This article is part of the article series "Perl One-Liners Explained."
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Perl One LinersThis is the second part of a seven-part article on Perl one-liners. In this part I will create various one-liners for line numbering. See part one for introduction of the series.

Perl one-liners is my attempt to create "perl1line.txt" that is similar to "awk1line.txt" and "sed1line.txt" that have been so popular among Awk and Sed programmers.

The article on Perl one-liners will consist of at least seven parts:

The one-liners will make heavy use of Perl special variables. A few years ago I compiled all the Perl special variables in a single file and called it Perl special variable cheat-sheet. Even tho it's mostly copied out of perldoc perlvar, it's still handy to have in front of you. Print it!

Awesome news: I have written an e-book based on this article series. Check it out:

And here are today's one-liners:

Line Numbering

9. Number all lines in a file.

perl -pe '$_ = "$. $_"'

As I explained in the first one-liner, "-p" causes Perl to assume a loop around the program (specified by "-e") that reads each line of input into the " $_ " variable, executes the program and then prints the " $_ " variable.

In this one-liner I simply modify " $_ " and prepend the " $. " variable to it. The special variable " $. " contains the current line number of input.

The result is that each line gets its line number prepended.

10. Number only non-empty lines in a file.

perl -pe '$_ = ++$a." $_" if /./'

Here we employ the "action if condition" statement that executes "action" only if "condition" is true. In this case the condition is a regular expression "/./", which matches any character except newline (that is, it matches a non-empty line); and the action is " $_ = ++$a." $_" ", which prepends variable " $a " incremented by one to the current line. As we didn't use strict pragma, $a was created automatically.

The result is that at each non-empty line " $a " gets incremented by one and prepended to that line. And at each empty line nothing gets modified and the empty line gets printed as is.

11. Number and print only non-empty lines in a file (drop empty lines).

perl -ne 'print ++$a." $_" if /./'

This one-liner uses the "-n" program argument that places the line in " $_ " variable and then executes the program specified by "-e". Unlike "-p", it does not print the line after executing code in "-e", so we have to call "print" explicitly to get it printed.

The one-liner calls "print" only on lines that have at least one character in them. And exactly like in the previous one-liner, it increments the line number in variable " $a " by one for each non-empty line.

The empty lines simply get ignored and never get printed.

12. Number all lines but print line numbers only non-empty lines.

perl -pe '$_ = "$. $_" if /./'

This one-liner is similar to one-liner #10. Here I modify the " $_ " variable that holds the entire line only if the line has at least one character. All other lines (empty ones) get printed without line numbers.

13. Number only lines that match a pattern, print others unmodified.

perl -pe '$_ = ++$a." $_" if /regex/'

Here we again use the "action if condition" statement but the condition in this case is a pattern (regular expression) "/regex/". The action is the same as in one-liner #10. I don't want to repeat, see #10 for explanation.

14. Number and print only lines that match a pattern.

perl -ne 'print ++$a." $_" if /regex/'

This one-liner is almost exactly like #11. The only difference is that it prints numbered lines that match only "/regex/".

15. Number all lines, but print line numbers only for lines that match a pattern.

perl -pe '$_ = "$. $_" if /regex/'

This one-liner is similar to the previous one-liner and to one-liner #12. Here the line gets its line number prepended if it matches a /regex/, otherwise it just gets printed without a line number.

16. Number all lines in a file using a custom format (emulate cat -n).

perl -ne 'printf "%-5d %s", $., $_'

This one-liner uses the formatted print "printf" function to print the line number together with line. In this particular example the line numbers are left aligned on 5 char boundary.

Some other nice format strings are "%5d" that right-aligns line numbers on 5 char boundary and "%05d" that zero-fills and right-justifies the line numbers.

Here my Perl printf cheat sheet might come handy that lists all the possible format specifiers.

17. Print the total number of lines in a file (emulate wc -l).

perl -lne 'END { print $. }'

This one-liner uses the "END" block that Perl probably took as a feature from Awk language. The END block gets executed after the program has executed. In this case the program is the hidden loop over the input that was created by the "-n" argument. After it has looped over the input, the special variable " $. " contains the number of lines there was in the input. The END block prints this variable. The " -l " parameter sets the output record separator for "print" to a newline (so that we didn't have to print "$.\n").

