From 48d078ffcda54643e87e421f0d3d51fc7c865829 Mon Sep 17 00:00:00 2001 From: Thibaut Horel Date: Tue, 7 Oct 2014 09:17:23 -0400 Subject: Adding lecture notes --- lecture/lec3.tex | 361 +++++++++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 361 insertions(+) create mode 100644 lecture/lec3.tex (limited to 'lecture/lec3.tex') diff --git a/lecture/lec3.tex b/lecture/lec3.tex new file mode 100644 index 0000000..0f548ab --- /dev/null +++ b/lecture/lec3.tex @@ -0,0 +1,361 @@ +\documentclass[11pt]{article} +\usepackage{amsmath,amssymb,amsthm} + +\DeclareMathOperator*{\E}{\mathbb{E}} +\let\Pr\relax +\DeclareMathOperator*{\Pr}{\mathbb{P}} + +\newcommand{\eps}{\varepsilon} +\newcommand{\inprod}[1]{\left\langle #1 \right\rangle} +\newcommand{\R}{\mathbb{R}} +\renewcommand{\H}{\mathcal{H}} + +\newcommand{\handout}[5]{ + \noindent + \begin{center} + \framebox{ + \vbox{ + \hbox to 5.78in { {\bf CS 224: Advanced Algorithms } \hfill #2 } + \vspace{4mm} + \hbox to 5.78in { {\Large \hfill #5 \hfill} } + \vspace{2mm} + \hbox to 5.78in { {\em #3 \hfill #4} } + } + } + \end{center} + \vspace*{4mm} +} + +\newcommand{\lecture}[4]{\handout{#1}{#2}{#3}{Scribe: #4}{Lecture #1}} + +\newtheorem{theorem}{Theorem} +\newtheorem{corollary}[theorem]{Corollary} +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{observation}[theorem]{Observation} +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{claim}[theorem]{Claim} +\newtheorem{fact}[theorem]{Fact} +\newtheorem{assumption}[theorem]{Assumption} +\theoremstyle{definition} +\newtheorem*{example}{Example} +\newtheorem{definition}[theorem]{Definition} + +% 1-inch margins, from fullpage.sty by H.Partl, Version 2, Dec. 15, 1988. +\topmargin 0pt +\advance \topmargin by -\headheight +\advance \topmargin by -\headsep +\textheight 8.9in +\oddsidemargin 0pt +\evensidemargin \oddsidemargin +\marginparwidth 0.5in +\textwidth 6.5in + +\parindent 0in +\parskip 1.5ex + +\begin{document} + +\lecture{Lecture 3 --- Tuesday Sep 09, 2014}{Fall 2014}{Prof.\ Jelani Nelson}{Thibaut Horel} + +\section{Overview} + +In the previous lecture we finished covering data structures for the +predecessor problem by studying Fusion Trees. + +In this lecture, we cover several notions related to hashing: using hashing +for load balancing, $k$-wise independent families of hashing functions and the +dictionary problem for which we show two solutions: chaining with linked lists +and linear probing. + +\section{Hashing} + +In hashing, we have a family $\H$ of hashing functions mapping $[u]$ to $[m]$ +(where $[u]$ denotes the set $\{0,\ldots, u-1\}$). For a set $S\subseteq [u]$ +with $|S| = n$, we want that $h$ picked at random in $\H$ behaves ``nicely'' on +$S$. + +\begin{example}[load balancing] +We have $n$ jobs that need to be assigned to $m$ machines and each job has a +job ID in $[u]$. We want to evenly spread the job over the machines. We pick +$h$ at random in a family of hashing functions $\H$. When a new job $j\in[u]$ +comes in, we assign it to machine $h(j)\in[m]$. This is usually referred to as +the ``balls and bins'' random process. +\end{example} + +We would like to say that $\Pr(\exists \text{ machine receiving more than +$\frac{m}{n}$ jobs})$ is small. First, let us assume that the random variables +$h(i)$ for $i\in [u]$ are independent. We will need the following concentration +inequality. + +\begin{theorem}[Chernoff bound] + Let $X_1, \ldots, X_n$ be $n$ independent random variables with + $X_i\in\{0,1\}$. Let us write $X=\sum_{i=1}^n X_i$, then: + \begin{displaymath} + \Pr\big(X\geq (1+\delta)\E[X]\big)\leq + \left(\frac{e^\delta}{(1+\delta)^{1+\delta}}\right)^{\E[X]}. + \end{displaymath} +\end{theorem} + +The Chernoff bound can be proved using Markov's inequality. + +\begin{lemma}[Markov's