Category Archives: Examples

These are updates to my examples database.

Lets Learn About XOR Encryption

One of the more common things about this generation is the constant desire to write up (type) their thoughts. So many of the conversations from my high school days were long lasting, but quickly forgotten. Today’s generation is much more likely to blog, tweet, write status updates or simply open up a notepad file and write up their thoughts after such a conversation.

When we feel that our thoughts are not ready for public eyes (maybe you want to run your idea by the Patent and Trademark Office before speaking about it) we may seek some form of security to ensure that they stay private. An old fashioned way of doing this was to write in a diary and enclosed it within a lock and key. The mathematical field of encryption also tries to grant privacy by encoding messages so that only people with the necessary information can read them.

The type of encryption I want to speak about today is called XOR encryption. It is based on the logical operation called “exclusive or” (hence the name XOR). The exclusive or operation is true between two logical statements if exactly one of the two statements is true, but not both statement. This can be represented with the following truth table

Input 1 Input2 XOR Result
T T F
T F T
F T T
F F F

XOR Encryption is particularly useful in this day and age because we every character we type is understood by the computer as a sequence of zeros and ones. The current standard encoding that is used is Unicode (also known as UTF-8). Under this encoding the letter ‘a’ is represented as the binary string ‘01100001’. Similarly every letter, number and special character can be represented as its own binary string. These binary strings are just an assignment of numbers to these characters so that we can to help represent them in the computer. The numbers can the be thought of in base 10, which is how we generally think about numbers, or in base 2 which is how computers generally work with numbers (or a number of other ways). The way we would use these binary strings in encoding is first by translating a text from human-readable text to machine readable text via its binary string. For example, the word “Invincible”, we would get the following binary strings:

Letter Unicode in base 10 Unicode in base 2
I 73 01001001
n 110 01101110
v 118 01110110
i 105 01101001
n 110 01101110
c 99 01100011
i 105 01101001
b 98 01100010
l 108 01101100
e 101 01100101

To encrypt the message we need a key to encode the message and will simply perform an XOR operation on the key and every character in the string. Similarly, do decrypt the message we perform XOR operation on the key and every character in the encoded message. This means that the key (much like a normal key to a diary) must be kept private and only those whom the message is to be shared between have access to it.

I’m working on a new design for the site. Instead of linking to a new site with the JavaScript, I have placed it below inside the WordPress writeup. Try it out and let me know what you think. I hope you enjoy.


Message:
Key:



Discrete-time Markov Chains

Much of how we interact with life could be described as transitions between states. These states could be weather conditions (whether we are in a state of “sunny” or “rainy”), the places we may visit (maybe “school”, “the mall”, “the park” and “home”), our moods (“happy”, “angry”, “sad”). There are a number of other ways to model states and even the possibility of infinitely many states.

Markov Chains are based on the principle that the future is only dependent on the immediate past. so for example, if I wished to predict tomorrow’s weather using a Markov Chain, I would need to only look at the weather for today, and can ignore all previous data. I would then compare the state of weather for today with historically how weather has changed in between states to determine the most likely next state (i.e what the weather will be like tomorrow). This greatly simplifies the construction of models.

To use Markov Chains to predict the future, we first need to compute a transition matrix which shows the probability (or frequency) that we will travel from one state to another based on how often we have done so historically. This transition matrix can be calculated by looking at each element of the history as an instance of a discrete state, counting the number of times each transition occurs and dividing each result by the number of times the origin state occurs. I’ll next give an example and then I’ll focus on explaining the Finite Discrete State Markov Chain tool I built using javascript.

Next, I want to consider an example of using Markov Chains to predict the weather for tomorrow. Suppose that we have observed the weather for the last two weeks. We could then use that data to build a model to predict tomorrow’s weather. To do this, lets first consider some states of weather. Suppose that a day can be classified in one of four different ways: {Sunny, Cloudy, Windy, Rainy}. Further, suppose that over the last two weeks we have seen the following pattern.

