Preprocessing — where it appears
Named by 24 essays across 7 fields — each of them below, with the objects they name alongside it.
An index larger than what it indexes
A suffix array over 16,384 characters is 229,376 bits, and it cannot answer a single question without the 81,920 bits of text beside it. Nearly four times the text, to search the text. Every index on this site had been weighed at zero until somebody put one on a scale.
A search that runs backwards
Twenty-three occurrences of a six-character pattern in sixteen thousand characters, found in twelve rank queries and zero character comparisons. Not few comparisons — none. The algorithm never asks whether two symbols are equal, and it knows how many matches there are before it has located one.
The shift the pattern already knows
On a text of 8,192 characters the naive scan examines 520,256 of them and Knuth–Morris–Pratt examines 16,321 — a factor of 32, and 16,321 is 99.6% of the 2n that no input can push it past. On ordinary random text the same two algorithms examine 20,862 and 20,833. Both measurements are of the same pair of algorithms and only one of them is the reason anybody uses the second.
The shift a set of patterns allows
Aho-Corasick reads every character of the text exactly once, whatever the number of patterns. Commentz-Walter reads backwards inside a window and steps over what it can, and on eight patterns of ten characters it looks at sixty-two per cent of a twenty-thousand-character text. The rule that does the skipping is not the one everybody implements.
An estimate borrowed from an easier problem
On a grid where every step costs one, the straight-line distance to the goal cuts a search from 543 cells to 325. On terrain where steps cost between one and nine it cuts 1,572 to 1,550, because it still believes every step costs one. Four exact distance tables, computed once, cut the same search to 252 — and cost 6,328 reads to build, so they pay for themselves on the fifth query.
The index that is the text
A suffix array sorts all 4,097 suffixes of a text — 8.4 million characters of string, in total — and examines exactly zero characters doing it. It then answers a search in 91 characters where a scan costs 1,472, and the whole thing pays for itself at six queries. Both halves of that are worth the same amount of attention, and the first is the one that is usually skipped.
The table that walks every pair
The exact shift rules cost 769,724 character comparisons to build for 128 patterns and the published ones cost 5,604. The scan they are both built for reads 41,580 characters, so one of the two constructions is eighteen times the work it is there to save.
The rule that pays on a long enough text
With two patterns, the cheap tables cost 106 steps and the scan reads 13,084 characters; the exact tables cost 594 and the scan reads 12,306. Below thirty-two thousand characters the cheap tables win the total, above it the extra skipping pays for them, and with thirty-two patterns there is no crossing at all.
The ceiling the shortest pattern sets
A matcher that skips is described as faster than one that reads every character, and the description leaves out what decides it. No shift can exceed the shortest pattern in the set, so adding one two-character pattern to fifteen of sixteen characters takes a run from reading fifty-eight per cent of the text to reading all of it twice.
The shift somebody published
The exact rules for shifting a multi-pattern window are a definition that quantifies over every pattern at every offset. The 1979 rules are two tables read off the trie's own failure links, they are computed in one pass, and on this pattern set they agree with the definition at every node.
Where the exact rules pay now
With a construction as cheap as the published one, the exact shift rules pay for themselves past eight thousand characters of text at two patterns, four thousand at four, and never at thirty-two — because by thirty-two patterns the two rules make identical decisions.
The table the links already knew
The exact good-suffix rule costs 757,058 character comparisons to build from its definition at 128 patterns, and 3,824 from the trie's failure links. Same table, checked at every node — a factor of 198, and the definition was never the algorithm.
Where 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 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.
How 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%.
A placement after the traffic moved
Four landmarks fitted to twenty queries that all end at one cell expand two thirds of the corners' cells there. Move the destination ten cells and the advantage is gone; move it thirty and the fitted placement expands nearly twice the corners' cells. Choosing again costs 317,000 cells of search — the saving of 8,600 queries — so on streams whose destination drifts, a policy that refits on a counter never beats fitting once and keeping the result, and past a drift of about ten cells nothing fitted beats the corners.