inequality] + Let $X$ be a nonnegative, random variable, then: + \begin{displaymath} + \forall\lambda > 0,\; \Pr (X\geq\lambda)\leq \frac{\E[X]}{\lambda}. + \end{displaymath} +\end{lemma} + +\begin{proof}[Proof (Chernoff bound)] + By Markov's inequality, we have: + \begin{displaymath} + \Pr (X \geq \lambda) = \Pr\big(e^{tX} \geq + e^{t\lambda}\big)\leq\frac{\E\big[e^{tX}\big]}{e^{t\lambda}}. + \end{displaymath} + By independence: + \begin{displaymath} + \E\big[e^{tX}\big] = \prod_{i=1}^n\E\big[e^{tX_i}\big] + =\prod_{i=1}^n\big(p_ie^t+(1-p_i)\big) + \leq\prod_{i=1}^ne^{p_i(e^t-1)} = e^{(e^t-1)\E[X]}, + \end{displaymath} + where the inequality uses $1+x\leq e^x$. We then conclude the proof by + choosing $\lambda = (1+\delta)\E[X]$ and $t=\log(1+\delta)$. +\end{proof} + +Using the Chernoff bound, we can prove the following proposition. + +\begin{proposition} + For the balls and bins random process, with $m=n$, then for any $c > 1$: + \begin{displaymath} + \Pr\left(\exists \text{ machine with more than $\frac{c\log n}{\log\log + n}$ jobs}\right) + \leq \frac{1}{\mathrm{poly}(n)}. + \end{displaymath} +\end{proposition} + +\begin{proof} + Focus on some machine $t$ and define: + \begin{displaymath} + X_i= \begin{cases} + 1 &\text{if $h(i) = t$}\\ + 0&\text{otherwise} + \end{cases} + \end{displaymath} + Then, $X=\sum_{i=1}^n X_i$ is the number of jobs assigned to machine $t$ + and $\E[X]=1$. Applying the Chernoff bound: + \begin{displaymath} + P\left(X\geq \frac{c\log n}{\log \log n}\right) \leq + \left(\frac{e}{c\log n/\log\log n}\right)^{\frac{c\log + n}{\log\log n}} = \left(\frac{e}{k}\right)^k + \end{displaymath} + where $k=\frac{c\log n}{\log \log n}$. It is easy to see that for any + $\eps>0$ and for $n$ large enough, $\left(\frac{e}{k}\right)^k \leq + \frac{1}{n^{c-\eps}}$. This + implies, by using a union bound: + \begin{displaymath} + \Pr(\exists\text{ overloaded machine}) + \leq \sum_{t=1}^n \Pr (t\text{ is overloaded}) + \leq \frac{1}{n^{c-\eps-1}} + \end{displaymath} + Chosing $\eps < c-1$ is enough to conclude the proof. +\end{proof} + +We propose an alternative proof which does not rely on the Chernoff bound. + +\begin{proof} + Using union bounds: + \begin{align*} + \Pr(\exists\text{ overloaded machine}) + &\leq n \Pr(\text{machine 1 is overloaded})\\ + &=\Pr(\exists \text{ $k$ items mapping to machine 1})\\ + &\leq \sum_{\substack{T\subseteq[u]\\|T|=k}} \Pr (\text{items in $T$ all + map to 1})\\ + &\leq\sum_{\substack{T\subseteq[u]\\|T|=k}} \frac{1}{n^k} + = \frac{\binom{n}{k}}{n^k}\leq \frac{1}{k!} + \leq\left(\frac{1}{(k/2)}\right)^{k/2} +\end{align*} +and we conclude as in the previous proof. +\end{proof} + +Note that the second proof does not require full independence, but only that +sets of $k$ elements are mapped to independent random variables. This motivates +the following definitions. + +\begin{definition} + The random variables $X_1,\ldots,X_n$ are \emph{$k$-wise independent} iff + for any set of indices $1\leq i_1\leq\ldots\leq i_k\leq n$, the random + variables $X_{i_1},\ldots,X_{i_k}$ are independent. +\end{definition} + +\begin{definition} + A set of hash functions $\H$ is a \emph{$k$-wise independent family} iff + the random variables $h(0), h(1),\ldots,h(u-1)$ are $k$-wise independent + when $h\in\H$ is drawn uniformly at random. +\end{definition} + +\begin{example} + The set $\H$ of all functions from $[u]$ to $[m]$ is $k$-wise independent + for all $k$. Note that this family of functions is impractical for many + applications since you need $O(u\log m)$ bits to store a totally random + function. For a hash table, this is already larger than the space of the + data structure itself, $O(n)$. +\end{example} + +\begin{example}[$u=m=p$, $p$ prime] We define $\H = \big\{h: