Using the LINQ sequence operators,
you can transliterate the description
from the previous paragraph directly
into code:
end that describes what the slice repre-
sents. This means that by defining the
charting API to be LINQ friendly, you
can create charts by writing a query over
the Google image charts API:

var file = System.IO.File. ReadAllText(“words.txt”); var words = file.Split(delimiters)

.Where(w⇒ !w.IsNullOrWhiteSpace())

.Select(w⇒ w.ToLower());

 

Instead of using the sequence operators directly, LINQ also provides a more “declarative” query comprehension syntax. Using comprehensions, you can rewrite the code as follows:

 

var words = from w in file.Split(delimiters) where !w.IsNullOrWhiteSpace() select w.ToLower();

to truly understand
the power of LinQ,
let’s take a step
back and investigate
its origins and
mathematical
foundations. Don’t
worry, you need
knowledge of only
high school-level
mathematics.

var chart = new Pie(from w in top5 select new Slice( w.Count){ Legend = r.Word }) {Title = “Top 5 words” }; var image = await chart;

Once you have converted the file into
a sequence of individual words, you can
find the number of occurrences of each
word by first grouping the collection by
each word and then counting the num-
ber of elements in each group (which
contains all occurrences of that word):
The await keyword is used in an
unorthodox way to make the expensive
coercion from Google.Linq.Charts.
Pie into an image that requires a net-
work round-trip explicit.

var wordcount = from w in words group by w into group select new{ Word = group.Key,

Count = group.Count() };

Without using query comprehension syntax, the code would look like this:

var wordcount = words.GroupBy(w⇒w).

Select(group⇒ new{ Word = group.Key, Count = group.Count() };

 

To find the top five most frequent words, you can order each record by Count and take the first five elements:

 

var top5 = wc.OrderByDescending(p⇒

p.Count).Take( 5);

 

Now that you have a collection of the top five words in the file, you can visualize them in a pie chart, as in Figure 1. A pie chart is really nothing more than a collection of slices, where each slice consists of a number that represents the proportion of the total pie and a leg-

Datacentric interpretation

Relational algebra, which forms the formal basis for SQL, defines a number of constants and constructors for sets of values {Σ}, such as the empty set ∅∈{Σ}; injection of a value into a singleton collection {_}∈Σ→{Σ}; and the union of two sets into a new combined set ∪∈{Σ}x{Σ}→{Σ}. There are also a number of relational operators such as projection, which applies a transformation to each element in a set π∈(Σ→Λ) x{Σ}→{Λ}; selection, which selects only those elements in a set that satisfy a given property σ∈(Σ→)x{Σ}→{Σ}; Cartesian product, which pairs up all the elements of a pair of sets X∈{Σ} x{Λ}→{ΣxΛ}; and cross-apply, which generates a secondary set of values for each element in a first set ∈(Σ→{Λ}) x{Σ}→{Λ}.

Figure 2 depicts the relational algebra operators using clouds to denote sets of values. An SQL compiler translates queries expressed in the familiar SELECT-FROM-WHERE syntax into relational algebra expressions; to optimize the query it applies algebraic