25 Rewriting R code in C++

25.1 Introduction

Sometimes R code just isn’t fast enough. You’ve used profiling to figure out where your bottlenecks are, and you’ve done everything you can in R, but your code still isn’t fast enough. In this chapter you’ll learn how to improve performance by rewriting key functions in C++. This magic comes by way of the Rcpp package (Eddelbuettel and François 2011) (with key contributions by Doug Bates, John Chambers, and JJ Allaire).

Rcpp makes it very simple to connect C++ to R. While it is possible to write C or Fortran code for use in R, it will be painful by comparison. Rcpp provides a clean, approachable API that lets you write high-performance code, insulated from R’s complex C API.

Typical bottlenecks that C++ can address include:

  • Loops that can’t be easily vectorised because subsequent iterations depend on previous ones.

  • Recursive functions, or problems which involve calling functions millions of times. The overhead of calling a function in C++ is much lower than in R.

  • Problems that require advanced data structures and algorithms that R doesn’t provide. Through the standard template library (STL), C++ has efficient implementations of many important data structures, from ordered maps to double-ended queues.

The aim of this chapter is to discuss only those aspects of C++ and Rcpp that are absolutely necessary to help you eliminate bottlenecks in your code. We won’t spend much time on advanced features like object-oriented programming or templates because the focus is on writing small, self-contained functions, not big programs. A working knowledge of C++ is helpful, but not essential. Many good tutorials and references are freely available, including http://www.learncpp.com/ and https://en.cppreference.com/w/cpp. For more advanced topics, the Effective C++ series by Scott Meyers is a popular choice.


  • Section 25.2 teaches you how to write C++ by converting simple R functions to their C++ equivalents. You’ll learn how C++ differs from R, and what the key scalar, vector, and matrix classes are called.

  • Section 25.2.5 shows you how to use sourceCpp() to load a C++ file from disk in the same way you use source() to load a file of R code.

  • Section 25.3 discusses how to modify attributes from Rcpp, and mentions some of the other important classes.

  • Section 25.4 teaches you how to work with R’s missing values in C++.

  • Section 25.5 shows you how to use some of the most important data structures and algorithms from the standard template library, or STL, built-in to C++.

  • Section 25.6 shows two real case studies where Rcpp was used to get considerable performance improvements.

  • Section 25.7 teaches you how to add C++ code to a package.

  • Section 25.8 concludes the chapter with pointers to more resources to help you learn Rcpp and C++.


We’ll use Rcpp to call C++ from R:

You’ll also need a working C++ compiler. To get it:

  • On Windows, install Rtools.
  • On Mac, install Xcode from the app store.
  • On Linux, sudo apt-get install r-base-dev or similar.

25.2 Getting started with C++

cppFunction() allows you to write C++ functions in R:

When you run this code, Rcpp will compile the C++ code and construct an R function that connects to the compiled C++ function. There’s a lot going on underneath the hood but Rcpp takes care of all the details so you don’t need to worry about them.

The following sections will teach you the basics by translating simple R functions to their C++ equivalents. We’ll start simple with a function that has no inputs and a scalar output, and then make it progressively more complicated:

  • Scalar input and scalar output
  • Vector input and scalar output
  • Vector input and vector output
  • Matrix input and vector output

25.2.1 No inputs, scalar output

Let’s start with a very simple function. It has no arguments and always returns the integer 1:

The equivalent C++ function is:

We can compile and use this from R with cppFunction()

This small function illustrates a number of important differences between R and C++:

  • The syntax to create a function looks like the syntax to call a function; you don’t use assignment to create functions as you do in R.

  • You must declare the type of output the function returns. This function returns an int (a scalar integer). The classes for the most common types of R vectors are: NumericVector, IntegerVector, CharacterVector, and LogicalVector.

  • Scalars and vectors are different. The scalar equivalents of numeric, integer, character, and logical vectors are: double, int, String, and bool.

  • You must use an explicit return statement to return a value from a function.

  • Every statement is terminated by a ;.

25.2.2 Scalar input, scalar output

The next example function implements a scalar version of the sign() function which returns 1 if the input is positive, and -1 if it’s negative:

In the C++ version:

  • We declare the type of each input in the same way we declare the type of the output. While this makes the code a little more verbose, it also makes clear the type of input the function needs.