h(x) + = \sum_{i=0}^{k-1} a_ix^i\bmod p\big\}$ where $a_i\in\{0,\ldots,p-1\}$. By + using polynomial interpolation, we know that for any \mbox{$(x_1,\ldots, + x_k, y_1,\ldots,y_k)\in[p]^{2k}$} there exists a unique polynomial $h\in\H$ + such that $h(x_1) = y_1,\ldots, h(x_k) = y_k$. Since there are $p^k$ + elements in $\H$, if $h\in\H$ is drawn uniformly at random, $\Pr(h(x_1) + = y_1,\ldots,h(x_k) = y_k) = 1/p^k$, which shows that $\H$ is $k$-wise + independent. It is also easy to see that this family is not + \mbox{$k+1$-wise} independent: again using polynomial interpolation, once + $h(x_1),\ldots,h(x_k)$ are fixed, $h(x_{k+1})$ is entirely determined. +\end{example} + +\begin{example}[$u=p$, $p$ prime\footnote{In practice, we do not lose by + requiring $u$ to be prime. Indeed, by Bertrand's postulate, there always + exists a prime between $u$ and $2u$.}] + We define $\H = \big\{h: h(x) = \big(\sum_{i=0}^{k-1}a_i x^i \bmod p\big) + \bmod m\big\}$. We see that $\Pr(h(x_1) = y_1,\ldots, h(x_k) = y_k)\leq + \left(\frac{2}{m}\right)^k$. Note that in this example, + $h(x_1),\ldots,h(x_k)$ does not behave exactly like $k$ independent uniformly + distributed variables, but this bound is sufficient in most applications. +\end{example} + +\section{Dictionary problem} + +This is a data structural problem with two operations: +\begin{itemize} + \item \texttt{update(k,v)}: associates key $k$ with value $v$. + \item \texttt{query(k)}: returns value $v$ associated with key $k$ or + \texttt{null} if $k$ is not in the dictionary. +\end{itemize} +The keys and the values are assumed to live the in same universe $[u]$. + +\subsection{Chaining with linked lists} +The dynamic dictionary problem can be solved by hash tables with chaining. In +this solution the hash table is an array $A$ of size $m$. Each element in the +array is a pointer to a linked list containing (key, value) pairs. + +We pick $h\in\H$ from a $2$-wise independent family. The operations on key $k$ +are made in the linked list pointed to by $A\big[h(k)\big]$. + +The running time of an operation on key $k$ is linear in the length of the +linked list at $h(k)$. It is possible to show that $\E_h[\text{length of list +at $h(k)$}]= O(1)$ if $m\geq n$. + +For details of the analysis, see Lecture Notes 12 from the Spring 2014 offering of CS 124, or see \cite[Section 11.2]{CLRS}. + +\subsection{Perfect hashing} + +In this problem there is some set $S\subset[u]$ where $|S| = n$, and we would like to find a hash function $h:[u]\rightarrow[m]$ (hashing into the $m$ tables of some array) which acts injectively on $S$. This was mentioned in class, but no details were given -- for the interested reader, see the lecture notes on perfect hashing from the Spring 2014 offering of CS 124, or read \cite[Section 11.5]{CLRS}, or see the original paper \cite{FKS84}. + +\subsection{Linear probing} + +In real architectures, sequential memory accesses are much faster. A solution +to the dictionary problem with better spacial locality is \emph{linear +probing}. In this solution, the hash table is an array $A$, but unlike the +solution with chaining and linked lists, the values are stored directly in +$A$. More specifically, the value associated with key $k$ is stored at $h(k)$. +If this cell is already taken, we start walking to the right, one cell at +a time\footnote{We wrap back to the beginning of the array when we reach its +right edge.}, until we find and empty cell and insert the value in it. + +\paragraph{History.