Another way to do the same is:

perl -le 'print $n=()=<>'

This is a tricky one, but easy to understand if you know about Perl contexts. In this one-liner the " ()=<> " part causes the <> operator (the diamond operator) to evaluate in list context, that causes the diamond operator to read the whole file in a list. Next, " $n " gets evaluated in scalar context. Evaluating a list in a scalar context returns the number of elements in the list. Thus the " $n=()=<> " construction is equal to the number of lines in the input, that is number of lines in the file. The print statement prints this number out. The " -l " argument makes sure a newline gets added after printing out this number.

This is the same as writing the following, except longer:

perl -le 'print scalar(()=<>)'

And completely obvious version:

perl -le 'print scalar(@foo=<>)'

Yet another way to do it:

perl -ne '}{print $.'

This one-liner uses the eskimo operator "}{" in conjunction with "-n" command line argument. As I explained in one-liner #11, the "-n" argument forces Perl to assume a " while(<>) { } " loop around the program. The eskimo operator forces Perl to escape the loop, and the program turns out to be:

while (<>) {
}{                    # eskimo operator here
    print $.;

It's easy to see that this program just loops over all the input and after it's done doing so, it prints the " $. ", which is the number of lines in the input.

18. Print the number of non-empty lines in a file.

perl -le 'print scalar(grep{/./}<>)'

This one-liner uses the "grep" function that is similar to the grep Unix command. Given a list of values, " grep {condition} " returns only those values that match condition. In this case the condition is a regular expression that matches at least one character, so the input gets filtered and the "grep{/./}" returns all lines that were non empty. To get the number of characters we evaluate the list in scalar context and print the result. (As I mentioned in the previous one-liner list in scalar context evaluates to number of elements in the list).

A golfer's version of this one-liner would be to replace "scalar()" with " ~~ " (double bitwise negate), thus it can be shortened:

perl -le 'print ~~grep{/./}<>'

This can be made even shorter:

perl -le 'print~~grep/./,<>'

19. Print the number of empty lines in a file.

perl -lne '$a++ if /^$/; END {print $a+0}'

Here I use variable $a to count how many empty lines have I encountered. Once I have finished looping over all the lines, I print the value of $a in the END block. I use " $a+0 " construction to make sure " 0 " gets output if no lines were empty.

I could have also modified the previous one-liner:

perl -le 'print scalar(grep{/^$/}<>)'

Or written it with " ~~ ":

perl -le 'print ~~grep{/^$/}<>'

These last two versions are not as effective, as they would read the whole file in memory. Where as the first one would do it line by line.

20. Print the number of lines in a file that match a pattern (emulate grep -c).

perl -lne '$a++ if /regex/; END {print $a+0}'

This one-liner is basically the same as the previous one, except it increments the line counter $a by one in case a line matches a regular expression /regex/.

Perl one-liners explained e-book

I've now written the "Perl One-Liners Explained" e-book based on this article series. I went through all the one-liners, improved explanations, fixed mistakes and typos, added a bunch of new one-liners, added an introduction to Perl one-liners and a new chapter on Perl's special variables. Please take a look:

Have Fun!

Have fun with these one-liners. These were really easy this time. The next part is going to be about various calculations.

Can you think of other numbering operations that I did not include here?

A Year of BloggingHoly smokes! It has now been two years since I started this blog. It seems almost like yesterday when I posted the "A Year of Blogging" article. And now it's two! With this post I'd like to celebrate the 2nd birthday and share various interesting statistics that I managed to gather.

During this year (July 20, 2008 - July 26, 2009) I wrote 55 posts, which received around 1000 comments. According to StatCounter and Google Analytics my blog was visited by 1,050,000 unique people who viewed 1,700,000 pages. Wow, 1 million visitors! That's very impressive!

Here is a Google Analytics graph of monthly page views for the last year (click for a larger version):

Catonmat.Net Page Views Per Month (Second Year of Blogging)

In the last three months I did not manage to write much and you can see how that reflected on the page views. A good lesson to be learned is to be persistent and keep writing articles consistently.

Here is the same graph with two years of data, showing a complete picture of my blog's growth:

Catonmat.Net Page Views Per Month (Two Years of Blogging)

I like this seemingly linear growth. I hope it continues the same way the next year!

