Algorithms & Data Structures
the bound is a claim, and it is checkable
Series

Distribution — the series

6 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. mean 494599th 60304,2825,4136,543comparisonsruns600 independent random inputs, n = 512worst run 1.32× the mean

    The words "on average" are not a number

    Quicksort is Θ(n log n) on average. Six hundred runs at n = 512 give a distribution with a mean of 4,945 comparisons, a median of 4,908, and a worst case 32% above the mean. The average is a summary of that picture, it is the least interesting thing in it, and it is almost always the only thing reported.

    part 1 · wrong
  2. first-element pivot130,816 on sortedmedian of three5088 meanrandom pivot4937 meanblue bar: range over 400 random inputs · marker: the already-sorted inputn = 512a bad case that cannot be chosen is a different kind of bad case

    What randomising the pivot buys

    Quicksort taking the first element as its pivot costs 130,816 comparisons on an already sorted array of 512 — 27 times its cost on random data, and exactly the quadratic behaviour the algorithm exists to avoid. Randomising the pivot costs 5,490 on the same input. Randomisation does not make the bad case impossible; it makes it unchoosable.

    part 2 · wrong
  3. 1234560.000.250.500.75formulameasuredload factor αprobes per insertion8,192 slots, mean of 8 fills, seeds stated in lib/structures.jstwo routes agree to 2.3%

    The probe formula nobody checks

    The expected number of probes to insert into a hash table under linear probing is ½(1 + 1/(1−α)²). It is quoted constantly, it is correct, and applied to a table of 256 slots at 95% load it overstates the measured cost by nearly half — because it is an asymptotic result and a real table is not asymptotic.

    part 2 · wrong
  4. quicksort on sorted input — comparisonsfirst-element pivot2,096,128random pivot25,318a search tree built from sorted keys — heightbinary search tree2,047treap26a hash table on keys chosen to collide — worst bucketthe low bits of the key2,048multiply–shift, a at random10n = 2,048, bars on a logarithmic scaledeterministic above, randomised below

    Expected is not average

    Quicksort on 2,048 sorted keys costs 2,096,128 comparisons with a first-element pivot and 25,318 with a random one. A binary search tree on the same keys is 2,047 deep; a treap is 26. A hash table on keys computed against its hash puts all 2,048 in one bucket; one drawn from a family puts at most 10 there. Three problems, one distinction.

    part 3 · bounds
  5. randomised pivot164 random bits2.96 ±23%median of mediansno random bits7.94 ±2%0.04.38.6comparisons per element — bar is the range over seeds, tick is the mean30 seeds, n = 4,001, median2.68× the mean, 14× the spread

    What derandomising costs

    Randomised selection finds the median of twenty thousand elements in 3.21 comparisons per element and median-of-medians takes 8.15 — two and a half times as many for the same answer, both linear. The number that decides between them is not either of those. It is that the first varies by 28% from seed to seed and the second by 1.6%.

    part 4 · bounds
  6. average 8.001,0242,048bucket, 0 to 255keys in the bucketmultiply–shift · keys built for this hashworst bucket 2,048 against 8.0

    The adversary who knows the seed

    Every randomised figure on this site is drawn from a stated seed, so that the numbers in the captions are the numbers on the reader's screen. That is also the exact condition under which none of the guarantees those figures demonstrate applies. A published seed is a published function.

    part 4 · wrong

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