Day 1 Sunny
Day 2 Sunny
Day 3 Cloudy
Day 4 Rain
Day 5 Sunny
Day 6 Windy
Day 7 Rain
Day 8 Windy
Day 9 Rain
Day 10 Cloudy
Day 11 Windy
Day 12 Windy
Day 13 Windy
Day 14 Cloudy

We can look at this data and calculate the probability that we will transition from each state to each other state, which we see below:

Rain Cloudy Windy Sunny
Rain 0 1/3 1/3 1/3
Cloudy 1/2 0 1/2 0
Windy 2/5 1/5 2/5 0
Sunny 0 1/3 1/3 1/3

Given that the weather for today is cloudy, we can look at the transition matrix and see that historically the days that followed a cloudy day have been Rainy and Windy days each with probability of 1/5. We can see this more mathematically by multiplying the current state vector (cloudy) [0, 1, 0, 0] by the above matrix, where we obtain the result [1/2, 0, 1/2, 0].

In similar fashion, we could use this transition matrix (lets call it T) to predict the weather a number of days in the future by looking at Tn. For example, if we wanted to predict the weather two days in the future, we could begin with the state vector [1/2, 0, 1/2, 0] and multiply it by the matrix T to obtain [1/5, 4/15, 11/30, 1/6].

We can also obtain this by looknig at the original state vector [0, 1, 0, 0] and multiplying it by T2.

T2 =

1

3/10 8/45 37/90 1/9
1/5 4/15 11/30 1/6
13/50 16/75 59/150 2/15
3/10 8/45 37/90 1/9

When we multiply the original state vector by T2 we arrive at this same answer [1/5, 4/15, 11/30, 1/6]. This matrix T2 has an important property in that every state can reach every other state.

In general, if we have a transition matrix where for every cell in row i and column j, there is some power of the transition matrix such that the cell (i, j) in that matrx is nonzero, then we say that every state is reachable from every other state and we call the Markov Chain regular.

Regular Markov Chains are important because they converge to what’s called a steady state. These are state vectrs x = [x0, …, xn] such that xTn = x for very large values of n. The steady state tells us how the Markov Chain will perform over long periods of time. We can use algebra and systems of linear equations to solve for this steady state vector.

For the Javascript program I’ve written, I have generated a set of painting samples for a fictional artist. The states are the different colors and the transitions are the colors that the artist will use after other colors. as well as the starting and ending colors. Given this input, we can form a Markov Chain to understand the artist’s behavior. This Markov Chain can then be used to solve for the steady state vector or to generate random paintings according to the artist’s profile. Be sure to check it out and let me know what you think.


Show Work?





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Topological Sort

One of the things I generally say about myself is that I love learning. I can spend hours upon hours reading papers and algorithms to better understand a topic. Some of these topics are stand alone segments that I can understand in one sitting. Sometimes, however, there is a need to read up on some preliminary work in order to fully understand a concept.

Lets say that I was interested in organizing this information into a new course. The order I present these topics is very important. Knowing which topics depend on one another allows me to use the topological sorting algorithm to determine an ordering for the topics that respects the preliminary work.

The input for the topoligical sorting algorithm is a Directed Acyclic Graph (DAG). This is a set of relationships between pairs of topics, where if topic 1 must be understood before topic 2, we would add the relationship (topic 1, topic 2) to the graph. DAGs can be visualized by a set of nodes (points) representing the topics. Relationships like the one above (topic 1, topic 2) can then represented by a directed arc originating at topic 1 and flowing in the direction of topic 2. We say that the graph is “Acyclic” because there cannot be a cycle in the topic preliminaries. This amounts to us saying that a topic cannot be a prerequisite for itself. An example of a DAG is shown in the image above.

With the topics represented as a DAG, the topologial ordering algorithm works by searching the set of nodes for the one with no arcs coming into it. This node (or these nodes is multiple are present) represents the topic that can be covered next without losing understanding of the material. Such a node is guaranteed to exist by the acyclic property of the DAG. Once the node is selected, we can remove this node as well as all arcs that originate at this node from the DAG. The algorithm then repeats the procedure of searching for a nod with no arcs coming into it. This process repeats until there are no remaining nodes from which to choose.

Now lets see how the topological sort algorithm works on the graph above. We will first need to count the in-degree (the number of arcs coming into) each node.