  • The if syntax is identical — while there are some big differences between R and C++, there are also lots of similarities! C++ also has a while statement that works the same way as R’s. As in R you can use break to exit the loop, but to skip one iteration you need to use continue instead of next.

25.2.3 Vector input, scalar output

One big difference between R and C++ is that the cost of loops is much lower in C++. For example, we could implement the sum function in R using a loop. If you’ve been programming in R a while, you’ll probably have a visceral reaction to this function!

In C++, loops have very little overhead, so it’s fine to use them. In Section 25.5, you’ll see alternatives to for loops that more clearly express your intent; they’re not faster, but they can make your code easier to understand.

The C++ version is similar, but:

  • To find the length of the vector, we use the .size() method, which returns an integer. C++ methods are called with . (i.e., a full stop).

  • The for statement has a different syntax: for(init; check; increment). This loop is initialised by creating a new variable called i with value 0. Before each iteration we check that i < n, and terminate the loop if it’s not. After each iteration, we increment the value of i by one, using the special prefix operator ++ which increases the value of i by 1.

  • In C++, vector indices start at 0, which means that the last element is at position n - 1. I’ll say this again because it’s so important: IN C++, VECTOR INDICES START AT 0! This is a very common source of bugs when converting R functions to C++.

  • Use = for assignment, not <-.

  • C++ provides operators that modify in-place: total += x[i] is equivalent to total = total + x[i]. Similar in-place operators are -=, *=, and /=.

This is a good example of where C++ is much more efficient than R. As shown by the following microbenchmark, sumC() is competitive with the built-in (and highly optimised) sum(), while sumR() is several orders of magnitude slower.

25.2.4 Vector input, vector output

Next we’ll create a function that computes the Euclidean distance between a value and a vector of values:

In R, it’s not obvious that we want x to be a scalar from the function definition, and we’d need to make that clear in the documentation. That’s not a problem in the C++ version because we have to be explicit about types:

This function introduces only a few new concepts:

  • We create a new numeric vector of length n with a constructor: NumericVector out(n). Another useful way of making a vector is to copy an existing one: NumericVector zs = clone(ys).

  • C++ uses pow(), not ^, for exponentiation.

Note that because the R version is fully vectorised, it’s already going to be fast.

On my computer, it takes around 5 ms with a 1 million element y vector. The C++ function is about 2.5 times faster, ~2 ms, but assuming it took you 10 minutes to write the C++ function, you’d need to run it ~200,000 times to make rewriting worthwhile. The reason why the C++ function is faster is subtle, and relates to memory management. The R version needs to create an intermediate vector the same length as y (x - ys), and allocating memory is an expensive operation. The C++ function avoids this overhead because it uses an intermediate scalar.

25.2.5 Using sourceCpp

So far, we’ve used inline C++ with cppFunction(). This makes presentation simpler, but for real problems, it’s usually easier to use stand-alone C++ files and then source them into R using sourceCpp(). This lets you take advantage of text editor support for C++ files (e.g., syntax highlighting) as well as making it easier to identify the line numbers in compilation errors.

Your stand-alone C++ file should have extension .cpp, and needs to start with:

And for each function that you want available within R, you need to prefix it with:

You can embed R code in special C++ comment blocks. This is really convenient if you want to run some test code:

The R code is run with source(echo = TRUE) so you don’t need to explicitly print output.

To compile the C++ code, use sourceCpp("path/to/file.cpp"). This will create the matching R functions and add them to your current session. Note that these functions can not be saved in a .Rdata file and reloaded in a later session; they must be recreated each time you restart R.

For example, running sourceCpp() on the following file implements mean in C++ and then compares it to the built-in mean():

NB: if you run this code, you’ll notice that meanC() is much faster than the built-in mean(). This is because it trades numerical accuracy for speed.

For the remainder of this chapter C++ code will be presented stand-alone rather than wrapped in a call to cppFunction. If you want to try compiling and/or modifying the examples you should paste them into a C++ source file that includes the elements described above. This is easy to do in RMarkdown: all you need to do is specify engine = "Rcpp".

25.2.6 Exercises

  1. With the basics of C++ in hand, it’s now a great time to practice by reading and writing some simple C++ functions. For each of the following functions, read the code and figure out what the corresponding base R function is. You might not understand every part of the code yet, but you should be able to figure out the basics of what the function does.

  2. To practice your function writing skills, convert the following functions into C++. For now, assume the inputs have no missing values.

