6.851: Advanced Data Structures (Spring'12)

Prof. Erik Demaine     TAs: Tom Morgan, Justin Zhang

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[+] Integer: fusion trees: sketching, parallel comparison, most significant set bit Scribe Notes [src]

This lecture describes a tour-de-force in integer data structures, called fusion trees (so named because they were published a year after the 1989 cold-fusion "scandal", and were perhaps just as surprising--though more correct). Fusion trees solve predecessor and successor among n w-bit integers in O(logw n) time per operation on the word RAM. The basic idea is to build a B-tree with branching factor wε. The tricky part is to build a wε-size node supporting constant-time predecessor/successor.

We'll see three major techniques for doing so. First, sketching reduces the w1+ε bits in a node down to w “essential” bits, by reducing each word down to w1−ε “essential” bits. The impressive part is sketching a word in constant time using a single multiplication. Second, parallel comparison lets us compare all of these sketches with a single query in constant time. Third, we'll see how to find the most significant set bit of a w-bit word in constant time on a word RAM, by using most of the fusion techniques again; this problem has tons of applications beyond fusion trees as well. (As a result, most significant set bit is a built-in instruction on most architectures; see e.g. StackOverflow.)

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Lecture notes, page 1/8[previous page][next page][PDF]

Lecture notes, page 1/8[previous page][next page][PDF]

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