Here are the top 5 referring sites that my visitors came from:

And here are the top 5 referring blogs:

I found that just a handful of blogs had linked to me during this year. The main reason, I suspect, is that I do not link out much myself... It's something to improve upon.

If you remember, I ended the last year's post with the following words (I had only 1000 subscribers at that time):

I am setting myself a goal of reaching 5000 subscribers by the end of the next year of blogging (July 2009)! I know that this is very ambitious goal but I am ready to take the challenge!

I can proudly say that I reached my ambitious goal! My blog now has almost 7000 subscribers! If you have not yet subscribed, click here to do it!

Here is the RSS subscriber graph for the whole two years:

RSS Subscriber Count, Two Years of Blogging

Several months ago I approximated the subscriber data with an exponent function and it produced a good fit. Probably if I had continued writing articles at the same pace I did three months ago, I'd have over 10,000 subscribers now.

Anyway, let's now turn to the top 10 most viewed posts:

The article that I liked the most myself but which didn't make it to top ten was the "Set Operations in Unix Shell". I just love this Unix stuff I did there.

I am also very proud for the following three article series that I wrote:

  • 1. Review of MIT's Introduction to Algorithms course (14 parts).
  • 2. Awk One-Liners Explained (4 parts: 1, 2, 3, 4).
  • 3. Sed One-Liners Explained (3 parts: 1, 2, 3)

Finally, here is a list of ideas that I have thought for the third year of blogging:

  • Publish three e-books on Awk One-Liners, Sed One-Liners and Perl One-Liners.
  • Launch mathematics, physics and general science blog.
  • Write about mathematical foundations of cryptography and try to implement various cryptosystems and cryptography protocols.
  • Publish my review of MIT's Linear Algebra course (in math blog, so the main topic of catonmat stays computing).
  • Publish my review of MIT's Physics courses on Mechanics, Electromagnetism, and Waves (in physics blog).
  • Publish my notes on how I learned the C++ language.
  • Write more about computer security and ethical hacking.
  • Write several book reviews.
  • Create a bunch of various fun utilities and programs.
  • Create at least one useful web project.
  • Add a knowledge database to catonmat, create software to allow easy publishing to it.
  • If time allows, publish reviews of important computer science publications.

I'll document everything here as I go, so if you are interested in these topics stay with me by subscribing to my rss feed!

And to make things more challenging again, I am setting a new goal for the next year of blogging. The goal is to reach 20,000 subscribers by July 2010!

Hope to see you all on my blog again! Now it's time for this delicious cake:

Second Birthday Portal Game Cake

This article is part of the article series "MIT Introduction to Algorithms."
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MIT AlgorithmsThis is a happy and sad moment at the same time - I have finally reached the last two lectures of MIT's undergraduate algorithms course. These last two lectures are on a fairly new area of algorithm research called "cache oblivious algorithms."

Cache-oblivious algorithms take into account something that has been ignored in all the lectures so far, particularly, the multilevel memory hierarchy of modern computers. Retrieving items from various levels of memory and cache make up a dominant factor of running time, so for speed it is crucial to minimize these costs. The main idea of cache-oblivious algorithms is to achieve optimal use of caches on all levels of a memory hierarchy without knowledge of their size.

Cache-oblivious algorithms should not be confused with cache-aware algorithms. Cache-aware algorithms and data structures explicitly depend on various hardware configuration parameters, such as the cache size. Cache-oblivious algorithms do not depend on any hardware parameters. An example of cache-aware (not cache-oblivious) data structure is a B-Tree that has the explicit parameter B, the size of a node. The main disadvantage of cache-aware algorithms is that they are based on the knowledge of the memory structure and size, which makes it difficult to move implementations from one architecture to another. Another problem is that it is very difficult, if not impossible, to adapt some of these algorithms to work with multiple levels in the memory hierarchy. Cache-oblivious algorithms solve both problems.

Lecture twenty-two introduces the terminology and notation used in cache-oblivious algorithms, explains the difference between cache-oblivious and cache-aware algorithms, does a simple memory analysis of several simple algorithms and culminates with a cache-oblivious algorithm for matrix multiplication.

The final lecture twenty-three is the most difficult in the whole course and shows cache-oblivious binary search trees and cache-oblivious sorting called funnel sort.

