Algorithms *

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.

The new issue of Data Science Digest is here! Hop to learn about the latest news, articles, tutorials, research papers, and event materials on DataScience, AI, ML, and BigData. All sections are prioritized for your convenience. Enjoy!

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 :)

<Promo> If you interested to work as a backend engineer, there is an open position in my squad. Prior knowledge of Golang is not required. I am NOT an HR and DO NOT represent the company in any capacity. However, I can share my personal experience as a backend engineer working in the company. </Promo>

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.

The flag of the Netherlands consists of three colors: red, white and blue. Given balls of these three colors arranged randomly in a line (it does not matter how many balls there are), the task is to arrange them such that all balls of the same color are together and their collective color groups are in the correct order.

For simplicity instead of colors red, white, and blue we will be dealing with ones, twos and zeroes.

Let's start with our intuition. We have an array of zeroth, ones, and twos. How would we sort it? Well, we could put aside all zeroes into some bucket, all ones into another bucket, and all twos into the third. Then we can fetch all items from the first bucket, then from the second, and from the last bucket, and restore all the items. This approach is perfectly fine and has a great performance. We touch all the elements when we iterate through the array, and then we iterate through all the elements once more when we "reassamble" the array. So, the overall time complexity is O(n) + O(n) ~= O(n). The space complexity is also O(n) as we need to store all items in the buckets.

Can we do better than that? There is no way to improve our time complexity. However, we can think of a more efficient algorithm in regard to space complexity. How would we solve the problem without the additional buckets?

Let's make a leap of faith and pretend that somehow we were able to process a part of the array. We iterate through part of the array and put encountered zeroes and ones at the beginning of the array, and twos at the end of the array. Now, we switched to the next index i with some unprocessed value x. What should we do there?

This is the third part of a series covering the implementation of algorithms in Go. In this article, we discuss the Merge Interval algorithm. Usually, when you start learning algorithms you have to deal with some problems like finding the least common denominator or finding the next Fibonacci number. While these are indeed important problems, it is not something that we solve every day. What I like about the Merge Interval algorithm is that we apply it in our everyday life, usually without even noticing that we are solving an algorithmic problem.

Let's say that we need to organize a meeting for our team. We have three colleagues Jay, May, and Ray and their time schedule look as follows (a colored line represents an occupied timeslot):

There’s a lot of talk about machine learning nowadays. A big topic – but, for a lot of people, covered by this terrible layer of mystery. Like black magic – the chosen ones’ art, above the mere mortal for sure. One keeps hearing the words “numpy”, “pandas”, “scikit-learn” - and looking each up produces an equivalent of a three-tome work in documentation.

I’d like to shatter some of this mystery today. Let’s do some machine learning, find some patterns in our data – perhaps even make some predictions. With good old Python only – no 2-gigabyte library, and no arcane knowledge needed beforehand.

Interested? Come join us.

This is the second part of the article covering the Sliding Window Pattern and its implementation in Go, the first part can be found here.

Let's have a look at the following problem: we have an array of words, and we want to check whether a concatenation of these words is present in the given string. The length of all words is the same, and the concatenation must include all the words without any overlapping. Would it be possible to solve the problem with linear time complexity?

Let's start with string catdogcat and target words cat and dog.

How can we handle this problem?