# Disjoint Set and Tarjan’s Off-line Lowest Common Ancestor Algorithm

## Improving lowest common ancestor algorithms with disjoint set off-line.

Tarjan’s Off-line Lowest Common Ancestor Algorithm is an interesting application of the disjoint set structure for optimizing the performance of determining the lowest common ancestor(LCA) of two nodes within a tree, which also involves concepts such as caching and recursion. In preparation for this story, please refer to my previous story on Lowest Common Ancestor for context.

Ok, let’s get started!

# Disjoint Set

In order to understand this algorithm, we first have to have a good grasp on the concept of disjoint sets…

Disjoint sets are collections of nodes, where there is no overlap between sets.

Sounds simple, right? Actually, it is, it’s just that how you enforce this rule when you build up your disjoint sets requires a bit of artistry. …

# Binary Tree and Lowest Common Ancestor

## What is “Lowest Common Ancestor (LCA)”?

It is the lowest level parent shared by two nodes within a tree.

Let’s take a look at an example:

Within the above binary tree, nodes 8 and 10 are highlighted. What are their shared parents?

It’s quite obvious that the shared parents are 5, 7, and 9.

But the shared parent at the lowest level is 9, and it is referred to as the lowest common ancestor(LCA).

## Lowest Common Ancestor(LCA) in a Binary Search Tree(BST)

Let’s warm up with a binary search tree.

A binary search tree is a special case of a binary tree, where the left subtree only contains smaller nodes and right subtree only contains bigger nodes. …

# Multivariate Normal Distribution

## What about multivariate normal distributions? Are they basically a few normally distributed variables bundled together? Let’s take a look.

Normal distribution, also called gaussian distribution, is one of the most widely encountered distributions. One of the main reasons is that the normalized sum of independent random variables tends toward a normal distribution, regardless of the distribution of the individual variables (for example you can add a bunch of random samples that only takes on values -1 and 1, yet the sum itself actually becomes normally distributed as the number of sample you have becomes larger). This is known as the central limit theorem. …

# Shortest Paths and Dijkstra’s Algorithm

## The most classic graph algorithm on planet Earth, explained.

A classic formulation of the problem would be:

If I were given a graph of nodes, each node is connected to several other nodes and the connections are of varying distance, what is the shortest path to every node in the graph if I start from one of the nodes in the graph?

# Depth First

In order to implement Dijkstra’s algorithm we first need to define a node and an edge:

`class Node:    def __init__(self, value):        self.value = value        self.edges = set()        self.distance = -1    @staticmethod    def add_edge(a, b, dist):        a.edges.add((b, dist))        b.edges.add((a, dist))`

In this case, I have defined a node that contains a value (an id basically), a distance initialized to be -1 (infinity). Additionally, the Node contains a list of edges, each is a tuple of the target node the edge connects to (from self) and the distance to the target node. …

# Sharpe Ratio, Sortino Ratio and Calmar Ratio

## In this short story, we are going to examine the deficiencies of Sharpe ratio, and how we can complement it with Sortino Ratio and Calmar Ratio to gain a clearer picture of the performance of a portfolio.

In portfolio performance analysis, sharpe ratio is the usually the first number that people look at. However, it does not tell us the whole story (nothing does…). So, let’s spend some time looking at a few more metrics that can be very helpful at times.

## Sharpe Ratio Revisited

Sharpe ratio is the ratio of average return divided by the standard deviation of returns annualized. We had an introduction to it in a previous story.

Let’s take a look at it again with a test price time series.

`import pandas as pdimport numpy as npfrom pandas.tseries.offsets import BDaydef daily_returns(prices):    res = (prices/prices.shift(1) - 1.0)[1:]    res.columns = ['return']    return resdef sharpe(returns, risk_free=0):    adj_returns = returns - risk_free    return (np.nanmean(adj_returns) * np.sqrt(252)) \        / np.nanstd(adj_returns, ddof=1)def test_price1():    start_date = pd.Timestamp(2020, 1, 1) + BDay()    len = 100    bdates = [start_date + BDay(i) for i in range(len)]    price = [10.0 + i/10.0 for i in range(len)]    return pd.DataFrame(data={'date': bdates,                              'price1': price}).set_index('date')def test_price2():    start_date = pd.Timestamp(2020, 1, 1) + BDay()    len = 100    bdates = [start_date + BDay(i) for i in range(len)]    price = [10.0 + i/10.0 for i in range(len)]    price[40:60] = [price for i in range(20)]    return pd.DataFrame(data={'date': bdates,                              'price2': price}).set_index('date')def test_price3():    start_date = pd.Timestamp(2020, 1, 1) + BDay()    len = 100    bdates = [start_date + BDay(i) for i in range(len)]    price = [10.0 + i/10.0 for i in range(len)]    price[40:60] = [price - i/10.0 for i in range(20)]    return pd.DataFrame(data={'date': bdates,                              'price3': price}).set_index('date')def test_price4():    start_date = pd.Timestamp(2020, 1, 1) + BDay()    len = 100    bdates = [start_date + BDay(i) for i in range(len)]    price = [10.0 + i/10.0 for i in range(len)]    price[40:60] = [price - i/8.0 for i in range(20)]    return pd.DataFrame(data={'date': bdates,                              'price4': price}).set_index('date')price1 = test_price1()return1 = daily_returns(price1)price2 = test_price2()return2 = daily_returns(price2)price3 = test_price3()return3 = daily_returns(price3)price4 = test_price4()return4 = daily_returns(price4)print('price1')print(f'sharpe: {sharpe(return1)}')print('price2')print(f'sharpe: {sharpe(return2)}')print('price3')print(f'sharpe: {sharpe(return3)}')print('price4')print(f'sharpe…`