Use this supplementary reading material by professor Demaine to understand the material better: Cache-oblivious algorithms and data structures (.pdf).

Lecture 22: Cache Oblivious Algorithms I

Lecture twenty-two starts with an introduction to the modern memory hierarchy (CPU cache L1, L2, L3, main memory, disk cache, etc.) and with the notation and core concepts used in cache-oblivious algorithms.

A powerful result in cache-oblivious algorithm design is that if an algorithm is efficient on two levels of cache, then it's efficient on any number of levels. Thus the study of cache-obliviousness can be simplified to two-level memory hierarchy, say the CPU cache and main memory, where the accesses to cache are instant but are orders of magnitude slower to main memory. Therefore the main question cache-oblivious algorithm analysis tries to address is how many memory transfers (MTs) does a problem of size N take. The notation used for this is MT(N). For an algorithm to be efficient, the number of memory transfers should be as small as possible.

Next the lecture analysis the number of memory transfers for basic array scanning and array reverse algorithms. Since array scanning is consequential, N elements can be processed with O(N/B) accesses, where B is the block size - number of elements that are automatically fetched as N-th element is accessed. That is MT(N) = O(N/B) for array scanning. The same bound holds for reversing an array, since it can be viewed two scans - one from the beginning and one from the end.

Next it's shown that the classical binary search (covered in lecture 3) is not cache efficient, but order statistics problem (covered in lecture 6) is cache efficient.

Finally the lecture describes a cache efficient way to multiply matrices by storing them block-wise in memory.

You're welcome to watch lecture twenty-two:

Topics covered in lecture twenty-two:

  • [00:10] Introduction and history of cache-oblivious algorithms.
  • [02:00] Modern memory hierarchy in computers: Caches L1, L2, L3, main memory, disk cache.
  • [06:15] Formula for calculating the cost to access a block of memory.
  • [08:18] Amortized cost to access one element in memory.
  • [11:00] Spatial and temporal locality of algorithms.
  • [13:45] Two-level memory model.
  • [16:30] Notation: total cache size M, block size B, number of blocks M/B.
  • [20:40] Notation: MT(N) - number of memory transfers of a problem of size N.
  • [21:45] Cache-aware algorithms.
  • [22:50] Cache-oblivious algorithms.
  • [28:35] Blocking of memory.
  • [32:45] Cache-oblivious scanning algorithm (visitor pattern).
  • [36:20] Cache-oblivious Array-Reverse algorithm.
  • [39:05] Memory transfers in classical binary search algorithm.
  • [43:45] Divide and conquer algorithms.
  • [45:50] Analysis of memory transfers in order statistics algorithm.
  • [01:00:50] Analysis of classical matrix multiplication (with row major, column major memory layout).
  • [01:07:30] Cache oblivious matrix multiplication.

Lecture twenty-two notes:

MIT Algorithms Lecture 22 Notes Thumbnail. Page 1 of 2.
Lecture 22, page 1 of 2: modern memory hierarchy, spacial and temporal locality, two-level memory model, blocking of memory, basic algorithms: parallel scan.

MIT Algorithms Lecture 22 Notes Thumbnail. Page 2 of 2.
Lecture 22, page 2 of 2: basic algorithms: binary search, divide and conquer algorithms, order statistics, matrix multiplication, block algorithms.

Lecture 23: Cache Oblivious Algorithms II

This was probably the most complicated lecture in the whole course. The whole lecture is devoted to two subjects - cache-oblivious search trees and cache-oblivious sorting.

While it's relatively easy to understand the design of cache-oblivious way of storing search trees in memory, it's amazingly difficult to understand the cache-efficient sorting. It's called funnel sort which is basically an n-way merge sort (covered in lecture 1) with special cache-oblivious merging function called k-funnel.

You're welcome to watch lecture twenty-three:

Topics covered in lecture twenty-three:

  • [01:00] Cache-oblivious static search trees (binary search trees).
  • [09:35] Analysis of static search trees.
  • [18:15] Cache-aware sorting.
  • [19:00] Sorting by repeated insertion in binary tree.
  • [21:40] Sorting by binary merge sort.
  • [31:20] Sorting by N-way mergesort.
  • [36:20] Sorting bound for cache-oblivious sorting algorithms.
  • [38:30] Cache-oblivious sorting.
  • [41:40] Definition of K-Funnel (cache-oblivious merging).
  • [43:35] Funnel sort.
  • [54:05] Construction of K-Funnel.
  • [01:03:10] How to fill buffer in k-funnel.
  • [01:07:30] Analysis of fill buffer.