A refit that changes one landmark
Choosing four landmarks again after traffic moves costs hundreds of thousands of cells of search, and on drifting streams it never paid. Keeping three and re-choosing the fourth costs a fifth as much — 0.19 of a full selection, not the twelfth the earlier page guessed. On drifting streams, refitting on the counter goes from a loss of up to 13% to within 2% of fitting once, and it wins where the destination moves six cells. When the destination jumps, it is the cheapest policy measured, 12% under the corners at one jump in 10,000 queries — unless the jumps come as often as one in 2,500.
The candidates near where the queries end
Re-choosing one landmark after the traffic moves tried every candidate as the replacement, and most of them could not win. Tried only on the four candidates nearest the new destination, the refit costs 0.085 of a full selection — the twelfth an earlier page guessed — against 0.236, and its placements serve fresh queries within 3% as well. Where the full search's winner lies further away, the refit was barely worth making. A refit that cheap still cannot follow a destination that jumps every 2,500 queries; what holds the counter back was never the refit's price.
Starting from the corners
A landmark search that refits one landmark when its traffic moves lost to the fixed corners when the destination jumped often, and the accounting blamed the one full selection every refitting policy makes at the start. Starting from the corners instead, and never selecting in full, closes 82% of that gap at one jump in 2,500 queries and is the cheapest policy of all at one in 5,000. It gives the full selection back its advantage when jumps are rare, and on traffic that drifts a step at a time it gives up fitting altogether. Its counter fires when the cost rises, and from the corners concentrated traffic raises nothing: the corners are no worse for it, only no better.
What the approximation gives up
Compared at every one of the 3,736 decisions a search could ask about, the published shift rules and the exact ones agree at all of them on a set of 128 patterns. At two patterns they differ at five of 84, by up to four positions — and the run reads 6.3% more characters.
A refit priced before it is made
A landmark placement that starts at a maze's corners and refits one landmark when a counter sees recent queries getting dearer closed most of its gap on traffic that jumps — and on traffic that drifts it never fired, because from the corners concentrated traffic raises no cost. The chance it missed is cost that could fall. Pricing that chance directly — the current placement on the last twenty queries against four nearby candidates, every two thousand queries, refitting when the estimated saving pays for the refit — costs 120.5 cells a query at one jump in 2,500 against the counter's 128.9 and the corners' 126.8, and 107.4 on a slow drift against 103.3 for a full selection fitted once. It fails where jumps come every thousand queries, and on one stream it stops firing after its placement has aged.
A horizon the stream can supply
A watch that refits a landmark placement when an estimated saving pays lost where traffic jumped every thousand queries, because its estimate assumed each saving would last a whole window. The horizon proposed to correct it, learned from the intervals between the watch's own refits, is the window every time: the watch looks once a window, so its refits are never closer together than one. A horizon learned from the traffic instead — how long it stays within ten cells of one destination — is safe on every stream and saves 3.4% at one jump in 2,500. At one in 1,000 it reaches 150.7 cells a query against the corners' 142.7, and a horizon read from the future only ties them. At that pace the looking costs what the refits save.
A watch that looks less
A landmark watch that refits when an estimated saving pays loses to the plain corners when traffic jumps every thousand queries, because its looking costs more than its refits save. Using its learned horizon to look less often there — doubling the interval when the horizon falls below half of it, halving it when the horizon comes back — was predicted to tie the corners at that pace and cost nothing elsewhere. It ties nothing: 150.5 cells a query against the corners' 142.7, since the looks it skips are the looks that found the chances. And at one jump in 2,500, where the watch had won, it loses by twenty cells, because the rule cannot come back down. Halving needs a horizon as long as the stretched interval, and at 8,000 queries no stream here supplies one. A two-state rule that can come back does better and still loses at both paces. The looking was never the waste; at one jump in 1,000 there is nothing to watch for.
The ratio that was an implementation
This collection published a factor of forty-two between two shift rules' preprocessing. Sixty-nine per cent of the denominator was a table the published rule never reads, and the numerator was a definition rather than a construction. The corrected ratio is 1.6.
Named alongside it
The objects these essays reach for when they reach for this one.
Break-evenMeasurementParameter choiceShortest pathHeuristic searchLandmarkHonest limitShift ruleMulti patternQuery distributionCommentz-walterPattern matching