Two Topics in Adaptive Algorithms:

Hulls and Strings

Robert Fraser

University of Waterloo

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What to expect

• Convex Hulls– Adaptive sorting, measures of difficulty– Techniques– Adaptivity of output sensitive

techniques

• String Matching– Online string matching– Boyer Moore Horspool technique– Adaptive string matching

Adaptive Sorting

• All kinds of techniques and measures (see Estivill-Castro & Wood, 89)

• I love Straight Merge Sort!

(Odd-even sms is pretty good too…)

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Planar Convex Hulls

• Many techniques– Gift wrapping– Graham Scan– Output sensitive– Adaptive (Levcopolous et al., Barbay & Chen)– New adaptive techniques?

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Adaptivity of CSY-hull

• Ω(n log h)

• Add a step CHECKHULL to give adaptivity

• Considering performance on a good instance however, you get CUPS

• Could also determine difficulty of interior points…

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Adaptivity wrt Max!

6Max

Com

paris

ons

QuicksortLP-odd-even sms

LP sort

Odd-even sms

Straight mergesort

Adaptivity wrt Runs!

7Runs

Com

paris

ons

Quicksort

LP-odd-even sms

LP sort

Odd-even sms

Straight mergesort

Straight mergesortwith reverse!

Busiest slide ever?

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New York GIS data

264346 points, 33 on hull

sm: 4346700

oesm: 4351874

lpoesm: 6056577

lp: 9408131

gs: 5792637

dis: 263950

max: 2074199

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String Matching Techniques

• Naïve – standard sliding window – θ(n·m)

• Linear time – as discussed by KMP– O(n+m)

• Expected sublinear – We’ll look at the standard Boyer-Moore

Horspool, but there are many!– O(n log m/m +m) expected, O(n·m) worst case

Boyer Moore Horpsool

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0 1 2 3 4 5 6 7 8 9 10

b b b c b a a c c a a

a c c a a

String Matching with More Jumps

• Try to maximize the jumps with Horspool-type searching

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Extensions

• Uneven distribution of data

• 2D string matching

• Applications (bioinformatics)

• Works best on small alphabets with many repeats

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Thanks

• Questions?

Supported by NSERC and the University of

Waterloo