Node | Indegree
—————-
0 | 2
1 | 2
2 | 0
3 | 2
4 | 2
5 | 2
6 | 0
7 | 2
8 | 3

Node to be removed (i.e. node with the minimum indegree): Node 2.
Arcs connected to node 2: (2, 5), (2, 3)
Resulting Indegree Count:
Node | Indegree
—————-
0 | 2
1 | 2
3 | 1
4 | 2
5 | 1
6 | 0
7 | 2
8 | 3

Node to be removed: Node 6:
Arcs connected to node 6: (6, 1), (6, 3), (6, 4), (6, 5), (6, 7), (6, 8)
Resulting Indegree Count:
Node | Indegree
—————-
0 | 2
1 | 1
3 | 0
4 | 1
5 | 0
7 | 1
8 | 2

Node to be removed: Node 3
Arcs connected to node 3: (3, 0), (3, 8)
Resulting Indegree Count:
Node | Indegree
—————-
0 | 1
1 | 1
4 | 1
5 | 0
7 | 1
8 | 1

Node to be removed: Node 5
Arcs connected to node 5: (5, 0), (5, 8)
Resulting Indegree Count:
Node | Indegree
—————-
0 | 0
1 | 1
4 | 1
7 | 1
8 | 0

Node to be removed: Node 0:
Arcs connected to node 0: (0, 1), (0, 4)
Resulting Indegree Count:
Node | Indegree
—————-
1 | 0
4 | 0
7 | 1
8 | 0

Node to be removed: Node 1
Arcs connected to node 1: none
Resulting Indegree Count:
Node | Indegree
—————-
4 | 0
7 | 1
8 | 0

Node to be removed: Node 4
Arcs connected to node 4: none
Resulting Indegree Count:
Node | Indegree
—————-
7 | 1
8 | 0

Node to be removed: Node 8
Arcs connected to node 8: (8, 7)
Resulting Indegree Count:
Node | Indegree
—————-
7 | 0

Node to be removed: Node 7
Arcs connected to node 7: none
Resulting Indegree Count:
Node | Indegree
—————-

Since there are no nodes remaining, we have arrived at a topological ordering. Going through this iteration, we can see that we arrived at the ordering (2, 6, 3, 5, 0, 1, 4, 8, 7). There were several occasions where there were multiple nodes with indegree of 0 and we could have selected an alternative node. This would have given us a different topological ordering of the nodes, but it would still be valid.

There are more learning opportunities and an interactive demonstration of the algorithm at Topological Sort Examples at LEARNINGlover.

The RSA Algorithm

I can remember back when I was in school, still deciding whether I wanted to study pure or applied mathematics. One of the common questions I would receive from those in applied mathematical realms would sound like “What’s the point of doing mathematics with no real world applications?”. Generally my response to these questions was about the intrinsic beauty of mathematics, no different from an artist painting not for some desire to be a millionaire, but because of an burning desire to paint. Whether their paintings would one day be on the walls of a Smithsonian museum or sit on their mother’s refrigerator is generally outside of the thought process of the artist. So too, would I argue about the thought process of a pure mathematician.

When I was an undergrad and learned about the RSA algorithm (named for Ron Rivest, Adi Shamir, and Leonard Adleman who discovered the algorithm) it helped me explain this concept a lot better. The algorithm is based on prime numbers and the problem of finding the divisors of a given number. Many mathematicians throughout the ages have written papers on the beauty of prime numbers (see Euclid, Eratosthenes, Fermat, Goldbach, etc). For a large period in time one of the beautiful things about prime numbers was that they were so interesting in themselves. There were questions about how to check if a number is prime, question of patterns in primes, famous conjectures like the Goldbach conjecture and the twin prime conjecture, quick ways of finding prime numbers or numbers that are almost always prime, etc. In short, this was an active area of research that much of the applied world was not using. This all changed in 1977 when Rivest, Shamir and Adleman published the RSA algorithm.

The algorithm is in the area called public key cryptography. These algorithms differ from many of the previous cryptography algorithms, namely symmetric key cryptography. Whereas symmetric key cryptography depends uses the same device (key) to encode as to decode, public key cryptography creates two keys – one for encoding that is generally shared with others, and another for decoding which is kept private. These two keys in generally relate to a mathematical problem that is very difficult to solve.

In my example script for the RSA Algorithm, I show two people who want to communicate, Alice and Bob. Bob wants people to be able to send him messages securely so he decides to use the RSA algorithm. To do this, he first needs to think of two prime numbers, p1 and p2.
From these, he computes the following:
n = p1 * p2

Next, he computes Euler’s function on this n which can be calculated as
(n) = (p1 – 1) * (p2 – 1)

Then Bob looks for a number e that is relatively prime to . This is what he will use as the encryption key.

