Algorithms & Data Structures
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The thread: Measured, not assumed — page 9

Page 9 of 9, continuing the same thread in the same order.
equal lengths5.00 bits100.0%lengths as one over rank4.15 bits83.4%a few long, many short4.31 bits86.7%one document holding most of the text1.69 bits39.5%log₂ 3232 documents · 8,192 characters100.0% to 39.5% What is taught wrongly

A saving quoted without its collection

A check asking whether a compressed document array is smaller than the plain one passes on the rounding whenever the document count is not a power of two. It would report a saving of nothing as sixteen per cent, on a collection that has no redundancy at all.

 extensions attempted9,206 dead7,594 livebit-vector ranksthe loop: 168,000the descent: 24,0868 patterns · 1 error · sigma 2054.8% dead · 6.98x Structures

The branches that find nothing

An approximate search over a twenty-symbol alphabet attempts sixteen thousand eight hundred extensions and nine thousand two hundred of them produce an empty interval. That is a full rank walk whose entire result is the discovery that nothing was there.

 02040255075100125one sampled position in every …of the index, per centplain marks: 18.5%Elias-Fano: 3.8%16,384 characters14.9% off at one in 32 The index that replaces the text

What the locating apparatus becomes

The two parts that answer "where" are half an index at a dense sampling and a fifth at a sparse one, and the fifth does not fall further. Represent the marks properly and it keeps falling, to under four per cent.

 a loop over the alphabet1,478,400the compound walk, inside the loop134,40011xone descent per node18,91678x6 patterns · 1 error · sigma 2155.8% of the extensions were dead The index that replaces the text

Three savings on one structure

A factor of eleven on an extension, a factor of seven on a branching search, and a sixth of the bits. Applied to one bidirectional index they do not give a factor of seventy-seven, and the reason is that two of the three are the same saving.

 rounds, if every ready cell ran at oncerow order, stored by rows65,793anti-diagonal order, stored by rows513anti-diagonal order, stored by diagonals513reads that miss the cacherow order, stored by rows6.3%anti-diagonal order, stored by rows31.1%anti-diagonal order, stored by diagonals8.7%fully associative · 32 lines × 8 elements · LRU263,169 reads and writes per order What the machine does

The table stored the way it is filled

Store an edit-distance table by anti-diagonals instead of by rows, and the anti-diagonal fill keeps its 513 rounds while its cache misses fall from 31.1% of reads to 8.7%. It does not fall to row order's 6.3%, and the gap is not noise — on caches of four and eight lines the two rates are 9.4% and 6.3%, exactly three to two, because a cell reads from two earlier diagonals and only one earlier row. The same layout turns row order into the order that strides, at 28.3%. How a table is stored and the order it is filled in are one decision, and its price is the number of earlier fronts the recurrence reads.

 02040608000.50011.502errors permittedbranches that find nothing, per cent5.75x6.98x9.22xthe label is thesaving at that budget8 patterns · 8 characters31.3% to 74.9% What the libraries do

A 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.

 a fixed-length codeleaves in order␣acdefghilmnoprstuvwythe best ordered treeleaves in order␣acdefghilmnoprstuvwythe best tree of any shapeleaves in frequency order␣eahnrstdiloucfgmpvwythe symbols, in the order the descent reports them21 symbols in 2,048 positionsone set, 2 of 3 sorted What is taught wrongly

One set, three orders

The symbols an interval holds do not depend on the tree's shape. The order they come out in does, and a search that accumulates a running count as it reads them computes a plausible number that is wrong on eighty per cent of queries.

 acdefghijklmnopqrstuvwyzroot24 of 26 symbols present54 ranks · 51 nodes What is taught wrongly

A walk that does not prune

Remove the emptiness test and the descent visits every node of the tree, returns exactly the same symbols with exactly the same intervals, and costs sixty per cent more. No test of the answer can see it.

 plain bit vector19,475compressed blocks4,55723.4%Elias–Fano3,90520.1%positions, priced as Elias–Fano3,59118.4%16,385 rows · one in 32 markedElias-Fano at 20.1% Structures

A position split in two

Write each sorted position as a high part and a low part. Store the low parts packed and the high parts as a bit vector in which the k-th one sits at position (p >> w) + k. A select on that vector and a low read recover any position.

 both halves locate250,584100.0%the reverse half counts only226,50490.4%and its marks are Elias-Fano210,93484.2%the saving spent on sampling241,47596.4%16,384 characters · one in 3284.2% smaller, or the same size and faster The other axis

The ladder, and the rung that spends

Two hundred and fifty thousand bits, then two hundred and twenty-six, then two hundred and eleven. The fourth rung takes the whole saving and buys a four-times denser sampling with it, landing at ninety-six per cent of where it started and locating several times faster.

 01,024120 phrases · 1,024 characters60 drawn, every one pointing right Structures

Every copy points right

A greedy self-referential parse chooses each phrase's source from text already produced, so an occurrence copied from another lies strictly to its right. That one fact removes a visited set from a propagation, exactly as a left-first walk removed an array one strand ago.

 "t than"8 found · set 8 · frontier 4"of her"7 found · set 7 · frontier 4" that "16 found · set 16 · frontier 11" every"31 found · set 31 · frontier 19"the ev"8 found · set 8 · frontier 4"o of c"8 found · set 8 · frontier 5" than "15 found · set 15 · frontier 9"ime ra"7 found · set 7 · frontier 4the visited set, pale; the sweep's frontier, dark17,715 phrases examined either way1.67x on what is held The index that replaces the text

The same occurrences, less bookkeeping

Two traversals examine identically many phrases and report identically many occurrences. What differs is that one holds every occurrence found so far in a set and the other holds an ordered list and a cursor.

