1. Description

One of the earliest Programming Praxis challenges is called RPN Calculator . Our task is to write a RPN module capable of evaluating simple RPN expressions and return the right result . The exact requirment:

Implement an RPN calculator that takes an expression like 19 2.14 + 4.5 2 4.3 / - * which is usually expressed as (19 + 2.14) * (4.5 - 2 / 4.3) and responds with 85.2974. The program should read expressions from standard input and print the top of the stack to standard output when a newline is encountered. The program should retain the state of the operand stack between expressions.

Programming Praxis

The natural algorithm to resolve this exercise is stack-based :

The first step is to tokenize the expression .If the expression is “19 2.14 + 4.5 2 4.3 / – *” after the tokenization the resulting structure will be an array of operands and operators, for example an array of Strings {“19”, “2.14”, “+”, “4.5”, “2”, “4.3”, “/”, “-“, “*”} .
At the second step we iterate over the array of tokens .If the token:

  • Is an operand (number, variable) – we push in onto the stack .
  • Is an operator (we apriopri know the operator takes N arguments), we pop N values from the stack, we evaluate the operator with those values as input parameters, we push the result back onto the stack .
If the RPN expression is valid, after we apply the previous steps, the stack would contain only one value, which is the result of the calculation .

2. Implementation

I will implement the solutions for this challenge using three different languages: Scala, python and Java .

2.1 Scala Solution

And if we compile and run the above code: Continue reading

After a failed attempt to grasp Haskell, but somewhat seduced by the concise and elegant ways of Functional Programming, I’ve started to look for alternatives . Knowing Java, and having an everyday interaction with the JVM the obvious choices were Clojure or Scala . I never had the chance to try Clojure, probably because right now I enjoy Scala so much (the butterflies…) . Of course I cannot say that Scala scales, or if Scala is the next Java, or Scala is the answer, but from what I’ve experienced so far, things look good and I am optimistic that some day Scala will receive the attention it deserves .

There are a lot good books on Scala, but probably the most popular is “Programming in Scala, A comprehensive step-by-step guide” by Martin Odersky (Scala’s creator), Lex Spoon, and Bill Venners .

In a chapter there, the authors give us an insight on how to create Functional Objects -> objects that are immutable . The materialization of the concept is a Rational class, that encapsulates the idea of rational numbers . It allows us to be add, subtract, multiply, compare etc. rational numbers .  At some point in the example the fraction is reduced to the lowest terms.

Reducing the fraction to its lowest terms  (also known as normalization) is probably one of the simplest programming exercises we all start with . The actual problem is to determine the greatest common divisor of the fraction’s denominator and numerator, and divide those numbers by it . Not long ago I’ve written a solution for this exercise in C, but we all know C is far from being an elegant peace of technology.

Inspired (a lot!) by the example in the book, I’ve come with this solution:

Fraction.scala

MainObject.scala

To compile and run the above we can use scalac (the equivalent of javac) as: Continue reading

1. Algorithm Description

Insertion Sort is one of the simplest, but less effective, sorting algorithms . It works very similar to the way (us humans) sort a hard of playing cards:

1. At first none of our cards are sorted . So we start with an “empty” space in our hand were we are going to insert cards one at a time .
2. We take a card from our hand and we put it in the “special place” were we planned to keep our cards sorted . We do this for every card in our initial hand, until we finish them off .
3. Finding the right position for the insertion is not hard . At this point we can apply two tactics :

  • We compare the card we are planning to insert with all the cards that are already in sorted state O(n);
  • We use binary search to determine the right position (O(logn)) .

In our case that “special place” were we are going to insert cards one at a time, is not an additional array, but the array itself . We know for sure that an array of size 1 is always sorted, so we consider that first sorted sub-array, is the block composed by the first element .

For example, we want to sort the following array (with bold, are the elements already sorted):

8 0 3 11 5 -1 14 10 1 1 -2 Array is unsorted .
8 0 3 11 5 -1 14 10 1 1 -2 The first sorted block is {8} .
0 8 3 11 5 -1 14 10 1 1 -2 We compare {0} with {8} we move {0} at the new position, we shift {8} .
0 3 8 11 5 -1 14 10 1 1 -2 We compare {3} with {8}, {0}, we move {3} at the new position, we shift {8} .
0 3 8 11 5 -1 14 10 1 1 -2 We compare {11} with {8}, {11} remains at the same position .
0 3 5 8 11 -1 14 10 1 1 -2 We compare {5} with {11}, {8} and {3}, we move {5} at the new position, we shift {8}, {11} .
-1 0 3 5 8 11 14 10 1 1 -2 We compare {-1} with {11}, {8}, …, {0}, we move {-1} at the new position, we shift {0}, {3}, … {11} .
-1 0 3 5 8 11 14 10 1 1 -2 We compare {14} with {11}, we move {11} at the new position .
-1 0 3 5 8 10 11 14 1 1 -2 We compare {10} with {14}, {11}, {8}, we move {10} at the new position, we shift {11}, {14}.
-1 0 1 3 5 8 10 11 14 1 -2 We compare {1} with {14}, {11}, …, {0}, we move {1} at the new position, we shift {3}, {5}, …, {14} .
-1 0 1 1 3 5 8 10 11 14 -2 We compare {1} with {14}, {11}, …, {1}, we move {1} at the new position, we shift {3}, {5}, …, {14} .
-2 -1 0 1 1 3 5 8 10 11 14 We compare {-2} with {14}, {11}, …, {-1}, we move {-2} at the new position, we shift {-1}, {0}, …, {14} .

