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hvXE6s=CZss@>JCs4mv4mIXUT55{Cr@)GnzX{y*K5n_0,\ 0\leq f(n)1 \end{align} - which we can implement using divide-and-conquer as follows: \begin{lstlisting} @@ -3330,13 +3287,11 @@ \subsubsection{Example: concatenate0} \end{lstlisting} The running time of the quicksort is given by - \begin{align} T_{best} &\in \Theta (n\log n) \\ T_{average} &\in \Theta (n\log n) \\ T_{worst} &\in \Theta (n^2) \end{align} - The quicksort can also be randomized by picking the pivot, {\ft A[r]}, at random: \begin{lstlisting} def quicksort(A,p=0,r=-1): @@ -3455,14 +3410,12 @@ \subsubsection{Example: concatenate0} \end{lstlisting} In the average and worst cases, it runs as fast as the quicksort, but in the best case, it is linear: - \begin{align} T_{best} &\in \Theta (n) \\ T_{average} &\in \Theta (n\log n) \\ T_{worst} &\in \Theta (n\log n) \\ T_{memory} &\in \Theta (1) \end{align} - \index{priority queue} A heap can be used to implement a priority queue, for example, storage from which we can efficiently extract the largest element. @@ -3491,14 +3444,12 @@ \subsubsection{Example: concatenate0} \end{lstlisting} The running times for {\ft heap\_pop} and {\ft heap\_push} are the same: - \begin{align} T_{best} &\in \Theta (1) \\ T_{average} &\in \Theta (\log n) \\ T_{worst} &\in \Theta (\log n) \\ T_{memory} &\in \Theta (1) \end{align} - Here is an example: \begin{lstlisting} @@ -3613,13 +3564,11 @@ \subsection{Binary search trees} height of the left subtree minus the height of the right subtree is more or less the same. The rebalance operation can be done in $O(\log n)$. For an AVL tree, the time for inserting or removing an element is given by - \begin{align} T_{best} &\in \Theta (1) \\ T_{average} &\in \Theta (\log n) \\ T_{worst} &\in \Theta (\log n) \end{align} - \index{k-tree}\index{B-tree} Until now, we have considered binary trees (each node has two children and @@ -3749,7 +3698,6 @@ \subsection{Breadth-first search} T_{worst} &\in \Theta (n_E+n_V) \\ T_{memory} &\in \Theta (n) \end{align} - \subsection{Depth-first search} \index{depth-first search} @@ -3787,7 +3735,6 @@ \subsection{Depth-first search} T_{worst} &\in \Theta (n_E+n_V) \\ T_{memory} &\in \Theta (1) \end{align} - \subsection{Disjoint sets} \index{disjoint sets} @@ -3892,7 +3839,6 @@ \subsection{Minimum spanning tree: Kruskal} T_{worst} &\in \Theta (n_E\log n_V) \\ T_{memory} &\in \Theta (n_E) \end{align} - We provide an example of application in the next subsection. \goodbreak\subsection{Minimum spanning tree: Prim} @@ -3952,7 +3898,6 @@ \subsection{Minimum spanning tree: Kruskal} T_{worst} &\in \Theta (n_E+n_V\log n_V) \\ T_{memory} &\in \Theta (n_E) \end{align} - One important application of the minimum spanning tree is in evolutionary biology. Consider, for example, the DNA for the genes that produce hemoglobin, a molecule responsible for the transport of oxygen in blood. This protein is present in every animal, and the gene is also present in the DNA of every known animal. Yet its DNA structure is a little different. One can select a pool of animals and, for each two of them, compute the similarity of the DNA of their hemoglobin genes using the {\ft lcs} algorithm discussed later. One can then link each two animals by a metric that represents how similar the two animals are. We can then run the Prim or the Kruskal algorithm to find the minimum spanning tree. The tree represents the most likely evolutionary tree connecting those animal species. Actually, three genes are responsible for hemoglobin ({\it HBA1}, {\it HBA2}, and {\it HBB}). By performing the analysis on different genes and comparing the results, it is possible to establish a consistency check of the results.~\cite{evolutionary} Similar studies are performed routinely in evolutionary biology. They can also be applied to viruses to understand how viruses evolved over time.~\cite{korber} @@ -4011,7 +3956,6 @@ \subsection{Single-source shortest paths: Dijkstra} T_{worst} &\in \Theta (n_E+n_V\log n_V) \\ T_{memory} &\in \Theta (n_E) \end{align} - An application of the Dijkstra is in solving a maze such as the one built when discussing disjoint sets. To use the Dijkstra algorithm, we need to generate a maze, take the links representing torn-down walls, and use them to build an undirected graph. This is done by symmetrizing the links (if $i$ and $j$ are connected, $j$ and $i$ are also connected) and adding to each link a length (1, because all links connect next-neighbor cells): @@ -4132,7 +4076,6 @@ \subsection{Huffman encoding} We can ask how good is this compression factor. The maximum theoretical best compression factor is given by the Shannon {\it entropy}, defined as - \begin{equation} E = -\sum_u w_i \log_2 w_i \end{equation} @@ -4351,7 +4294,6 @@ \subsection{Continuous Knapsack} \begin{equation} T_{worst}(x)\in O(n\log n) \end{equation} - \subsection{Discrete Knapsack} \index{discrete knapsack} @@ -4457,7 +4399,6 @@ \subsection{Discrete Knapsack} \end{tabular} \label{table0} \end{equation} - STAGE TWO: we maximize the return of investment considering offers from firm $% 1$ and the prior table. We fill a table $f[1,k]$ with the maximum return of investment if we invest $k$ million dollars in firm $0$ and firm $1$: @@ -4477,7 +4418,6 @@ \subsection{Discrete Knapsack} \end{tabular} \label{table1} \end{equation} - STAGE THREE: we maximize the return of investment considering offers from firm $2$ and the preceding table. We fill a table $f[2,k]$ with the maximum return of investment if we invest $k$ million dollars in firm $0$, firm $1$, @@ -4507,11 +4447,9 @@ \subsection{Discrete Knapsack} We can generalize eq.(\ref{stage1}) and eq.(\ref{stage2}) for any number of investment firms (decision stages): - \begin{equation} f[i,k]=\max_{j|c[i,j]