# New York City Coronavirus Stats

## I live in Manhattan and I love New York, so when the superbug invaded the island, I was concerned. Luckily there is a wealth of data collected on NYC everyday, so let’s dig in!

New York government provides a large variety of datasets related to the city, making New York one of the best cities to do data analysis on (I love NYC!). For example: NYC OpenData. From driver applications data to rodent inspection, you can pretty much find any data you can think of…

The Coronavirus data we are going to look at today comes from NYC Health Department. It is a daily updated Github repository that contains NYC coronavirus cases data. All data shown below are as of date 2020–09–29.

# Rates by Poverty Level

• Higher poverty level corresponds to higher rates, as expected.
• Low poverty has a lower ratio of hospitalization vs case rate, presumably financially better off people tend to stay at home than going to hospital. …

# Graph Algorithms in Python: Locks and Graphs

## Graph algorithms are some of the most fascinating algorithms you’ll encounter. In this story we will implement such an algorithm to solve a problem involving combination locks.

Graphs are everywhere. from database to parallel processing, graphs underlie architectures of all kinds in real life. The concept behind graph is so ubiquitous that people develop such an intuitive understanding of it, sometimes they don’t actually realize they are dealing with graphs! So now, we are going to play with a couple of graph algorithms that involves a discussion around depth first vs breath first traversal.

# Locks

The problem is simple. Suppose I have a combination lock of 4 digits, such as 0000, 0001, etc. I start out at a particular combination of digits, say 1010, and I want to reach another combination of digits, say 9876. At each step, I can increase or decrease one digit by one, like a real lock you use at the gym. …

# Beta, Covariance and Stock Returns

## Explores the relationships of beta and covariance in the context of stock performance.

The fundamental idea behind beta and linear correlation, of course, goes back to the least square approximation that we all know and love.

Briefly reviewing the idea behind linear regression:

Suppose I have an independent variable y, for example number of views I get for my story, and a dependent variable x, the amount of time I spent on the story. I make a guess that the relationship between the two variables are linear:

The beta then, is the slope of the line that best fit x to y. There is a constant alpha as well.

I am sure everyone has seen them countless times, if you have only two data points (x_1, y_1), (x_2, y_2), there is a unique solution to beta, essentially you are drawing a linear segment between the two points. But if you have more than two points, then there is no unique solution to the fit. But in general, there is a solution, (beta, alpha), such that the line defined by them minimizes the square of the difference of the line and the y variable values, that’s the least square solution.

# Triangular Numbers

## In programming, the series that correspond to the triangular numbers seem to occur quite often. In this short story, I would like to discuss it a little with a couple fun algorithms to give some intuitions behind it.

I think most people have seen the triangular numbers at some point, they are the results of the following series:

As you can see, it is quite simple. I remember the first time I learned it was in middle school, it is usually written more casually like this:

# Closed Form

as you may have guessed, this series has a closed form that’s quite easy to remember:

expanding it:

As you can see it is of order O(n²), many algorithms reduce down to the triangular numbers in performance, for example bubble sort, thus are of order n².

How do you arrive at this closed form? …

# Pyramid Maker

## Recursion is beautiful, and when applied in combination it is truly a form of art. In this story, we are going to implement an interesting algorithm to create pyramids with recursion.

Ancient civilizations around the world have built giant pyramids for worship and other holy purposes. The most famous, perhaps, are the pyramids built by the Egyptians. These huge structures are built with white, limestone surfaces so that they appear to be shining when seen from a distance. They are so magnificent, in fact, that many people today still believe that they were built by the aliens. So, today we are going to build them in code with recursion. It is very fitting.

# Problem:

Suppose I am given a base like this:

[a, b, c ]

and that the following triangles are… 