 124816326410step width, cellscache lines a stepstored by diagonalsby diagonals, each line-alignedstored by rows512 × 512, lines of 8 cellsone step of an anti-diagonal What the machine does

Eight cells at once

The anti-diagonal fill order exists because its cells do not depend on one another, and every table filled here has been walked one cell at a time anyway. Computed eight at a time, a step touches 5.71 cache lines on the layout that stores the table by diagonals and 10.87 on the one that stores it by rows — and per cell the first keeps falling to 0.42 while the second stops at 1.27. The prediction that a diagonal step would touch three or four lines was wrong, and line-aligning each diagonal only takes it to 4.94.

 01e+42e+43e+44e+450100one sampled position in every …bitsplain, one bit a rowcompressed blocksElias-Fano16,384 rowscrosses at one in 4 What is taught wrongly

Where the sparse representation loses

At every row marked, Elias–Fano costs twice the plain vector. The crossing is at one row in four, which is a sampling rate a real index uses — so the choice between the two is a choice, not an improvement.

 bits heldthe model3,591the structure3,905one rank, in operationsbinary search10 stepsselect and walk1 select + 0.7016,385 rows · one in 32 marked14x on the operation What is taught wrongly

A price with no structure under it

A class in this collection has charged Elias–Fano's price for several strands and stores an array of positions searched by binary search. The accounting is right about space to a bit per thousand and wrong about one operation by a factor of eight.

 0102030100200one sampled position in every …of both halves saved, per centplain marks: 13.5%Elias-Fano: 1.8%16,384 characters17.6x against 2.3x What the libraries do

The floor was the marks

A saving reported as about a sixth of a bidirectional index, falling to an eighth and levelling off. Represent one array properly and it falls to a fiftieth instead — most of what was being dropped was a badly encoded bit vector.

 factor against the published loopthe compound walk11xthe enumeration78xthe two, multiplied860xnothing measures thisan index with both78xthe shortfall is 11x, and the walk's factor is 11x6 patterns · 1 error · sigma 21the larger of the two, not the product What is taught wrongly

Two factors that do not multiply

Eleven times and seventy-eight times against the same baseline, so an index with both should be eight hundred and sixty. It is seventy-eight, and the shortfall is eleven — the first factor, exactly, because the second operation already contains it.

 a loop over the alphabet107.9the compound walk10.0the cheapest hereone descent per step19.016 exact patterns of 12ranks per step · sigma 21 What the libraries do

The saving that is a loss

An operation that is seventy-eight times cheaper on a branching search costs twice as much on an exact one. It reports every symbol present in order to hand back the one that was asked for, and a search that knows its character needs none of the rest.

 020406080255075100125one sampled position in every …of the plain index, per centsize: 84.7%67.5 steps16,384 characters · 8-character patternssize falls, the walk lengthens Structures

Bits 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.

 phrase lengths2,71222.6%phrase sources2,71222.6%phrase literals1,1309.4%boundary orders5,42445.3%three permutations of 226 elements: 1,808 bits each, and one is the inverse of another226 phrases · 4,096 characters45.3% in the orders The other axis

Half an index is three permutations

A phrase index stores a length, a source and a literal per phrase — and three orderings of its boundaries, at forty-five per cent of the structure. One of the three is the inverse of another, and nothing needs both at once.

 0a1b2split3split4a5a6b7split8a9b10split11a12b13split14accept9 character tests · 5 splits · 1 accepting(a|b)*a(a|b)(a|b)(a|b)15 states Structures

Two states per operator

Thompson's construction adds a bounded number of states per rule and no rule copies a sub-machine, so the machine is linear in the expression and is built in linear time. Every exponential in this field is somewhere else.

 0100200300400102030copies of the basebits held during one querythe visited set: 420the frontier: 2666-character patterns20x to 1.58x What is taught wrongly

A constant factor, not a term

The substitution was expected to remove a quantity proportional to the answer. Both traversals hold a quantity proportional to the answer, and what it removes is a factor of 1.6 that shrinks as the collection grows.

 phrases examined per probethe linear scan207.0sorted, with a running maximum10.9 — 19xthe running maximum costs 2,821 bits — 21.6% of the index40 of 40 positions checked, and the two agree at all of them238 phrases · 6,144 characters19x on the scan What the libraries do

The scan the order does not touch

Two hundred and seven phrases examined per probe, against eleven. A running maximum of the source regions' right ends lets a leftward walk stop for good, and it costs twenty-two per cent more bits.

 10010³10⁴10³10⁴10⁵characters of textoperations, construction includedcrosses at 256the NFAthe DFA64 DFA statescrosses at 256 characters What the libraries do

Where 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.

 10100states the cache holdsoperations a characterthe plain NFA: 52.5the whole set fits: 19.94,096 characters · k = 8the knee is at 512 states What is taught wrongly

A cache below the reachable set

A lazy machine with a cache of two hundred and fifty-six states costs fifty-one operations a character and a plain non-deterministic simulation costs fifty-three. At five hundred and twelve it costs eleven. The line is flat across two orders of magnitude and then falls off a cliff.

 1010³10⁴10⁵10⁶characters of textstepsbacktrackingThompson's NFAopen points: capped at4,000,000(a|a)*b12,158x at 20 characters What is taught wrongly

The folklore is about a matcher

Four million steps against three hundred and twenty-nine, on a twenty-character expression matched against twenty characters. One of the two machines doubles with every character of the input and the other does not, and only one of them is what a regular expression is.

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