The thread: The subproblem is the unit
The cost is the number of subproblems
The edit-distance recurrence, written down literally, makes 29,737 calls on a six-letter word and a seven-letter word. Written down with a table beside it, it makes 56. Nothing about the arithmetic changed, and the class did.
When the algorithm is a tableThe same table, filled two ways
Top-down and bottom-up compute identical cells and return identical answers. One of them asks the table half a million questions and recurses four hundred frames deep; the other asks none and recurses none — and on a knapsack it fills twenty-two times as many cells as anything can reach.
When the algorithm is a tableA band as wide as the answer
If two strings are close, the optimal route stays near the diagonal and nine cells in ten cannot be on it. A band of three finds the right answer on a pair 300 characters long — and a band of thirty-two is needed before anything can prove it.
When the algorithm is a tableThe cells are not the cost
This field opened by pricing a dynamic program in subproblems — 29,737 calls became 56 cells and the class changed. That is right when a cell is cheap. A table over intervals has 8,385 cells and considers 349,504 transitions to fill them, and the cubic in its bound is inside the cell rather than in the table.
When the algorithm is a tableA cell that has to know where it is
A gap of four characters is usually one event, not four. No recurrence over a single table can charge it that way, because the price of a gap character depends on how the cell above it was reached and a cell holding one number has thrown that away. The repair is three tables, and it costs exactly three times the cells.
When the algorithm is a tableThe argmin that cannot go backwards
The same triangular table, the same ninety-one cells, the same tree at the end of it — and 364 transitions one way against 156 the other. At 256 keys the ratio is 38. What removes the factor is not a property of the recurrence but a property of the numbers it is given, and the recurrence does not mention them.
When the algorithm is a tableThe order that has a depth
One hundred cells, filled in three orders, producing one table. Row order takes ninety-one steps and anti-diagonal order takes nineteen. Nineteen is not a property of the order — it is the longest chain of cells in the recurrence itself, no schedule can get under it, and every count taken until now was a total that could not see it.
The data that is not a numberA distance that is a path through a grid
How far apart two strings are is a shortest-path problem on a grid whose every edge is drawn by the recurrence — and finding a string in a text costs 4,988 character comparisons where measuring how far it is from one costs 96,000.
The data that is not a numberThe filter that feeds the table
A self-index answers exact queries and nothing else. Approximate matching needs a table with twenty thousand columns in it. The pigeonhole joins them — cut the pattern into k+1 pieces and at least one occurs exactly, and the index that cannot answer the question decides where to ask it.
When the algorithm is a tableThe cells that were never worth having
Two three-hundred-character strings over twenty-six letters give a table of 90,601 cells, and 3,421 of them are pairs of positions whose characters agree. Only those can lengthen anything. A method that enumerates exactly those computes a twenty-sixth of the table — and on a two-letter alphabet it computes half of it and is worse than the table it replaced.
The data that is not a numberThe row that starts at zero
The same 1,413 cells, filled by the same recurrence in the same order, answer 148 and 0. One line of initialisation decides which question the table was asked, and only one of the two answers is about whether the pattern is there.
When the algorithm is a tableA triangle stored in a square
An interval table has a cell for every range of keys and nothing below its diagonal, and it can be stored as a square array, as packed rows, or as packed diagonals — the last matching the order it is filled in. On sixty-four keys, with every read replayed through a small cache, the square misses 39.7% of its reads, packed rows 38.8%, and packed diagonals 78.4%. Storing a table in the order it is written is storing it in the order it is not read.
What is taught wronglyA table wider than its input
The knapsack table has (n+1)(W+1) cells and is called polynomial. Adding one character to the input doubles it — across four settings the table grows sixty-four times while the input it is written from grows by half.
When the algorithm is a tableThe bound the search finds for itself
A spelling checker that computes the full edit-distance table against every word in a 2,424-word vocabulary fills 156,714 cells for each misspelt query. Bound each table by the best distance found so far, and abandon it the moment a whole row exceeds that bound, and the same search fills 40,273 and finds the same words. Meet the candidates nearest in length first and it fills 26,203, starting a table for exactly the words a search that knew the answer in advance would start. The last factor of 1.7 is the price of not knowing, and it is largest when the misspelling is smallest.
When the algorithm is a tableThe columns the candidates share
Three thousand tables against one query, and most of them begin the same way. Stored as a trie, the 2,424-word vocabulary has 7,710 distinct prefixes holding 17,239 letters, and a search that computes one column per prefix reads 61,449 cells against 156,714 — before it applies any bound at all. Apply the bound at a prefix instead of at a word and it reads 16,958, beating a list search that was told the answer in advance.
When the algorithm is a tableThe bound a block can and cannot have
Knuth's condition turns an interval table's cubic fill into a quadratic one by bounding each cell's best split between its two neighbours'. A blocked fill cannot use it a cell at a time, and the two cells that bound a block lie outside the block — one to its left, one below it. The schedule has finished both for ten per cent of the blocks, the bound then removes eleven per cent of the splits, and it removes half a per cent of the cache misses, because the splits it skips are the ones already in the cache.