Algorithms & Data Structures
the bound is a claim, and it is checkable
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The thread: Distributions, not averages

The expected cost of a randomised algorithm is the least interesting thing about it. What matters is how tightly the distribution concentrates, and that is what makes the bad cases stop mattering.
110100257amortised 2.000256512append numbercost of that append (log scale)growth factor 2, cost = 1 write + a copy of the array when it resizes9 resizes in 512 appends Structures

What amortised means

Appending to a dynamic array is O(1) amortised. It is also, on 512 appends, an operation that costs one unit 503 times and 257 units once. The amortised bound is a true statement about the sequence and a false one about any append in it, and the picture that shows why is a sawtooth nobody draws.

 mean 494599th 60304,2825,4136,543comparisonsruns600 independent random inputs, n = 512worst run 1.32× the mean What is taught wrongly

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.

 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 is taught wrongly

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.

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