Exponential time

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Also named here as randomised algorithm — the same set of essays touches all of them, so they are one junction rather than several.

A walk that beats trying everything

To decide whether clauses of three letters can all be satisfied, the obvious method tries all 2ⁿ assignments. Uwe Schöning's method, from 1999, starts at a random assignment and wanders: pick a clause that is false, flip one of its letters at random, and repeat three times as many times as there are letters. A single try usually fails, but it succeeds with chance at least about (3/4)ⁿ, so about (4/3)ⁿ tries are enough — and the reason is a walk on a line that goes the wrong way two times in three.

logic · Class diagrams

A letter the clauses already decide

Set the letters of a formula one at a time, in a random order, and guess each one with a coin — unless some clause has already had its other two letters set to false, in which case the clause decides. A try succeeds when every guess is right, so what matters is how many letters get guessed. On a formula with one solution, every letter has a clause that forces it in at least one order out of three, so at most two thirds of the letters are guessed on average, and about 1.587ⁿ tries suffice. It is slower than the random walk. Refined, it is the fastest method known.

logic · Class diagrams

Randomised algorithmSatisfiabilityClauseCritical clauseExhaustive searchExpectationHamming distanceRandom walkUnit propagationVenn diagram

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