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Acknowledgment is given for using some contents from Wikipedia. Computers can store and process vast amounts of data.

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Asymptotic notation

Acknowledgment is given for using some contents from Wikipedia. Computers can store and process vast amounts of data. Formal data structures enable a programmer to mentally structure large amounts of data into conceptually manageable relationships.

Sometimes we use data structures to allow us to do more: for example, to accomplish fast searching or sorting of data. Other times, we use data structures so that we can do less : for example, the concept of the stack is a limited form of a more general data structure.

These limitations provide us with guarantees that allow us to reason about our programs more easily. Data structures also provide guarantees about algorithmic complexity — choosing an appropriate data structure for a job is crucial for writing good software. Because data structures are higher-level abstractions, they present to us operations on groups of data, such as adding an item to a list, or looking up the highest-priority item in a queue.

When a data structure provides operations, we can call the data structure an abstract data type sometimes abbreviated as ADT. Abstract data types can minimize dependencies in your code, which is important when your code needs to be changed. Because you are abstracted away from lower-level details, some of the higher-level commonalities one data structure shares with a different data structure can be used to replace one with the other.

Our programming languages come equipped with a set of built-in types, such as integers and floating-point numbers, that allow us to work with data objects for which the machine's processor has native support. These built-in types are abstractions of what the processor actually provides because built-in types hide details both about their execution and limitations. For example, when we use a floating-point number we are primarily concerned with its value and the operations that can be applied to it.

Consider computing the length of a hypotenuse:. The machine code generated from the above would use common patterns for computing these values and accumulating the result. In fact, these patterns are so repetitious that high-level languages were created to avoid this redundancy and to allow programmers to think about what value was computed instead of how it was computed. A programming language is both an abstraction of a machine and a tool to encapsulate-away the machine's inner details.

For example, a program written in a programming language can be compiled to several different machine architectures when that programming language sufficiently encapsulates the user away from any one machine. In this book, we take the abstraction and encapsulation that our programming languages provide a step further: When applications get to be more complex, the abstractions of programming languages become too low-level to effectively manage.

Thus, we build our own abstractions on top of these lower-level constructs. We can even build further abstractions on top of those abstractions. Each time we build upwards, we lose access to the lower-level implementation details. While losing such access might sound like a bad trade off, it is actually quite a bargain: We are primarily concerned with solving the problem at hand rather than with any trivial decisions that could have just as arbitrarily been replaced with a different decision.

When we can think on higher levels, we relieve ourselves of these burdens. Each data structure that we cover in this book can be thought of as a single unit that has a set of values and a set of operations that can be performed to either access or change these values. The data structure itself can be understood as a set of the data structure's operations together with each operation's properties i.

Big-oh notation is a common way of expressing a computer code's performance. The notation creates a relationship between the number of items in memory and the average performance for a function. The first data structure we look at is the node structure. A node is simply a container for a value, plus a pointer to a "next" node which may be null. In some languages, structures are called records or classes. Some other languages provide no direct support for structures, but instead allow them to be built from other constructs such as tuples or lists.

Here, we are only concerned that nodes contain values of some form, so we simply say its type is "element" because the type is not important. In some programming languages no type ever needs to be specified as in dynamically typed languages, like Scheme, Smalltalk or Python.

In other languages the type might need to be restricted to integer or string as in statically typed languages like C. In any of these cases, translating the pseudocode into your own language should be relatively simple. Principally, we are more concerned with the operations and the implementation strategy than we are with the structure itself and the low-level implementation.

The above implementation meets this criteria, because the length of time each operation takes is constant. Another way to think of constant time operations is to think of them as operations whose analysis is not dependent on any variable. For now, it is safe to assume it just means constant time. Because a node is just a container both for a value and container to a pointer to another node, it shouldn't be surprising how trivial the node data structure itself and its implementation is.

Although the node structure is simple, it actually allows us to compute things that we couldn't have computed with just fixed-size integers alone. But first, we'll look at a program that doesn't need to use nodes. The following program will read in from an input stream; which can either be from the user or a file a series of numbers until the end-of-file is reached and then output what the largest number is and the average of all numbers:.

But now consider solving a similar task: read in a series of numbers until the end-of-file is reached, and output the largest number and the average of all numbers that evenly divide the largest number.

This problem is different because it's possible the largest number will be the last one entered: if we are to compute the average of all numbers that divide that number, we'll need to somehow remember all of them. We could use variables to remember the previous numbers, but variables would only help us solve the problem when there aren't too many numbers entered.

For example, suppose we were to give ourselves variables to hold the state input by the user. And further suppose that each of the variables had bits.