One he has e, he can calculate d, which is the multiplicative inverse of e in (n).
This means that e * d = 1 (mod (n)).

The public key that will be used for encryption will be the pair (e, n). This is what he posts publicly via his web page for others to communicate with him securely. Bob also has a private key pair (d, n) that he will use to decrypt messages.

Alice sees Bob’s public key and would like to communicate with him. So she uses it to encode a message. The formula she uses to encrypt her message is c = me mod n, where c is the encrypted message. Once Alice encrypts her message, she sends it to Bob.

Bob receives this encoded message and uses the private key (d, n) to decode the message from Alice. The formula to decrypt is m = cd mod n.

For a more illustrative idea of how this algorithm works as well as examples, be sure to visit Script for the RSA Algorithm.

The Depth-First-Search Algorithm

I remember when I was younger I used to play the game of hide-and-seek a lot. This is a game where a group of people (at least two) separate into a group of hiders and a group of seekers. The most common version of this that I’ve seen is having one person as the seeker and everyone as hiders. Initially, the seeker(s) is given a number to count towards and close their eyes while counting. The hiders then search for places to hide from the seeker. Once the seeker is finished counting, their job is to find where everyone is hiding or admitting that they cannot find all the seekers. Any seekers not found are said to have won, and seekers that are found are said to have lost.

I played this game a number of times in my childhood, but I remember playing it with a friend named Dennis in particular. Dennis had a certain way he played as seeker. While many of us would simply go to places we deemed as “likely” hiding spots in a somewhat random order, Dennis would always begin by looking in one area of the room, making sure that he had searched through every area connected to that area before going to a new area. He continued this process until he either found everybody or concluded that he had searched every spot he could think of and gave up.

It wasn’t until years later that I was able to note the similarity between Dennis’s way of playing hide-and-seek and the Depth-First-Search algorithm. The Depth-First-Search Algorithm is a way of exploring all the nodes in a graph. Similar to hide-and-seek, one could choose to do this in a number of different ways. Depth-First-Search does this by beginning at some node, looking first at one of the neighbors of that node, then looking at one of the neighbors of this new node. If the new node does not have any new neighbors, then the algorithm goes to the previous node, looks at the next neighbor of this node and continues from there. Initially all nodes are “unmarked” and the algorithm proceeds by marking nodes as being in one of three states: visited nodes are marked as “visited”; nodes that we’ve marked to visit, but have not visited yet are marked “to-visit”; and unmarked nodes that have not been marked or visited are “unvisited”.

Consider a bedroom with the following possible hiding locations: (1) Under Bed, (2) Behind Cabinet, (3) In Closet, (4) Under Clothes, (5) Behind Curtains, (6) Behind Bookshelf, and (7) Under Desk. We can visualize how the bedroom is arranged as a graph and then use a Breadth First Search algorithm to show how Brent would search the room. Consider the following bedroom arrangement, where we have replaced the names of each item by the number corresponding to that item. Node (0) corresponds to the door, which is where Dennis stands and counts while others hide.

Bedroom Items as a Graph

Now consider how a Breadth First Search would be run on this graph.

Bedroom Items as a Graph Colored by DFS

The colors correspond to the order in which nodes are visited in Depth-First-Search.

The way we read this is that initially Dennis would start at node 0, which is colored in Blue.
While Dennis is at node 0, she notices that nodes 1, 5, and 6 (under bed, behind curtains, and behind bookshelf) are the nearby and have not been checked yet so she places them on the “to visit” list.
Next, Dennis will begin to visit each node on the “to visit” list, and when a node is visited, she labels it as visited. At each location, she also takes note of the other locations she can reach from this location. Below is the order of nodes Dennis visits and how he discovers new locations to visit.

Order Visited Node Queue Adding Distance From Node 0
1 0 6,5,1 0
2 6 5,1 7,3,2 1
3 7 3,2,5,1 2
4 3 2,5,1 2
5 2 5,1 4 2
6 4 5,1 3
7 5 1 1
8 1 1

Here is a link to my Examples page that implements the Depth-First-Search Algorithm on Arbitrary Graphs.

