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diff --git a/lecture/main.tex b/lecture/main.tex deleted file mode 100644 index 0f548ab..0000000 --- a/lecture/main.tex +++ /dev/null @@ -1,361 +0,0 @@ -\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} |