While this is a very large number of combinations, a list of bit numbers would require even more combinations to be properly encoded. In general, the problem is said to require linear space. All programs that need only a finite number of variables can be solved in constant space. Instead of building-in limitations that complicate coding such as having only a constant number of variables , we can use the properties of the node abstraction to allow us to remember as many numbers as our computer can hold:.

Above, if n integers are successfully read there will be n calls made to make-node. Note that when we iterate the numbers in the chain, we are actually looking at them in reverse order.

For example, assume the numbers input to our program are 4, 7, 6, 30, and After EOF is reached, the nodes chain will look like this:. Such chains are more commonly referred to as linked-lists. However, we generally prefer to think in terms of lists or sequences , which aren't as low-level: the linking concept is just an implementation detail. While a list can be made with a chain, in this book we cover several other ways to make a list. For the moment, we care more about the abstraction capabilities of the node than we do about one of the ways it is used.

The above algorithm only uses the make-node, get-value, and get-next functions. If we use set-next we can change the algorithm to generate the chain so that it keeps the original ordering instead of reversing it. To do: pseudocode to do so; also TODO; should probably think of compelling, but not too-advanced, use of set-value.

The chains we can build from nodes are a demonstration of the principle of mathematical induction:. Step 2 above is called the Inductive Hypothesis ; let's show that we can prove it:. How is this so? Probably the best way to think of induction is that it's actually a way of creating a formula to describe an infinite number of proofs. The principle says that there is nothing to stop us from doing this repeatedly, so we should assume it holds for all cases.

Induction may sound like a strange way to prove things, but it's a very useful technique. Typically base cases are easy to prove because they are not general statements at all.

You can think of the contained value of a node as a base case, while the next pointer of the node as the inductive hypothesis. Just as in mathematical induction, we can break the hard problem of storing an arbitrary number of elements into an easier problem of just storing one element and then having a mechanism to attach on further elements.

As a first attempt, we might try to just show that this is true for 1. Even if you carried out this proof and showed it to be true for the first billion numbers, that doesn't nescessarily mean that it would be true for one billion and one or even a hundred billion.

This is a strong hint that maybe induction would be useful here. Let's say we want to prove that the given formula really does give the sum of the first n numbers using induction. The first step is to prove the base case; i.

Now for the inductive step. There is no single data structure that offers optimal performance in every case. In order to choose the best structure for a particular task, we need to be able to judge how long a particular solution will take to run. Or, more accurately, you need to be able to judge how long two solutions will take to run, and choose the better of the two. We don't need to know how many minutes and seconds they will take, but we do need some way to compare algorithms against one another.

Asymptotic complexity is a way of expressing the main component of the cost of an algorithm, using idealized not comparable units of computational work. Consider, for example, the algorithm for sorting a deck of cards, which proceeds by repeatedly searching through the deck for the lowest card. The asymptotic complexity of this algorithm is the square of the number of cards in the deck.

This quadratic behavior is the main term in the complexity formula, it says, e. The exact formula for the cost is more complex, and it contains more details than we need to understand the essential complexity of the algorithm. With our deck of cards, in the worst case, the deck would start out reverse-sorted, so our scans would have to go all the way to the end.

This is in fact an expensive algorithm; the best sorting algorithms run in sub-quadratic time. Now let us consider how we would go about comparing the complexity of two algorithms. Note that we have been speaking about bounds on the performance of algorithms, rather than giving exact speeds. The actual number of steps required to sort our deck of cards with our naive quadratic algorithm will depend upon the order in which the cards begin. The actual time to perform each of our steps will depend upon our processor speed, the condition of our processor cache, etc.

It's all very complicated in the concrete details, and moreover not relevant to the essence of the algorithm. It's a measure of the longest amount of time it could possibly take for the algorithm to complete.

We can assume that it represents the "worst case scenario" of a program. So, let's take an example of Big-O.

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In my last post , I described Big O notation, why it matters, and common search and sort algorithms and their time complexity essentially, how fast a given algorithm will run as data size changes. Now, with the basics down, we can begin to discuss data structures, space complexity, and more complex graphing algorithms. Previously, I used Big O notation to describe t i me complexity for some common search and sort algorithms. Big O is also used to describe space complexity.

Data Structures/All Chapters

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This book provides a more practical approach by explaining the concepts of machine learning algorithms and describing the areas of application for each algorithm. Jozef Gruska IV 1. Edelsbrunner et al. Randomized algorithms have the advantages of simplicity and better performance both in theory and often is a collection of articles written by renowned experts in practice.

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