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Where a source's text could not be read, the entry says so and gives what was checked instead.","knuthVol3":{"cite":"Donald E. Knuth, The Art of Computer Programming, Volume 3: Sorting and Searching, second edition, Addison-Wesley, 1998 (ISBN 0-201-89685-0)","read":"Not read: the book is not freely available. Checked: a bookseller's copy of the publisher's table of contents lists 'Sorting by Insertion' (page 80) and 'Sorting by Merging' (page 158) under internal sorting; the 5.2.1 and 5.2.4 numbers are the book's usual numbering and were not seen in that listing. Section 6.2.1 ('Searching an Ordered Table', Algorithm B, binary search) was not seen in any table of contents; the UNIX System III bsearch manual page and the illumos bsearch.c header (seen in web search results, not opened) state they are 'generalized from Knuth (6.2.1) Algorithm B', which confirms the section and algorithm name, not the bound.","citedFor":["5.2.1: straight insertion; the link between the moves it makes and the inversions of the input (unverified wording)","5.2.4: merge sorting (unverified wording; the worst-case count n log2 n - n + 1 for n a power of two is derived and measured here, not quoted)","6.2.1: Algorithm B, binary search with three-way comparisons; at most floor(log2 n) + 1 comparisons for a successful search (unverified wording; measured here exactly for every key)"]},"clrs":{"cite":"Cormen, Leiserson, Rivest and Stein, Introduction to Algorithms, fourth edition, MIT Press, 2022 (ISBN 978-0-262-04630-5), chapter 2 Getting Started","read":"Table of contents checked from a copy of the front matter: 2.1 Insertion sort (page 17), 2.2 Analyzing algorithms (page 25), 2.3 Designing algorithms (page 34, merge sort). The text of the sections was not read for this lesson.","citedFor":["2.1 and 2.2: insertion sort, its best case on sorted input and its worst case on reversed input; the book counts running time in its own cost model, so the exact comparison counts n - 1 and n(n-1)/2 are derived here and checked by the counters, not quoted","2.3: merge sort by divide and conquer"]},"derived":{"about":"Closed forms derived for this lesson, each checked exactly against the counters in search.test.mjs and in the closedForms block.","forms":["binary search average over all n successful searches = 1 + ((n + 1) q - 2^(q+1) + 2) / n, q = floor(log2 n): the tree of probes has its leaves on two adjacent levels, so its internal path length is the minimum, sum over k = 1..n of floor(log2 k)","insertion sort comparisons = inversions + (n - 1) - (keys that travel all the way to the front); on a random permutation of distinct keys the expectation is n(n-1)/4 + n - H_n, H_n the n-th harmonic number","merge sort with n = 2^k: merging two runs of m takes between m and 2m - 1 comparisons, so the total lies between (n/2) log2 n and n log2 n - n + 1, and every level writes n elements, n log2 n moves"]}},"inputs":{"about":"Seeded inputs (makeInput in search.mjs, the shared mulberry32 generator rng from tinyformer-ops.mjs): random is a Fisher-Yates permutation of 0..n-1; sorted and reversed are 0..n-1 forwards and backwards; few draws each value from 0 to 3. 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Linear search on the same arrays finds every key (search.test.mjs).","unsortedBinary":{"rows":[{"n":8,"seeds":10,"keys":8,"wrong":4,"wrongFraction":{"mean":0.6125,"min":0.5,"max":0.75},"maxComparisons":4,"bound":4},{"n":16,"seeds":10,"keys":16,"wrong":11,"wrongFraction":{"mean":0.7688,"min":0.6875,"max":0.8125},"maxComparisons":5,"bound":5},{"n":32,"seeds":10,"keys":32,"wrong":26,"wrongFraction":{"mean":0.85,"min":0.8125,"max":0.9063},"maxComparisons":6,"bound":6},{"n":64,"seeds":10,"keys":64,"wrong":57,"wrongFraction":{"mean":0.9109,"min":0.8906,"max":0.9375},"maxComparisons":7,"bound":7},{"n":128,"seeds":10,"keys":128,"wrong":118,"wrongFraction":{"mean