    1. all().

    2. cumprod(), cummin(), cummax().

    3. diff(). Start by assuming lag 1, and then generalise for lag n.

    4. range().

    5. var(). Read about the approaches you can take on Wikipedia. Whenever implementing a numerical algorithm, it’s always good to check what is already known about the problem.

25.3 Other classes

You’ve already seen the basic vector classes (IntegerVector, NumericVector, LogicalVector, CharacterVector) and their scalar (int, double, bool, String) equivalents. Rcpp also provides wrappers for all other base data types. The most important are for lists and data frames, functions, and attributes, as described below. Rcpp also provides classes for more types like Environment, DottedPair, Language, Symbol, etc, but these are beyond the scope of this chapter.

25.3.1 Lists and data frames

Rcpp also provides List and DataFrame classes, but they are more useful for output than input. This is because lists and data frames can contain arbitrary classes but C++ needs to know their classes in advance. If the list has known structure (e.g., it’s an S3 object), you can extract the components and manually convert them to their C++ equivalents with as(). For example, the object created by lm(), the function that fits a linear model, is a list whose components are always of the same type. The following code illustrates how you might extract the mean percentage error (mpe()) of a linear model. This isn’t a good example of when to use C++, because it’s so easily implemented in R, but it shows how to work with an important S3 class. Note the use of .inherits() and the stop() to check that the object really is a linear model.

25.3.2 Functions

You can put R functions in an object of type Function. This makes calling an R function from C++ straightforward. The only challenge is that we don’t know what type of output the function will return, so we use the catchall type RObject.

Calling R functions with positional arguments is obvious:

But you need a special syntax for named arguments:

25.3.3 Attributes

All R objects have attributes, which can be queried and modified with .attr(). Rcpp also provides .names() as an alias for the name attribute. The following code snippet illustrates these methods. Note the use of ::create(), a class method. This allows you to create an R vector from C++ scalar values:

For S4 objects, .slot() plays a similar role to .attr().

25.4 Missing values

If you’re working with missing values, you need to know two things:

  • How R’s missing values behave in C++’s scalars (e.g., double).
  • How to get and set missing values in vectors (e.g., NumericVector).

25.4.1 Scalars

The following code explores what happens when you take one of R’s missing values, coerce it into a scalar, and then coerce back to an R vector. Note that this kind of experimentation is a useful way to figure out what any operation does.

With the exception of bool, things look pretty good here: all of the missing values have been preserved. However, as we’ll see in the following sections, things are not quite as straightforward as they seem. Integers

With integers, missing values are stored as the smallest integer. If you don’t do anything to them, they’ll be preserved. But, since C++ doesn’t know that the smallest integer has this special behaviour, if you do anything to it you’re likely to get an incorrect value: for example, evalCpp('NA_INTEGER + 1') gives -2147483647.

So if you want to work with missing values in integers, either use a length 1 IntegerVector or be very careful with your code. Doubles

With doubles, you may be able to get away with ignoring missing values and working with NaNs (not a number). This is because R’s NA is a special type of IEEE 754 floating point number NaN. So any logical expression that involves a NaN (or in C++, NAN) always evaluates as FALSE:

(Here I’m using evalCpp() which allows you to see the result of running a single C++ expression, making it excellent for this sort of interactive experimentation.)

But be careful when combining them with Boolean values:

However, in numeric contexts NaNs will propagate NAs:

25.4.2 Strings

String is a scalar string class introduced by Rcpp, so it knows how to deal with missing values.

25.4.3 Boolean

While C++’s bool has two possible values (true or false), a logical vector in R has three (TRUE, FALSE, and NA). If you coerce a length 1 logical vector, make sure it doesn’t contain any missing values; otherwise they will be converted to TRUE. An easy fix is to use int instead, as this can represent TRUE, FALSE, and NA.

25.4.5 Exercises

  1. Rewrite any of the functions from the first exercise of Section 25.2.6 to deal with missing values. If na.rm is true, ignore the missing values. If na.rm is false, return a missing value if the input contains any missing values. Some good functions to practice with are min(), max(), range(), mean(), and var().

  2. Rewrite cumsum() and diff() so they can handle missing values. Note that these functions have slightly more complicated behaviour.