The Breadth-First-Search Algorithm

I remember when I was younger I used to play the game of hide-and-seek a lot. This is a game where a group of people (at least two) separate into a group of hiders and a group of seekers. The most common version of this that I’ve seen is having one person as the seeker and everyone as hiders. Initially, the seeker(s) is given a number to count towards and close their eyes while counting. The hiders then search for places to hide from the seeker. Once the seeker is finished counting, their job is to find where everyone is hiding or admitting that they cannot find all the seekers. Any seekers not found are said to have won, and seekers that are found are said to have lost.

I played this game a number of times in my childhood, but I remember playing it with a friend named Brenda in particular. Brenda had a certain way she played as seeker. While many of us would simply go to places we deemed as “likely” hiding spots in a somewhat random order, Brenda would always take a survey of the room, and no matter where she began searching, she would always make note of the locations close to her starting point and make sure she was able to give them all a look before she looked at locations that were close to the points she deemed close to the starting point. She continued this process until she either found everybody or concluded that she had searched every spot she could think of and gave up.

It wasn’t until years later that I was able to note the similarity between Brenda’s way of playing hide-and-seek and the Breadth-First-Search algorithm. The Breadth-First-Search algorithm is a way of exploring all the nodes in a graph. Similarly to hide-and-seek, one could choose to do this in a number of different ways. Breadth-First-Search does this by beginning at some node, looking first at each of the neighbors of the starting node, then looking at each of the neighbors of the neighbors of the starting node, continuing this process until there are no remaining nodes to visit. Initially all nodes are “unmarked” and the algorithm proceeds by marking nodes as being in one of three stages: visited nodes are marked as “visited”; nodes that we’ve marked to visit, but have not visited yet are marked “to-visit”; and unmakred nodes that have not been marked are “unvisited”.

Consider a bedroom with the following possible hiding locations: (1) Under Bed, (2) Behind Cabinet, (3) In Closet, (4) Under Clothes, (5) Behind Curtains, (6) Behind Bookshelf, and (7) Under Desk. We can visualize how the bedroom is arranged as a graph and then use a Breadth First Search algorithm to show how Brenda would search the room. Consider the following bedroom arrangement, where we have replaced the names of each item by the number corresponding to that item. Node (0) corresponds to the door, which is where Brenda stands and counts while others hide.

Bedroom Items as a Graph

Now consider how a Breadth First Search would be run on this graph.

Bedroom Items as a Graph

The colors correspond to the order in which nodes are visited in Breadth-First-Search.

The way we read this is that initially Brenda would start at node 0, which is colored in Blue.
While Brenda is at node 0, she notices that nodes 1, 5, and 6 (under bed, behind curtains, and behind bookshelf) are the nearby and have not been checked yet so she places them on the “to visit” list.
Next, Brenda will begin to visit each node on the “to visit” list, and when a node is visited, she labels it as visited. At each location, she also takes note of the other locations she can reach from this location. Below is the order of nodes Brenda visits and how she discovers new locations to visit.

Order Visited Node Queue Adding Distance From Node 0
1 0 1, 5, 6 0
2 1 5, 6 2, 4 1
3 5 6, 2, 4 1
4 6 2, 4 3, 7 1
5 2 4, 3, 7 2
6 4 3, 7 2
7 3 7 2
8 7 2

Here is a link to my Examples page that implements the Breadth-First-Search Algorithm on Arbitrary Graphs.

ID3 Algorithm Decision Trees

ID3 Algorithm Page Image

As I grow LEARNINGlover.com, I’m always thinking of different ways to expose my own personality through the site. This is partially because it is easier for me to talk about subjects where I am already knowledgeable, but it is more-so being done to help make some of these algorithms and concepts I encode more understandable, and sometimes relating foreign concepts to everyday life makes them easier to understand.

Today, I’d like to write about decision trees, and the ID3 algorithm for generating decision trees in particular. This is a machine learning algorithm that builds a model from a training data set consisting of a feature vector and an outcome. Because our data set consists of an outcome element, this falls into the category of supervised machine learning.

The model that the ID3 algorithm builds is called a decision tree. It builds a tree based on the features, or columns of the data set with a possible decision corresponding to each value that the feature can have. The algorithm selects the next feature by asking “which feature tells me the most about our data set?” This question can be answered first by asking how much “information” is in the data set, and then comparing that result with the amount of information in each individual feature.