The pseudo-code for the algorithm can be easily written as follows:

2. Algorithm Implemenation in Java

The Java implementation of the algorithm is pretty straight-forward (follow the comments): Continue reading

1. Algorithm Description

In the last article I’ve described a recursive version of the Merge Sort Algorithm . Of course every recursive algorithm can be written in an iterative manner . So today I am going to present the bottom-up version of the same algorithm, written in a non-recursive fashion .

The main idea of the bottom-up merge sort is to sort the array in a sequence of passes . During each pass the array is divided into smaller sub-arrays of a pseudo-fixed size (step) . Initially step is 1, but the value increases with every pass, as every adjacent sub-arrays are merged, thus step doubles .

Example.:

0. We consider an array with size < = 1 sorted .
1. The array that needs to be sorted is A = { 5, 2, 1, 12, 2, 10, 4, 13, 5} . At this point step = 1 .
2. At the first iteration array A is divided into blocks of size step = 1 . The resulting blocks (sub-arrays) are {5}, {2}, {1}, {12}, {2}, {10}, {4}, {13}, {5} .
3. step *= 2, thus step is now 2 . At this point we have a collection of sorted sub-arrays (because their size is = 1) . We will group the sub-arrays one-by-one and we will start merging them . After the merge, the resulting sub-arrays are: {2, 5}, {1,12}, {2,10}, {4, 13} and {5} . {5} remains unpaired as the array size is an odd number . We will take care of this block later .
4. step *= 2, thus step is now 4 . Now we have a collection of 4 blocks of size two and one block of size one . We will start to merge again the adjacent blocks, so the sub-arrays collection becomes: {1, 2, 5, 12}, {2, 4, 10, 13} and {5} .
5. step *= 2, thus step is now 8 . Now we have a collection of 2 blocks with size 4 and one block with size 1 . We will merge the adjacent blocks so the sub-arrays collection becomes {1, 2, 2, 4, 5, 10, 12, 13} and {5} .
6. We now have two blocks one of size 8 and one of size 1 . We will merge those blocks and obtain the resulting sorted array: {1, 2, 2, 4, 5, 5, 10, 12, 13} .

The pseudo code for the algorithm can be written as follows . We will start by writing the merge function . This is responsible with the merging of two already sorted blocks (sub-arrays) from a given array . The input parameters for this function are : the array itself, and the interval headers (stop, start) for the two already sorted blocks . The sentinel is a concept that simplifies the code . In our case we consider SENTINEL infinity (no value from the array is bigger or equal to the Sentinel) .

And the actual function responsible with the actual merge can be written as follows:

Where A is the unsorted-but-soon-to-be-sorted array, length represents the size of A, step is the current size of the block, startL and startR represent the starting indexes of the sorted blocks that are going to be merged .

2. Algorithm implementation in Java Continue reading

1. Description

In computer science many algorithms are recursive by nature and they usually follow an approach called divide-and-conquer . In this particular case dividing-and-conquering a problems means to break it into smaller identical sub-problems, resolve the smaller sub-problems recursively (by dividing the smaller sub-problems into even smaller problems), and eventually combine all their results to determine the solution for the initial problem .

The Merge Sort algorithm is based on this technique :

1. The merge sort algorithm works on arrays (or lists). If the array has its size < = 1, then we consider the array already sorted and no change is done . This is considered to be the limit condition of this recursive approach . [Limit Condition]
2. If size of the array (or list) > 1, then we divide the array into two equally-sized sub-arrays . [Divide] – O(1)
3. At this point we recursively sort both of the two arrays . [Conquer] 2*T*(n/2)
4. We merge the two sorted sub-arrays into one sorted array . [Merge] O(n)

We will start by defining the MERGE function . This is function is responsible with the merging of two sorted arrays. The resulting array (in our case portion of the initial array composed by the two merged sub-arrays) will be also sorted :

Where:

L The initial unsorted array .
llim The starting index of the left sub-array;
mlim The stopping index for right sub-array, and the starting index for the right sub-array;
rlim The stopping index for the right sub-array;
LEFTL, RIGHTL Additional helper arrays .
SENTINEL A value to simplify our code, there’s nothing “bigger” than the Sentinel .

After we define the MERGE function, the actual MERGE-SORT function can be written as:

2. Java implementation Continue reading