25.5 Standard Template Library

The real strength of C++ is revealed when you need to implement more complex algorithms. The standard template library (STL) provides a set of extremely useful data structures and algorithms. This section will explain some of the most important algorithms and data structures and point you in the right direction to learn more. I can’t teach you everything you need to know about the STL, but hopefully the examples will show you the power of the STL, and persuade you that it’s useful to learn more.

If you need an algorithm or data structure that isn’t implemented in STL, a good place to look is boost. Installing boost on your computer is beyond the scope of this chapter, but once you have it installed, you can use boost data structures and algorithms by including the appropriate header file with (e.g.) #include <boost/array.hpp>.

25.5.1 Using iterators

Iterators are used extensively in the STL: many functions either accept or return iterators. They are the next step up from basic loops, abstracting away the details of the underlying data structure. Iterators have three main operators:

  1. Advance with ++.
  2. Get the value they refer to, or dereference, with *.
  3. Compare with ==.

For example we could re-write our sum function using iterators:

The main changes are in the for loop:

  • We start at x.begin() and loop until we get to x.end(). A small optimization is to store the value of the end iterator so we don’t need to look it up each time. This only saves about 2 ns per iteration, so it’s only important when the calculations in the loop are very simple.

  • Instead of indexing into x, we use the dereference operator to get its current value: *it.

  • Notice the type of the iterator: NumericVector::iterator. Each vector type has its own iterator type: LogicalVector::iterator, CharacterVector::iterator, etc.

This code can be simplified still further through the use of a C++11 feature: range-based for loops. C++11 is widely available, and can easily be activated for use with Rcpp by adding [[Rcpp::plugins(cpp11)]].

Iterators also allow us to use the C++ equivalents of the apply family of functions. For example, we could again rewrite sum() to use the accumulate() function, which takes a starting and an ending iterator, and adds up all the values in the vector. The third argument to accumulate gives the initial value: it’s particularly important because this also determines the data type that accumulate uses (so we use 0.0 and not 0 so that accumulate uses a double, not an int.). To use accumulate() we need to include the <numeric> header.

25.5.2 Algorithms

The <algorithm> header provides a large number of algorithms that work with iterators. A good reference is available at https://en.cppreference.com/w/cpp/algorithm. For example, we could write a basic Rcpp version of findInterval() that takes two arguments a vector of values and a vector of breaks, and locates the bin that each x falls into. This shows off a few more advanced iterator features. Read the code below and see if you can figure out how it works.

The key points are:

  • We step through two iterators (input and output) simultaneously.

  • We can assign into an dereferenced iterator (out_it) to change the values in out.

  • upper_bound() returns an iterator. If we wanted the value of the upper_bound() we could dereference it; to figure out its location, we use the distance() function.

  • Small note: if we want this function to be as fast as findInterval() in R (which uses handwritten C code), we need to compute the calls to .begin() and .end() once and save the results. This is easy, but it distracts from this example so it has been omitted. Making this change yields a function that’s slightly faster than R’s findInterval() function, but is about 1/10 of the code.

It’s generally better to use algorithms from the STL than hand rolled loops. In Effective STL, Scott Meyers gives three reasons: efficiency, correctness, and maintainability. Algorithms from the STL are written by C++ experts to be extremely efficient, and they have been around for a long time so they are well tested. Using standard algorithms also makes the intent of your code more clear, helping to make it more readable and more maintainable.

25.5.3 Data structures

The STL provides a large set of data structures: array, bitset, list, forward_list, map, multimap, multiset, priority_queue, queue, deque, set, stack, unordered_map, unordered_set, unordered_multimap, unordered_multiset, and vector. The most important of these data structures are the vector, the unordered_set, and the unordered_map. We’ll focus on these three in this section, but using the others is similar: they just have different performance trade-offs. For example, the deque (pronounced “deck”) has a very similar interface to vectors but a different underlying implementation that has different performance trade-offs. You may want to try it for your problem. A good reference for STL data structures is https://en.cppreference.com/w/cpp/container — I recommend you keep it open while working with the STL.

Rcpp knows how to convert from many STL data structures to their R equivalents, so you can return them from your functions without explicitly converting to R data structures.

25.5.4 Vectors

An STL vector is very similar to an R vector, except that it grows efficiently. This makes vectors appropriate to use when you don’t know in advance how big the output will be. Vectors are templated, which means that you need to specify the type of object the vector will contain when you create it: vector<int>, vector<bool>, vector<double>, vector<String>. You can access individual elements of a vector using the standard [] notation, and you can add a new element to the end of the vector using .push_back(). If you have some idea in advance how big the vector will be, you can use .reserve() to allocate sufficient storage.