} Linear probing dates back at least to 1954 where it was +used in an assembly program written by Samuel, Amdahl and Boehme. It was first +analyzed in a note published by Knuth \cite{Knuth63} in the case where $h$ is +totally random ($\H$ is the set of all functions). In this paper, he showed +that when $m= (1+\eps)n$, then $\E[\text{time per operation}] += O\left(\frac{1}{\eps^2}\right)$. There was a breakthrough in a paper by Pagh, +Pagh and Ruzic \cite{ppr} where it was proved that for any $5$-wise independent +family, $\E[\text{time per operation}] = O(1)$. Finally Patrascu and Thorup +\cite{pt} showed that $4$-wise independent families do not guarantee a constant +running time per operation (that is, they provided an example of a $4$-wise +independent family for which the expected running time is not constant). + +Here, we will prove the following proposition. + +\begin{proposition}\label{prop:unfinished} + If $m\geq cn$ and $\H$ is $7$-wise independent, then $\E[\text{query time}] + = O(1)$. +\end{proposition} + +To prove this proposition, we need to able to compute the expected number of +cells we have to probe before finding an empty cell. We are going to show +that this expected number is constant. + +\begin{definition} + Given an interval $I$ of cells in $A$, we define $L(I)=|\{i: h(i)\in + I\}|$\footnote{We allow intervals to wrap around at the end of the table.}. +\end{definition} + +\begin{definition} + We say that $I$ is \emph{full} iff $L(I) \geq |I|$. +\end{definition} + +\begin{lemma} + If \texttt{query(x)} for some $x$ makes $k$ probes, then there exist at + least $k$ full intervals containing $h(x)$. +\end{lemma} + +\begin{proof} + Let us define $z_1$ the position of the first empty cell on the left of + $h(x)$ and $z_2$ the position of the first empty cell on the right of + $h(x)$. It is easy to see that each of the intervals $[z_1, h(x)], [z_1, + h(x)+1],\ldots,[z_1, z_2-1]$ are full. There are $z_2-h(x)$ such intervals. + But by the assumption of the lemma, $z_2 = h(x)+ k$ which conclues the + proof of the lemma. +\end{proof} + +\begin{proof}[Proof of Proposition~\ref{prop:unfinished}] +By applying the previous lemma: +\begin{align*} + \E[\text{time of \texttt{query(x)}}] + &\leq \E[\text{number of full intervals + containing $h(x)$}]\\ + &\leq \sum_{k\geq 1}\sum_{\substack{I, |I|=k\\h(x)\in I}} \Pr(I\text{ + full}) \leq \sum_{k\geq 1}k \max_{\substack{I, |I|=k\\ h(x)\in I}}\Pr(I\text{ full}) +\end{align*} +We need to show that the $\max$ in the previous sum is +$O\left(\frac{1}{k^3}\right)$ so that the sum converges. We assume that $m=2n$ +where $n$ is the number of different keys active. +\begin{displaymath} + \Pr(I\text{ full}) = \Pr(L(I)\geq |I|) + \leq\Pr\big(|L(I)-\E L(I)|\geq \E L(I)\big) +\end{displaymath} +where the inequality follows from $E[L(I)] = \frac{|I|}{2}$. Now we would like +to apply Chernoff, but we don't have independence. Instead we can raise both +sides of the inequality to some power and apply Markov's inequality. This is +called the \emph{methods of moments}. We will do this in the next lecture. +\end{proof} + +\bibliographystyle{alpha} +\begin{thebibliography}{PPR09} + +\bibitem[CLRS]{CLRS} +Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, Clifford Stein. +Introduction to Algorithms, 3$^{\text{rd}}$ edition, MIT Press, 2009. + +\bibitem[FKS84]{FKS84} +Michael L. Fredman, J\'{a}nos Koml\'{o}s, Endre Szemer\'{e}di. +Storing a Sparse Table with $O(1)$ Worst Case Access Time. +{\em J. ACM 31(3)}, pages 538--544, 1984. + +\bibitem[Knu63]{Knuth63} +Don Knuth. +\newblock Notes On ``Open'' Addressing, 1963. + +\bibitem[PPR09]{ppr} +Anna Pagh, Rasmus Pagh, and Milan Ruzic. +\newblock Linear probing with constant independence. +\newblock {\em {SIAM} J. Comput.}, 39(3):1107--1120, 2009. + +\bibitem[PT10]{pt} +Mihai Patrascu and Mikkel Thorup. +\newblock On the \emph{k}-independence required by linear probing and minwise + independence. +\newblock {\em ICALP}, pages 715--726, 2010. + +\end{thebibliography} + +\end{document} -- cgit v1.2.3-70-g09d2