":0.9461,"min":0.9219,"max":0.9609},"maxComparisons":8,"bound":8},{"n":256,"seeds":10,"keys":256,"wrong":246,"wrongFraction":{"mean":0.9691,"min":0.9609,"max":0.9805},"maxComparisons":9,"bound":9},{"n":512,"seeds":10,"keys":512,"wrong":501,"wrongFraction":{"mean":0.982,"min":0.9785,"max":0.9863},"maxComparisons":10,"bound":10},{"n":1024,"seeds":10,"keys":1024,"wrong":1011,"wrongFraction":{"mean":0.9902,"min":0.9873,"max":0.9941},"maxComparisons":11,"bound":11},{"n":2048,"seeds":10,"keys":2048,"wrong":2033,"wrongFraction":{"mean":0.9953,"min":0.9927,"max":0.9971},"maxComparisons":12,"bound":12},{"n":4096,"seeds":10,"keys":4096,"wrong":4076,"wrongFraction":{"mean":0.9969,"min":0.9951,"max":0.998},"maxComparisons":13,"bound":13},{"n":8192,"seeds":10,"keys":8192,"wrong":8176,"wrongFraction":{"mean":0.9985,"min":0.998,"max":0.999},"maxComparisons":14,"bound":14},{"n":16384,"seeds":3,"keys":16384,"wrong":16366,"wrongFraction":{"mean":0.9991,"min":0.9989,"max":0.9993},"maxComparisons":15,"bound":15},{"n":32768,"seeds":3,"keys":32768,"wrong":32750,"wrongFraction":{"mean":0.9996,"min":0.9995,"max":0.9997},"maxComparisons":16,"bound":16},{"n":65536,"seeds":3,"keys":65536,"wrong":65515,"wrongFraction":{"mean":0.9998,"min":0.9997,"max":0.9998},"maxComparisons":17,"bound":17}]},"verdict":"Binary search on unsorted input misses 61.3 percent of keys at n = 8 and 99.975 percent at n = 65536, while never using more than floor(log2 n) + 1 comparisons."},"timing":{"about":"Wall time in Node v25.2.1 on this machine only (13th Gen Intel(R) Core(TM) i7-13700KF, win32), one process at below-normal priority, median of 3 or 5 runs, random input seed 1. Searches in microseconds per lookup (1,000 keys; binary repeated 100 times), sorts in milliseconds per sort, including the counting. builtinSortMs is the engine's own typed-array sort, for scale. Another machine, engine or load gives other numbers; the comparison counts do not change.","rows":[{"n":1024,"linearSearchUs":0.3241,"binarySearchUs":0.0437,"insertionSortMs":0.4242,"mergeSortMs":0.0992,"builtinSortMs":0.0282},{"n":4096,"linearSearchUs":2.9458,"binarySearchUs":0.047,"insertionSortMs":6.759,"mergeSortMs":0.4755,"builtinSortMs":0.173},{"n":16384,"linearSearchUs":11.6553,"binarySearchUs":0.067,"insertionSortMs":73.646,"mergeSortMs":2.3695,"builtinSortMs":0.6892},{"n":65536,"linearSearchUs":41.3484,"binarySearchUs":0.0827,"insertionSortMs":1151.1831,"mergeSortMs":7.3449,"builtinSortMs":2.9949}]},"film":{"about":"What the film draws (Rerun knocking on shells one at a time against halving the beach). beach: n shells, the average knocks for a linear walk ((n + 1) / 2) against the most halvings binary search needs. curves: comparisons against n (random input, mean over seeds) for each method, for log-log plots.","beach":{"shells":65536,"linearAverage":32768.5,"linearWorst":65536,"binaryMax":17},"curves":{"linear":[[8,4.4285],[16,8.474],[32,16.423],[64,32.2515],[128,65.08],[256,127.5835],[512,253.1615],[1024,505.512],[2048,1046.1365],[4096,2057.384],[8192,4121.1295],[16384,8088.731],[32768,16436.862],[65536,32921.391]],"binary":[[8,2.625],[16,3.375],[32,4.2188],[64,5.125],[128,6.0703],[256,7.0391],[512,8.0215],[1024,9.0117],[2048,10.0063],[4096,11.0034],[8192,12.0018],[16384,13.001],[32768,14.0005],[65536,15.0003]],"insertion":[[8,20],[16,74.8],[32,274.8],[64,1105.3],[128,4160.6],[256,16666.6],[512,66152.5],[1024,260739.6],[2048,1042321.4],[4096,4192377.8],[8192,16794252.3],[16384,67240109.7],[32768,268687884],[65536,1074190182.4]],"merge":[[8,15.7],[16,45],[32,122.8],[64,304.6],[128,735.2],[256,1724.8],[512,3959.8],[1024,8951.5],[2048,19942.9],[4096,43967.3],[8192,96156],[16384,208730.9],[32768,450136],[65536,965754.8]]}},"cost":{"about":"Milliseconds this fill spent in each part (this machine only), and the fill's wall time in seconds.","totals":{"search":212.4716,"insertion":28660.8032,"merge":393.7371},"fillSeconds":34.771},"seconds":34.8}