The following code implements run length encoding (rle()). It produces two vectors of output: a vector of values, and a vector lengths giving how many times each element is repeated. It works by looping through the input vector x comparing each value to the previous: if it’s the same, then it increments the last value in lengths; if it’s different, it adds the value to the end of values, and sets the corresponding length to 1.

(An alternative implementation would be to replace i with the iterator lengths.rbegin() which always points to the last element of the vector. You might want to try implementing that.)

Other methods of a vector are described at https://en.cppreference.com/w/cpp/container/vector.

25.5.5 Sets

Sets maintain a unique set of values, and can efficiently tell if you’ve seen a value before. They are useful for problems that involve duplicates or unique values (like unique, duplicated, or in). C++ provides both ordered (std::set) and unordered sets (std::unordered_set), depending on whether or not order matters for you. Unordered sets tend to be much faster (because they use a hash table internally rather than a tree), so even if you need an ordered set, you should consider using an unordered set and then sorting the output. Like vectors, sets are templated, so you need to request the appropriate type of set for your purpose: unordered_set<int>, unordered_set<bool>, etc. More details are available at https://en.cppreference.com/w/cpp/container/set and https://en.cppreference.com/w/cpp/container/unordered_set.

The following function uses an unordered set to implement an equivalent to duplicated() for integer vectors. Note the use of seen.insert(x[i]).second. insert() returns a pair, the .first value is an iterator that points to element and the .second value is a Boolean that’s true if the value was a new addition to the set.

25.5.6 Map

A map is similar to a set, but instead of storing presence or absence, it can store additional data. It’s useful for functions like table() or match() that need to look up a value. As with sets, there are ordered (std::map) and unordered (std::unordered_map) versions. Since maps have a value and a key, you need to specify both types when initialising a map: map<double, int>, unordered_map<int, double>, and so on. The following example shows how you could use a map to implement table() for numeric vectors:

25.5.7 Exercises

To practice using the STL algorithms and data structures, implement the following using R functions in C++, using the hints provided:

  1. median.default() using partial_sort.

  2. %in% using unordered_set and the find() or count() methods.

  3. unique() using an unordered_set (challenge: do it in one line!).

  4. min() using std::min(), or max() using std::max().

  5. which.min() using min_element, or which.max() using max_element.

  6. setdiff(), union(), and intersect() for integers using sorted ranges and set_union, set_intersection and set_difference.

25.6 Case studies

The following case studies illustrate some real life uses of C++ to replace slow R code.

25.6.1 Gibbs sampler

The following case study updates an example blogged about by Dirk Eddelbuettel, illustrating the conversion of a Gibbs sampler in R to C++. The R and C++ code shown below is very similar (it only took a few minutes to convert the R version to the C++ version), but runs about 20 times faster on my computer. Dirk’s blog post also shows another way to make it even faster: using the faster random number generator functions in GSL (easily accessible from R through the RcppGSL package) can make it another two to three times faster.

The R code is as follows:

This is straightforward to convert to C++. We:

  • Add type declarations to all variables.

  • Use ( instead of [ to index into the matrix.

  • Subscript the results of rgamma and rnorm to convert from a vector into a scalar.

Benchmarking the two implementations yields:

25.6.2 R vectorisation versus C++ vectorisation

This example is adapted from “Rcpp is smoking fast for agent-based models in data frames”. The challenge is to predict a model response from three inputs. The basic R version of the predictor looks like:

We want to be able to apply this function to many inputs, so we might write a vector-input version using a for loop.

If you’re familiar with R, you’ll have a gut feeling that this will be slow, and indeed it is. There are two ways we could attack this problem. If you have a good R vocabulary, you might immediately see how to vectorise the function (using ifelse(), pmin(), and pmax()). Alternatively, we could rewrite vacc1a() and vacc1() in C++, using our knowledge that loops and function calls have much lower overhead in C++.

Either approach is fairly straightforward. In R:

(If you’ve worked R a lot you might recognise some potential bottlenecks in this code: ifelse, pmin, and pmax are known to be slow, and could be replaced with p * 0.75 + p * 0.5 * female, p[p < 0] <- 0, p[p > 1] <- 1. You might want to try timing those variations.)

