Algorithms *

A basic data structure in computer science is the “associative array” known as a “map”. This structure is called a “dictionary”. Dictionaries are being used when you have key-value pairs of the information. Inputs are called keys, and outputs are called values. A dictionary is the abstract data type that can store elements so that they can be positioned quickly by using keys. Dictionary is like a container that will have a searchable assortment of items. Each item in the dictionary is stored as a key-value pair. In a dictionary, we can store multiple items with the same key.

Dictionary consists of multiple elements in terms of key and value pair. Both key and value are considered as one single pair. This is called mapping. Elements of the dictionary are enclosed in curly brackets in terms of key and value pairs. Dictionaries enable us to work with key-value pairs. Key-value pairs are two linked values where the key is the unique identifier where we can discover our data and the value is that the information.

Dictionary maps key-value pairs. It is a collection data type that has key-value pairs. A dictionary does not contain any duplicate members.

It is unordered and stores data values like a map. Thus, it is similar to the real-life dictionary with distinct key values. In a dictionary, we use keys as indexes to access elements.

The dictionary helps us to organize the collection of data. It is a special data type. Its syntax is:

Data structures are classified into linear and non-linear data structures. A tree is a non-linear data structure. Data is stored hierarchically in a non-linear data structure. So the tree is a way of organizing data hierarchically. A tree grows from top to bottom. In a tree, there are different kinds of nodes that are linked with each other. A tree consists of the following elements:

It is a collection of edges and vertices. It can be used to display any form of network.

The following graph contains a total of 4 vertices and 5 edges. In this graph, vertices are A, B, C and D while edges are AB, BD, DC, CA and AD. Vertices are also known as nodes. The line connecting these vertices is the edge. Vertices are like objects and edges indicate the relation between those vertices. Data is stored in nodes. This data can be of numerical data type or any other data structure.

Recursion is a strategy that algorithms use to solve specific problems. A recursive algorithm is an algorithm that solves the main problem by using the solution of a simpler sub-problem of the same type. Recursion is a particular way of solving a problem by having a function calling itself repeatedly. It is always applied to a function only. By using recursion, we can reduce the size of the program or source code. In recursion, a function invokes itself. And the function that invokes itself is referred to as a recursive function.

Suppose we have a user-defined function named ‘recursion’, and it will be written in the main function. The compiler will execute the recursion function automatically, and it will search for a particular function definition. This function definition will be executed, and control will go back to the main function. If we call the same function inside the function definition, then the compiler will move on to function definition first. When the compiler executes the recursion function, we will be calling the same function.

On implementing streaming algorithms, counting of events often occurs, where an event means something like a packet arrival or a connection establishment. Since the number of events is large, the available memory can become a bottleneck: an ordinary -bit counter allows to take into account no more than events.
One way to handle a larger range of values using the same amount of memory would be approximate counting. This article provides an overview of the well-known Morris algorithm and some generalizations of it.

Another way to reduce the number of bits required for counting mass events is to use decay. We discuss such an approach here [3], and we are going to publish another blog post on this particular topic shortly.

In the beginning of this article, we analyse one straightforward probabilistic calculation algorithm and highlight its shortcomings (Section 2). Then (Section 3), we describe the algorithm proposed by Robert Morris in 1978 and indicate its most essential properties and advantages. For most non-trivial formulas and statements, the text contains our proofs, the demanding reader can find them in the inserts. In the following three sections, we outline valuable extensions of the classic algorithm: you can learn what Morris's counters and exponential decay have in common, how to improve the accuracy by sacrificing the maximum value, and how to handle weighted events efficiently.

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Dynamic programming is applied to solve optimization problems. In optimization, we try to find out the maximum or minimum solution of something. It will find out the optimal solution to any problem if that solution exists. If the solution does not exist, dynamic programming is not able to get the optimal solution.

Optimization problems are the ones that require either lowest or highest possible results. We attempt to discover all the possible solutions in dynamic programming and then choose the best optimal solution. Dynamic programming problems are solved by utilizing the recursive formulas though we will not use a recursion of programming the procedures are recursive. Dynamic programming pursues the rule of optimality. 

A dynamic programming working involves around following significant steps:

Asymptotic notations are used to represent the complexity or running time of an algorithm. It is a technique of defining the upper and lower limits of the run-time performance of an algorithm.  We can analyze the runtime performance of an algorithm with the help of asymptotic notations. Asymptotic notations are also used to describe the approximate running time of an algorithm.

Types of Asymptotic Notations

Following are the different types of asymptotic notations:

The mathematical model of signed sequences with repetitions (texts) is a multiset. The multiset was defined by D. Knuth in 1969 and later studied in detail by A. B. Petrovsky [1]. The universal property of a multiset is the existence of identical elements. The limiting case of a multiset with unit multiplicities of elements is a set. A set with unit multiplicities corresponding to a multiset is called its generating set or domain. A set with zero multiplicity is an empty set.