In order to execute this algorithm we need a way to measure both the amount the information in outcomes of the overall data set as well as how much each feature tells us about the data set. For the first, we will use entropy, which comes from the field of information theory and encoding. Entropy is based on the question of how many bits are necessary to encode the information in a set. The more information, the higher the entropy, and the more bits required to encode that information. Although we are not encoding, the correlation between high information and high entropy suits our purposes.

Entropy Formula

To understand how much each feature tells us about the outcomes of the data set we will build on the concept of entropy to define the information gain of a feature. Each feature has multiple options, so the dataset can be partitioned based on each possible value of this feature. Once we have this partition, we can calculate the entropy of each subset of the rows of data. We define the information gain of a feature as the sum over all possible outcomes of that feature can have of the entropy of that outcome multiplied by the probability of that outcome.

Information Gain Formula

To illustrate this algorithm, I decided to relate it to the question of whether we think of a character in a novel as a hero or villain. This is interesting because I try to read at least one book a month and as I’m reading, I often find myself asking this question about characters based on the traits of the characters as well as characters I’ve read about. In order to build an interactive script for this problem, I considered 25 possible character traits that could be present. A subset of these 25 character traits will be selected and a row will be generated grading a fictional character on a scale of 0 to 3 (0 meaning that they do not possess the trait at all, 3 meaning that the trait is very strong in their personality), and users will be asked whether they think a character with the given character traits should be listed as a hero or a villain. Then there is a button at the bottom of the script with the text “Build Tree” that executes the ID3 Algorithm and shows a decision tree that could be used to reach the set of decisions given by the user.

The possible features are:
Abstract, Adaptable, Aggressive, Ambition, Anxiety, Artistic, Cautious, Decisive, Honesty, Dutiful, Fitness, Intellect, Independent, Introverted, Lively, Open-minded, Orderly, Paranoid, Perfectionist, Romantic, Sensitive, Stable, Tension, Warmth and Wealthy

Once users select the option to build the tree, there will be several links outlining each step in the process to build this tree. These links will allow for users to expand the information relating to that step and minimize that information when done. Hopefully this will help users to understand each step more. I must say that as much fun as it has been writing this program, there were several questions when trying to explain it to others. Hopefully users get as much fun from using this tool as I had in creating it. As always feel free to contact me with any comments and or questions.

Ok, so here’s a link to the ID3 Algorithm Page. Please check it out and let me know what you think.

Clique Problem Puzzles

I still remember how I felt when I was first introduced to NP-Complete problems. Unlike the material I had learned up to that point, there seemed to be such mystery and intrigue and opportunity surrounding these problems. To use the example from Garey and Johnson’s book “Computers and Intractability: A Guide to the Theory of NP Completeness”, these were problems that not just one researcher found difficult, but that a number of researchers had been unable to find efficient algorithms to solve them. So what they did was show that the problems all had a special relationship with one another, and thus through this relationship if someone were to discover an algorithm to efficiently solve any one of these problems they would be able to efficiently solve all the problems in this class. This immediately got my mind working into a world where I, as a college student, would discover such an algorithm and be mentioned with the heavyweights of computer science like Lovelace, Babbage, Church, Turing, Cook, Karp and Dean.

Unfortunately I was a student so I did not have as much time to devote to this task as I would have liked. In my spare time though I would try to look at problems and see what kind of structure I found. One of my favorite problems was, The Clique Problem. This is a problem where we are given an undirected graph and seek to find a maximum subset of nodes in this graph that all have edges between them, i.e. a clique (Actually the NP-Complete version of this problem takes as input an undirected graph G and an integer k and asks if there is a clique in G of size k).

Although I now am more of the mindset that there do not exist efficient algorithms to solve NP-Complete problems, I thought it would be a nice project to see if I could re-create this feeling – both in myself and others. So I decided to write a program that generates a random undirected graph and asks users to try to find a maximum clique. To test users answers, I coded up an algorithm that works pretty well on smaller graphs, the Bron-Kerbosch Algorithm. This algorithm uses backtracking to find all maximal cliques, which then allows us to sort them by size and determine the largest.

Users should click on the numbers in the table below the canvas indicating the nodes they wish to select in their clique (purple indicates that the node is selected, gray indicates that it is not). Once they have a potential solution, they can press the “Check” button to see if their solution is optimal. If a user is having trouble and simply wishes to see the maximum clique, they can press the “Solve” button. And to generate a new problem, users can press the “New Problem” button.

So I hope users have fun with the clique problem puzzles, and who knows maybe someone will discover an algorithm that efficiently solves this problem and become world famous.

Unidirectional TSP Puzzles

As we’ve entered the late spring into early summer season, I’ve found myself wanting to go out more to sit and enjoy the weather. One of these days recently I sat in the park with a good book. On this occurrence, I decided not to go with a novel as I had just finished “Incarceron“, “The Archer’s Tale“, and “14 Stones” – all of which were good reads, but I felt like taking a break from the novels.

Just as a side note, 14 Stones is a free book available on smashwords.com and I’ve now read about 6 books from smashwords.com and haven’t been disappointed yet. My favorite is still probably “The Hero’s Chamber” because of the imagery of the book, but there are some well written ebooks available there by some good up and coming writers for a reasonable price, with some being free.

 

So with the desire to read, but not being in the mood for novels I decided to pick up one of my non-text but still educational books that make me think. This day it was “Programming Challenges“. I browsed through the book until I found one that I could lay back, look at the water, and think about how to solve it.

 

The programming puzzle the peaked my interest was called “Unidirectional TSP”. We are given a grid with m rows and n columns, with each cell showing the cost of using that cell. The user is allowed to begin in any cell in the first column and is asked to reach any cell in the last column using some minimum cost path. There is an additional constraint that once a cell is selected in a column, a cell in the next column can only be chosen from the row directly above, the same row, or the row directly below. There is a javascript version of this puzzle available here.

Fundamentally, the problem is asking for a path of shortest length. Many shortest length problems have a greedy structure, but this one gained my interest because the greedy solution is not always optimal in this case. So I took a moment to figure out the strategy behind these problems. Once I had that solution, I decided that it would be a good program to write up as a puzzle.

 

In this puzzle version, users will click the cells they wish to travel in each column in which case they will turn green (clicking again will turn them white again). Once the user clicks on a cell in the last column, they will be notified of whether or not they have chosen the minimum path. Or if users are unable to solve a puzzle, the “Solution” button can be pressed to show the optimal path and its cost. 

Learn About Descriptive Statistics

I had been meaning to write a script and blog post on descriptive statistics for some time now, but with work and winter weather and the extra work that winter weather brings, and now that the winter weather is over trying to get back into an exercise routine (running up a hill is such a challenging experience, but when I get to the top of that hill I feel like Rocky Balboa on the steps at the steps at the entrance of the Philadelphia Museum of Art), I haven’t had the time to devote to this site that I would have liked. Well, that’s not entirely true. I have still been programming in my spare time. I just haven’t been able to share it here. I went to a conference in February and in my down time, I was able to write a script on descriptive statistics that I think gives a nice introduction to the area.

Before I go into descriptive statistics though, lets talk about statistics, which is concerned with the collection, analysis, interpretation and presentation of data. Statistics can generally be broken down into two categories, descriptive statistics and infernalinferential statistics, depending on what we would like to do with that data. When we are concerned with visualizing and summarizing the given data, descriptive statistics gives methods to operate on this data set. On the other hand, if we wish to draw conclusions about a larger population from our sample, then we would use methods from inferential statistics.

In the script on descriptive statistics I’ve written, I consider three different types of summaries for descriptive statistics:

Measures of Central Tendency
Mean – the arithmetic average of a set of values
Median – the middle number in a set of values
Mode – the most used number in a set of values

Dispersion
Maximum – the largest value in the data set
Minimum – the smallest value in the data set
Standard Deviation – the amount of variation in a set of data values
Variance – how far a set of numbers is spread out

Shape
Kurtosis – how peaked or flat a data set is
Skewness – how symmetric a data set is

Plots
Histogram Plots – a bar diagram where the horizontal axis shows different categories of values, and the height of each bar is related to the number of observations in the corresponding category.
Box and Whisker Plots – A box-and-whisker plot for a list of numbers consists of a rectangle whose left edge is at the first quartile of the data and whose right edge is at the third quartile, with a left whisker sticking out to the smallest value, and a right whisker sticking out to the largest value.
Stem and Leaf Plots – A stem and leaf plot illustrates the distribution of a group of numbers by arranging the numbers in categories based on the first digit.