Or in C++:

We next generate some sample data, and check that all three versions return the same values:

The original blog post forgot to do this, and introduced a bug in the C++ version: it used 0.004 instead of 0.04. Finally, we can benchmark our three approaches:

Not surprisingly, our original approach with loops is very slow. Vectorising in R gives a huge speedup, and we can eke out even more performance (about ten times) with the C++ loop. I was a little surprised that the C++ was so much faster, but it is because the R version has to create 11 vectors to store intermediate results, where the C++ code only needs to create 1.

25.7 Using Rcpp in a package

The same C++ code that is used with sourceCpp() can also be bundled into a package. There are several benefits of moving code from a stand-alone C++ source file to a package:

  1. Your code can be made available to users without C++ development tools.

  2. Multiple source files and their dependencies are handled automatically by the R package build system.

  3. Packages provide additional infrastructure for testing, documentation, and consistency.

To add Rcpp to an existing package, you put your C++ files in the src/ directory and create or modify the following configuration files:


    LinkingTo: Rcpp
    Imports: Rcpp
  • Make sure your NAMESPACE includes:

    importFrom(Rcpp, sourceCpp)

    We need to import something (anything) from Rcpp so that internal Rcpp code is properly loaded. This is a bug in R and hopefully will be fixed in the future.

The easiest way to set this up automatically is to call usethis::use_rcpp().

Before building the package, you’ll need to run Rcpp::compileAttributes(). This function scans the C++ files for Rcpp::export attributes and generates the code required to make the functions available in R. Re-run compileAttributes() whenever functions are added, removed, or have their signatures changed. This is done automatically by the devtools package and by Rstudio.

For more details see the Rcpp package vignette, vignette("Rcpp-package").

25.8 Learning more

This chapter has only touched on a small part of Rcpp, giving you the basic tools to rewrite poorly performing R code in C++. As noted, Rcpp has many other capabilities that make it easy to interface R to existing C++ code, including:

  • Additional features of attributes including specifying default arguments, linking in external C++ dependencies, and exporting C++ interfaces from packages. These features and more are covered in the Rcpp attributes vignette, vignette("Rcpp-attributes").

  • Automatically creating wrappers between C++ data structures and R data structures, including mapping C++ classes to reference classes. A good introduction to this topic is the Rcpp modules vignette, vignette("Rcpp-modules").

  • The Rcpp quick reference guide, vignette("Rcpp-quickref"), contains a useful summary of Rcpp classes and common programming idioms.

I strongly recommend keeping an eye on the Rcpp homepage and signing up for the Rcpp mailing list.

Other resources I’ve found helpful in learning C++ are:

  • Effective C++ (Meyers 2005) and Effective STL (Meyers 2001).

  • C++ Annotations, aimed at knowledgeable users of C (or any other language using a C-like grammar, like Perl or Java) who would like to know more about, or make the transition to, C++.

  • Algorithm Libraries, which provides a more technical, but still concise, description of important STL concepts. (Follow the links under notes.)

Writing performance code may also require you to rethink your basic approach: a solid understanding of basic data structures and algorithms is very helpful here. That’s beyond the scope of this book, but I’d suggest the Algorithm Design Manual (Skiena 1998), MIT’s Introduction to Algorithms, Algorithms by Robert Sedgewick and Kevin Wayne which has a free online textbook and a matching Coursera course.

25.9 Acknowledgments

I’d like to thank the Rcpp-mailing list for many helpful conversations, particularly Romain Francois and Dirk Eddelbuettel who have not only provided detailed answers to many of my questions, but have been incredibly responsive at improving Rcpp. This chapter would not have been possible without JJ Allaire; he encouraged me to learn C++ and then answered many of my dumb questions along the way.


Eddelbuettel, Dirk, and Romain François. 2011. “Rcpp: Seamless R and C++ Integration.” Journal of Statistical Software 40 (8):1–18. https://doi.org/10.18637/jss.v040.i08.

Meyers, Scott. 2001. Effective STL: 50 Specific Ways to Improve Your Use of the Standard Template Library. Pearson Education. http://amzn.com/0201749629.

Meyers, Scott. 2005. Effective C++: 55 Specific Ways to Improve Your Programs and Designs. Pearson Education. http://amzn.com/0321334876.

Skiena, Steven S. 1998. The Algorithm Design Manual. Springer. http://amzn.com/0387948600.