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In computer science, a disjoint-set data structure, also called a union–find data structure or merge–find set, is a data structure that stores a collection of disjoint (non-overlapping) sets. Equivalently, it stores a partition of a set into disjoint subsets. It provides operations for adding new sets, merging sets (replacing them by their union), and finding a representative member of a set. The last operation makes it possible to find out efficiently if any two elements are in the same or different sets. While there are several ways of implementing disjoint-set data structures, in practice they are often identified with a particular implementation called a disjoint-set forest. This is a specialized type of forest which performs unions and finds in near-constant amortized time. To perform a sequence of m addition, union, or find operations on a disjoint-set forest with n nodes requires total time O(mα(n)), where α(n) is the extremely slow-growing inverse Ackermann function. Disjoint-set forests do not guarantee this performance on a per-operation basis. Individual union and find operations can take longer than a constant times α(n) time, but each operation causes the disjoint-set forest to adjust itself so that successive operations are faster. Disjoint-set forests are both asymptotically optimal and practically efficient. Disjoint-set data structures play a key role in Kruskal's algorithm for finding the minimum spanning tree of a graph. The importance of minimum spanning trees means that disjoint-set data structures underlie a wide variety of algorithms. In addition, disjoint-set data structures also have applications to symbolic computation, as well as in compilers, especially for register allocation problems. Disjoint-set forests were first described by Bernard A. Galler and Michael J. Fischer in 1964. In 1973, their time complexity was bounded to , the iterated logarithm of , by Hopcroft and Ullman. In 1975, Robert Tarjan was the first to prove the (inverse Ackermann function) upper bound on the algorithm's time complexity,.
[2]Paolo Prandoni, [3]LCAV, EPFL
The Karplus-Strong algorithm is a simple method to generate synthetic
sounds with arbitrary pitch. In this notebook we will explore a few
variations of the method to illustrate some basic notions of
discrete-time signal processing.
In [1]:
%matplotlib inline
import matplotlib
import matplotlib.pyplot as plt
import numpy as np
import IPython
In [2]:
plt.rcParams["figure.figsize"] [2]Paolo Prandoni, [3]LCAV, EPFL
The Parks-McClellan (PM) algorithm is a clever application of advanced
polynomial fitting techniques to the problem of FIR filter design. In
this notebook we will explore the key ideas behind the method by
considering the design of a simple Type-I lowpass filter.
NOTE: This notebook uses interactive widgets. If the interactive
controls do not work or show up, please fo [2]Paolo Prandoni, [3]LCAV, EPFL
In this notebook we will look at the practical issues associated to
plotting the DFT and in particular the DFT of real-world signals. We
will examine how to map the DFT coefficients to real-world frequencies
and we will investigate the frequency resolution of the DFT and the
effects of zero padding.
In [1]: