From their description:

Project Euler is a series of challenging mathematical/computer programming problems that will require more than just mathematical insights to solve. Although mathematics will help you arrive at elegant and efficient methods, the use of a computer and programming skills will be required to solve most problems.

I’ve done a few of the first problems, implemented them all in ruby so far (or just did the math required), and they’ve definitely been interesting and fun to work on.  4 down, 242 to go.  All of the problems are math related, and you are basically presented with the problem and a field to input your answer.  After you supply the correct answer you’re given a link to the forum (most of which are locked now) where people have discussed their approaches to solving the problem.

If you’re looking to brush up on some math skills, or just need some interesting problems to throw some code at, this site provides some great things to work on.

It’s a pretty good summary of the different levels of programmer competence in a lot of areas.  Mostly I’m finding it useful in seeing where I’m at and getting suggestions on what to look into next in an effort to become better at programming and software development.

The basic idea is that we want to be able to generate an identifier for a random object consistently, then we can associate this identifier with a value.  The limitation is that the identifier needs to be a string (limitation of Javascript’s internal associative array/object construct), and that native javascript objects don’t have an internal mechanism for generating an ID (unlike Java’s Object class, which provides a hashCode() method).

The process to solve this problem has two steps in my current implementation:

1. Store the ‘key’ object in an array
2. Use the ‘key’ object’s array index as the key value for the association

This way, I can tell if a passed in ‘key’ object has a value in my map by seeing if it’s in my internal array.  If it’s in there, I can look up it’s index’s value in my associative array (or anonymous object).  If I’m adding a new key/value pair, I insert the key object into my internal array and then use it’s index as the ‘key’ for my associative array.  Removing a key/value pair from the map involves the following steps:

1. Look up the index of the key object in my internal array
2. Remove the value that corresponds to the index from the associative array/anonymous object
3. Remove the key object from the internal array

Here’s an example of the code that inserts a key/value pair into the map, this is a snip from the code, which is more object-oriented:

// Internal holders
var _internalMap = {};
var _internalArray = new Array();
 
/*
 * Private function for getting the key for an object
 */
function getKey(obj) {
    var key = _internalArray.indexOf(obj);
    if(key == -1) {
        _internalArray.push(obj);
        key = _internalArray.indexOf(obj);
    }
    // "Keys" will be of the form "OR.x" where 'x' is the index in the array
    return "OR." + key;
};
 
/*
 * Internal put method
 */
put = function(key, value) {
    var objKey = getKey(key);
    _internalMap[objKey] = value;
};

The full implementation has get, remove, and size methods as well, and is built as a class, but this kind of shows the mechanism behind it all.  One thing to note is that this implementation relies on Javascript 1.6 (or higher) as the ‘indexOf()’ array method was introduced in Javascript 1.6

I’ve attached the new version (version 0.2), you can find it here: JsMap Version 0.2

This approach is working pretty well, especially if the size of the storage is set to a larger size to start with.   The insertions and hashing and lookups are fast, it’s the rehashing and collisions that cause lots of problems.

Cuckoo Hash Part 2 – Second ‘version’ of my cuckoo hash implementation, including the changes above.  It now uses a list of 499 commonly used english words, and inserts 6 permutations of the list (pattern: “word-i“, where i is 1-6), for a total insertion count of 2994.

Cuckoo hashing is a scheme in computer programming for resolving hash collisions of values of hash functions in a table

Very basically, a Cuckoo hash is a table, where the position of an element can be one of two places.  On insertion, if there is another element (call it ‘y’) in the candidate spot, it is replaced by the item you are inserting.  Then you re-add the ‘y’ element to the hash, using the second hash method.  Doing this could cause another element (call it ‘z’) to be displaced, in which case the cycle repeats until all elements are in their new spots.

To look up if an element is in the table, you need to check both possible spots.

You can run into a problem with cycles when inserting values if, for example,  h1(x) = h2(y) and  h2(x) = h1(y), but you can limit the number of re-insertions to avoid this.  If you reach this limit, you can change your hashing methods and re-insert everything again. This problem happens even more frequently if the load factor on your table is too high, so you can put checks in place to increase the size of the table if you detect that the load is getting high.  In my example, I set the top table load factor to 60%, at which point I double the size of the table and re-insert the elements.

The main problem I’m running into is that the insert method can call the rehash method if it loops too many times trying to place everything…but rehash calls insert to rebuild the table with new hashing methods.  This seems to lead to deeply recursive problems, even with a load set to 50%.  I’m not sure how to fix this right now, but I’ll keep working at it and take any suggestions.  One method might be to find better hashing algorithms, but I thought the one I was using was descent.

Anyway, since I’m still learning about it, here’s a few links to read:

• Cuckoo Hashing for Undergraduates, 2006, R. Pagh, 2006. (PDF)
• Some String hashing algorithms
• Cuckoo Hash Attempt 1 – My first attempt at implementing a Cuckoo Hash in Ruby

For example, for the input “foo” you could generate the indices 1, 4 and 6.  Then in your (fixed size) array, you set positions 1, 4 and 6 to “1″.     For the input “bar” you generate the indices 1, 4, and 8, and set those positions to “1″.  When you want to find out if an input is in the set, you generate the index values (via your hashing method) and check if those positions are set to 1 in the array.  If all of them are, the input was likely in the set.  If one or more of the indices were not set to 1, then the input was definitely not in the set.  You can vary the number of index positions your hashing function provides and the size of your bit-array in order to reduce the number of false positives.

After your bit-array is built, representing your set, it becomes a constant time operation to check if something is in the set.  Insertion of new items into the Bloom Filter is also a constant time operation.  One major downside to Bloom Filters that I can see is that there isn’t a way to remove something from the bloom filter without causing false negatives.

The two articles I found useful in learning about Bloom Filters were:

• Bloom Filter – Wikipedia Bloom Filter Article
• Article at Cool/Snap? blog

So when I was asked a question about summations, I was kind of stumped. Now that I’ve started reading a algorithms text book, I’m running into summations all over again. In my search to re-learn them, I’ve found the following sites useful:

• Paul’s Online Math Notes – Prof. Paul Dawkins, of Lamar University, has a great math site, and has a good, short, re-introduction to summations here.
• PlanetMath’s Summation Page – PlanetMath has a good page on summations that goes a little more into detail on some summation properties and useful formulas
• Summation Formulas – Good outline of summation formulas

What I haven’t been able to find is a page explaining how the solutions for common summations is found, such as the one for a simple summation:

I’m not sure how that formula is reached, but I’m sure I’ll be trying to figure it out soon.

Update: Finally found a good description of the steps in solving the summation above, here’s my attempt at explaining it:

The simple summation above can be thought of as the series:

It can also be thought of (in reverse) as:

Now, imagine writing each of the terms of those two series above eachother, then adding the ones that are above and below eachother.  You would end up with something like:

This can be simplified by doing the math inside the parens to:

Which is simplified even further to:

At this point, you’ve basically described adding the summation twice, or

Divide each side by two, and you’ve got the solution we were looking for:

I finally found this explanation at the Math Refresher weblog.  The article I linked to goes into a lot more detail, so I would suggest reading it if you want to know more.  I tried, but my eyes are getting tired from math at ths point in time, and his equation representations are a little hard to read.

The Byzantine Generals problem is an exercise in fault tolerance and dealing with conflicting or even malicious behavior/messages in a distributed system.

Here’s the article

Bigtable is a distributed storage system for managing structured data that is designed to scale to a very large size: petabytes of data across thousands of commodity servers. Many projects at Google store data in Bigtable, including web indexing, Google Earth, and Google Finance. These applications place very different demands on Bigtable, both in terms of data size (from URLs to web pages to satellite imagery) and latency requirements (from backend bulk processing to real-time data serving). Despite these varied demands, Bigtable has successfully provided a flexible, high-performance solution for all of these Google products. In this paper we describe the simple data model provided by Bigtable, which gives clients dynamic control over data layout and format, and we describe the design and implementation of Bigtable.

The paper isn’t too long (14 pages or so) and isn’t too hard to digest. You can read it here