from 3.97 word reads at runs of four to 0.35. The dearest saving is now the short run's, and it is still above the 3.1 at which a table of compared pairs breaks even.
When the algorithm flips a coinA walk that stops at two lines
A blocked filter's threshold table, its lines sized for the mean number of closed blocks and each line's surplus carried forward into the next lines' empty slots, was cheapest in bits — 3.113 a block — and dear in lines, 2.183 a lookup, because a carried value joins a queue with no slack and walks 12.8 lines. Capping the walk was predicted to keep most of the bits, about 3.13, for under 2.02 lines, two hundredths under a reserve sized for the spill. Capped at two lines, the walk sends 53% of its carried values to overflow, and the table costs 3.180 bits a block for 2.027 lines. That is a hundredth under the spill-sized reserve at the same lines and a few thousandths under the uncarried line's curve: at room for the mean most of what joins the queue would wait more than two lines, so the overflow becomes the reserve by another name. With room to spare, every cap loses to the reserve.
The data that is not a numberA layout the traffic does not move
A bidirectional index whose bit vectors code each block as an arrangement or a gap code saved bits at equal work once the choice was priced where a workload's ranks land. Traffic changes, so a layout fitted once was predicted to lose most of that gain when the searches moved from one half of a text's commonest patterns to the other's, and an index that recoded its blocks every thousand searches from their own counts was predicted to keep it. Neither happened, because nothing needed to be kept. Priced on the first half's patterns and never changed, the layout does 8.24 units of in-block work a rank on the second half's, 1% above a layout refitted to them for free; recoding every thousand searches rewrites 48 of 385 blocks, 0.71 bits a character, and ends no better. The busy blocks are where the text's common substrings sort, and both halves share them: 27 of the 39 busiest blocks under one half's patterns are among the 39 busiest under the other's.
When it does not fitA level scaled by the queries it meets
A levelled store's range-filter memory, divided among its levels by their sizes with a floor of half a bit a key, comes within a few points of the division fitted to its own queries from one bit a key up; at half a bit the shuffled and the clustered store want opposite divisions. Letting the floor step aside below one bit was predicted to fix the shuffled store and cost the clustered one, and it does exactly that: 9.3 points behind the fitted division on the shuffled store instead of 34.8, 26.1 on the clustered instead of 9.1. Scaling each level by the share of queries its fences can meet — the measure the span rule should have read — brings the shuffled store to 2.2 points, but on that store every level meets the same 91% of queries, and the plain size rule with its slope cut by a tenth does the same. On the clustered store the same scaling is the worst rule measured, 41.5 points behind: its newest level meets 3% of queries and is the level the fitted division protects.
What is taught wronglyWhat a quadratic construction was setting
A depth cap of one costs 90% of an 8,192-character text in phrases, and 86% of a 65,536-character one. The ladder's conclusions hold at thirty-two times the size — and the number the ladder could not reach, the deepest chain a real collection produces, turns out to be fourteen.
The index that replaces the textThe array the walk never reads
Document listing compares a chain entry against the start of a range. The comparison is true exactly when the document has not been reported yet — which the walk already knows, because it just wrote it down.
What a bound isA code word is at least one bit
A wavelet tree of plain vectors reaches the entropy by its shape, and a Huffman code word cannot be shorter than one bit. On a collection whose document array has an entropy of 1.69 the tree costs 1.98, and the gap is a floor rather than an inefficiency.
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 the machine doesWhat a character costs on four machines
Forty-four operations, one, forty-two, and a number that moves. Four machines for one language, with the construction charged separately from the steps, because a machine that is free per character paid eleven thousand operations before the first one.
CountingA record switched off by its runs
The two-ended selection sort's record of adjacent order, consulted through registers kept through confirmations, became cheap everywhere but on the shortest runs, where it saves too few comparisons to pay for any reading. The pass between neighbours already counts the runs, and a switch that keeps the record only when the pass's mean run reaches a threshold was predicted to find its best threshold at a mean of six keys, at three word reads a comparison. On even runs it does — and what the switch saves there is three thousandths of a sort. The risk the proposal named is the larger result: on inputs whose keys are half in short runs and half in long ones, the pass reads a mean of 4.4 to 4.9, under the threshold, while the long half saves the record up to 74,000 comparisons, and switching it off costs up to 31%. Over every input the best threshold is three: keep the record for everything but runs of two.
When the algorithm flips a coinA queue that drains at intervals
A blocked filter's threshold table, its lines sized for the mean number of closed blocks and each line's surplus carried into the next lines' free slots, is cheapest in bits and dear in lines, because at room for the mean the queue of carried values never drains. Capping the walk turned the cap into an overflow. Giving every eighth line a deviation of extra room was predicted to empty the queue there, bound every walk by eight lines, and cost about 3.15 bits a block for under 2.02 lines a lookup. It costs 3.123 bits for 2.103 lines. After a slack line the queue is empty a third of the time, walks reach 33 lines, and the reason is arithmetic: one line's deviation of room is not room for eight lines' wander, which grows as the square root of eight. The slack lines do buy something — a hundredth or two of a bit a block under spreading the same room evenly, at five of the six spacings — but they buy it at lines no other design here would accept.
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.
StructuresA document already in the answer
The same walk, with the chain's test replaced by a lookup in the answer so far. It reports the same documents at the same cost, and two exchanged lines make it lose nine of seventeen without failing.
The index that replaces the textThe index that does not notice
Three compressed indexes over the same characters. One is flat at six and a half bits a character however many copies the collection holds; the other two fall by factors of five and six. At one copy the two that fall are the largest of the three.
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.
The other axisWhat the chain cost
The chain of previous occurrences is one row number per row — exactly as wide as the suffix array it sits beside, and the largest single part of a document-listing apparatus. It is now absent, and what replaces it is one bit per document.
What is taught wronglyA boundary that costs nothing
A separator occurs nowhere else in the collection, so it must break a phrase that would have spanned it. Cutting a repetitive text into a hundred and twenty-eight documents raises its run count by two per cent, and cutting prose raises it by ten.
StructuresThe price of a boundary is what precedes it
A separator sorts before everything, so its rows sit at the top of the suffix array and hold the documents' last characters. What a document boundary costs the transform is the entropy of the character in front of it, and nothing else.
The index that replaces the textThe tree answers the question
The distinct documents in a range of rows are the distinct symbols of the document array in that range. A wavelet tree enumerates those in one descent, so the range minimum, the chain, the bitmap and the recursion all go at once.
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 doThe apparatus that is smaller than its index
Answering "which documents hold this" at a price proportional to the answer used to cost 2.70 times the index it sits beside. Two changes later it costs 0.84, and the largest thing left is an array that says which document each row belongs to.
The other axisTwo currencies for one separator
Giving every document its own boundary marker costs a fifth of the packed text and three per cent of the run count. Both numbers are right, they are about the same change, and which one a collection pays depends on a structure nobody named.
What is taught wronglyOne copy per document is one occurrence per document
The output-sensitive listing apparatus wins when a pattern occurs far more often than it occurs in documents. A collection of versions was supposed to be that case, and it is the one collection where the two numbers are equal by construction.