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\subsubsection{Example: concatenate0} T_{worst} &\in \Theta (n\log n) \\ T_{memory} &\in \Theta (1) \end{align} -\index{greedy algorithms} {\bf Greedy algorithms} \index{greedy algorithms} work in phases. In each phase, a decision is made @@ -4968,7 +4967,7 @@ \chapter{Numerical Algorithms} \end{equation} (the latter equality only holds if $f$ is differentiable in $x$). -A problem with a low condition number is said to be well-conditioned, while a problem with a high condition number is said to be ill-conditioned. XXX +A problem with a low condition number is said to be well-conditioned, while a problem with a high condition number is said to be ill-conditioned. We say that a problem characterized by a function $f$ is well conditioned in a domain $D$ if the condition number is less than 1 for every input in the domain. We also say that a problem is stable if it is well conditioned. @@ -8496,14 +8495,14 @@ \chapter{Probability and Statistics} This theorem is of fundamental importance for stochastic calculus. Notice that the theorem does not apply when the $X_i$ follow distributions that do not have a finite mean or a finite variance. -\index{Levy distributions} +\index{L'{e}vy distributions} -Distributions that do not follow the central limit are called {\it Levy distributions}. +Distributions that do not follow the central limit are called {\it L\'{e}vy distributions}. They are characterized by fat tails for the form \begin{equation} p(x) \stackunder{x\rightarrow\infty}{\sim} \frac1{\left|x\right|^{1+\alpha}}, \qquad 0<\alpha<2 \end{equation} -An example if the Pareto distribution. +An example is the Pareto distribution. \goodbreak\subsection{Error in the mean} \index{error in the mean} @@ -8746,7 +8745,7 @@ \chapter{Random Numbers and Distributions} \index{distribution!Bernoulli} Given the sequence $\{t_i\}$ with exponential distribution, we can build a -random sequence of zeros and ones (Bernoulli distribution) by applying the following formula, known as the Von Neumann procedure~\cite{bits}: +random sequence of zeros and ones (Bernoulli distribution) by applying the following formula, known as the von Neumann procedure~\cite{bits}: \begin{equation} x_i= \begin{cases} @@ -9936,7 +9935,7 @@ \chapter{Random Numbers and Distributions} xm * random.paretovariate(alpha) \end{lstlisting} -The Pareto distribution is an example of Levy distribution. The Central Limit theorem does not apply to it. +The Pareto distribution is an example of L\'{e}vy distribution. The Central Limit theorem does not apply to it. \goodbreak\subsection{In and on a circle} \index{distribution!circle} @@ -11310,7 +11309,7 @@ \section{Markov chain Monte Carlo (MCMC) and Metropolis} if i==100: break \end{lstlisting} -In this example, {\ft Q} is the function that generates random points in the domain (in the example, $[0,1]\times[0,1]$), and ${\ft P}$ is an example probability $p(x) = 6(x_0-x_1)^2$. Notice we used the Python {\ft yield} function instead of {\ft return}. This means the function is a generator and we can loop over its returned (yielded) values without having to generate all of them at once. They are generated as needed. +In this example, {\ft Q} is the function that generates random points in the domain (in the example, $[0,1]\times[0,1]$), and {\ft P} is an example probability $p(x) = 6(x_0-x_1)^2$. Notice we used the Python {\ft yield} function instead of {\ft return}. This means the function is a generator and we can loop over its returned (yielded) values without having to generate all of them at once. They are generated as needed. \index{decorrelation} @@ -13238,7 +13237,7 @@ \subsubsection{Practical rules to compute derivatives} \bibitem{lorenz} Edward N. Lorenz, ``Deterministic non-periodic flow''. Journal of the Atmospheric Sciences 20 (2) (1963) pp 130–141. DOI:10.1175/1520-0469 \bibitem{determinism} Ian Hacking, ``19th-century Cracks in the Concept of Determinism'', Journal of the History of Ideas, 44 (3) (1983) pp 455-475 JSTOR:2709176 \bibitem{decay} F. Cannizzaro, G. Greco, S. Rizzo, E. Sinagra, ``Results of the measurements carried out in order to verify the validity of the poisson-exponential distribution in radioactive decay events''. The International Journal of Applied Radiation and Isotopes 29 (11) (1978) pp 649. DOI:10.1016/0020-708X(78)90101-1 -\bibitem{bits} Yuval Perez, ``Iterating Von Neumann's Procedure for Extracting Random Bits''. The Annals of Statistics 20 (1) (1992) pp 590–597 +\bibitem{bits} Yuval Perez, ``Iterating von Neumann's Procedure for Extracting Random Bits''. The Annals of Statistics 20 (1) (1992) pp 590–597 \bibitem{lusher} Martin Luescher, ``A Portable High-Quality Random Number Generator for Lattice Field Theory Simulations'', Comput. Phys. Commun. 79 (1994) pp 100-110 \bibitem{randu} \url{http://demonstrations.wolfram.com/PoorStatisticalQualitiesForTheRANDURandomNumberGenerator} \bibitem{fishman} G. S. 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