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The previous work from ref [1] describes the method of transforming a sign sequence into algebra through an example of a linguistic text. Two other examples of algebraic structuring of texts of a different nature are given to illustrate the method.

This article is a part of Algorithms in Go series where we discuss common algorithmic problems and their solution patterns.

In this edition, we take a closer look at bit manipulations. Bit operations can be extremely powerful and useful in an entire class of algorithmic problems, including problems that at first glance does not have to do anything with bits.

Let's consider the following problem: six friends meet in the bar and decide who pays for the next round. They would like to select a random person among them for that. How can they do a random selection using only a single coin?

The solution to this problem is not particularly obvious (for me:), so let's simplify a problem for a moment to develop our understanding. How would we do the selection if there were only three friends? In other words, how would we "mimic" a three-sided coin with a two-sided coin?

Algebra and language (writing) are two different learning tools. When they are combined, we can expect new methods of machine understanding to emerge. To determine the meaning (to understand) is to calculate how the part relates to the whole. Modern search algorithms already perform the task of meaning recognition, and Google’s tensor processors perform matrix multiplications (convolutions) necessary in an algebraic approach. At the same time, semantic analysis mainly uses statistical methods. Using statistics in algebra, for instance, when looking for signs of numbers divisibility, would simply be strange. Algebraic apparatus is also useful for interpreting the calculations results when recognizing the meaning of a text.

Here I am going to cover my own approach to compilation of mathematical functions into Linq.Expression. What we are going to have implemented at the end:

1. Arithmetical operations, trigonometry, and other numerical functions

2. Boolean algebra (logic), less/greater and other operators

3. Arbitrary types as the function's input, output, and those intermediate

Hope it's going to be interesting!

Most solutions to algorithmic problems can be grouped into a rather small number of patterns. When we start to solve some problem, we need to think about how we would classify them. For example, can we apply fast and slow аlgorithmic pattern or do we need to use cyclic sortpattern? Some of the problems have several solutions based on different patterns. In this series, we discuss the most popular algorithmic patterns that cover more than 90% of the usual problems.

It is different from High-School Algorithms 101 Course, as it is not intended to cover things like Karatsuba algorithm (fast multiplication algorithm) or prove different methods of sorting. Instead, Algorithmic Patterns focused on practical skills needed for the solution of common problems. For example, when we set up a Prometheus alert for high request latency we are dealing with Sliding Window Pattern. Or let say, we organize a team event and need to find an available time slot for every participant. At the first glance, it is not obvious that in this case, we are actually solving an algorithmic problem. Actually, during our day we usually solve a bunch of algorithmic problems without realizing that we dealing with algorithms.

The knowledge about Algorithmic Patterns helps one to classify a problem and then apply the appropriate method.

But probably most importantly learning algorithmic patterns boost general programming skills. It is especially helpful when you are debugging some production code, as it trains you to understand the execution flow.

Patterns covered so far:

Sliding Window I

Sliding Window II

Merge Intervals

Dutch National Flag

Matrix Spiral

Iterative Postorder Traversal

Bit Manipulation

Stay tuned :)

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In this article, we discuss the postorder traversal of a binary tree. What does postorder traversal mean? It means that at first, we process the left subtree of the node, then the right subtree of the node, and only after that we process the node itself.

Why would we need to do it in this order? This approach solves an entire class of algorithmic problems related to the binary trees. For example, to find the longest path between two nodes we need to traverse the tree in a postorder manner. In general, postorder traversal is needed when we cannot process the node without processing its children first. In this manner, for example, we can calculate the height of the tree. To know the height of a node, we need to calculate the height of its children and increment it by one.

Let's start with a recursive approach. We need to process the left child, then the right child and finally we can process the node itself. For simplicity, let's just save the values into slice out.

After 210 days, 600 commits, tens of debugging nights, and thousands of messages in the project chat, I finally released AngouriMath 1.2.

This is an open-source symbolic algebra library for C# and F#, maybe it is interesting for someone?

Most solutions to algorithmic problems can be grouped into a rather small number of patterns. When we start to solve some problem, we need to think about how we would classify them. For example, can we apply fast and slowalgorithmic pattern or do we need to use cyclic sortpattern? Some of the problems have several solutions with different patterns. In this article of series Algorithms in Go we consider an algorithmic pattern that solves an entire class of the problems related to a matrix. Let's take one of such problems and see how we can handle it.

How can we traverse a matrix in a spiral order?

An essential part of any modern static code analyzer is data flow analysis. However, from an outside perspective, the use of data flow analysis and its benefit is unclear. Some people still consider static analysis a tool searching for something in code according to a certain pattern. Thus, we occasionally write blog posts to show how this or that technology, used in the PVS-Studio analyzer, helps to identify another interesting error. Today, we have such an article about the bug found in the Base64, one of the encoding standard implementations of binary data.