6.851: Advanced Data Structures (Spring'12)

Prof. Erik Demaine     TAs: Tom Morgan, Justin Zhang

[Home] [Lectures] [Assignments] [Project] [Problem Session] [Accessibility]

Lecture 12 Video     [previous] [next]

[+] 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.)

Download Video: 360p, 720p

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

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

The video above should play if your web browser supports either modern Flash or HTML5 video with H.264 or WebM codec. The lecture notes should advance automatically. If you have any trouble with playback, email Erik.