Lecture twenty-three notes:

MIT Algorithms Lecture 23 Notes Thumbnail. Page 1 of 2.
Lecture 23, page 1 of 2: static search trees, cache aware sorting.

MIT Algorithms Lecture 23 Notes Thumbnail. Page 2 of 2.
Lecture 23, page 2 of 2: cache oblivious sorting, k-funnels, funnel sort, fill-buffer algorithm and analysis.

Have fun with the cache oblivious algorithms! I'll do a few more posts that will summarize all these lectures and highlight key ideas.

If you loved this, please subscribe to my blog!

Fibonacci of PisaIn this article I'd like to show how the theory does not always match the practice. I am sure you all know the linear time algorithm for finding Fibonacci numbers. The analysis says that the running time of this algorithm is O(n). But is it still O(n) if we actually run it? If not, what is wrong?

Let's start with the simplest linear time implementation of the Fibonacci number generating algorithm in Python:

def LinearFibonacci(n):
  fn = f1 = f2 = 1
  for x in xrange(2, n):
    fn = f1 + f2
    f2, f1 = f1, fn
  return fn

The theory says that this algorithm should run in O(n) - given the n-th Fibonacci number to find, the algorithm does a single loop up to n.

Now let's verify if this algorithm is really linear in practice. If it's linear then the plot of n vs. running time of LinearFibonacci(n) should be a line. I plotted these values for n up to 200,000 and here is the plot that I got:

Quadratic performance of linear algorithm
Note: Each data point was averaged over 10 calculcations.

Oh no! This does not look linear at all! It looks quadratic! I fitted the data with a quadratic function and it fit nearly perfectly. Do you know why the seemingly linear algorithm went quadratic?

The answer is that the theoretical analysis assumed that all the operations in the algorithm executed in constant time. But this is not the case when we run the algorithm on a real machine! As the Fibonacci numbers get larger, each addition operation for calculating the next Fibonacci number "fn = f1 + f2 " runs in time proportional to the length of the previous Fibonacci number. It's because these huge numbers no longer fit in the basic units of computation in the CPU; so a big integer library is required. The addition of two numbers of length O(n) in a big integer library takes time of O(n).

I'll show you that the running time of the real-life linear Fibonacci algorithm really is O(n^2) by taking into account this hidden cost of bigint library.

So at each iteration i we have a hidden cost of O(number of digits of fi) = O(digits(fi)). Let's sum these hidden cost for the whole loop up to n:

Hidden bigint cost in linear fibonacci

Now let's find the number of digits in the n-th Fibonacci number. To do that let's use the well-known Binet's formula, which tells us that the n-th Fibonacci number fn can be expressed as:

Binet’s Fibonacci Formula

It is also well-known that the number of digits in a number is integer part of log10(number) + 1. Thus the number of digits in the n-th Fibonacci number is:

Digits in the n-th Fibonacci number

Thus if we now sum all the hidden costs for finding the n-th Fibonacci number we get:

Hidden integer library cost in linear fibonacci algorithm

There we have it. The running time of this "linear" algorithm is actually quadratic if we take into consideration that each addition operation runs proportionally to the length of addends.

Next time I'll show you that if the addition operation runs in constant time, then the algorithm is truly linear; and later I will do a similar analysis of the logarithmic time algorithm for finding Fibonnaci numbers that uses this awesome matrix identity:

Fibonacci Fibonnaci Matrix Identity

Don't forget to subscribe if you are interested! It's well worth every byte!

Bit HacksI decided to write an article about a thing that is second nature to embedded systems programmers - low level bit hacks. Bit hacks are ingenious little programming tricks that manipulate integers in a smart and efficient manner. Instead of performing some operation (such as counting the 1 bits in an integer) by looping over individual bits, these programming nuggets do the same with one or two carefully chosen bitwise operations.

To get things going I'll assume that you know what the two's complement binary representation of an integer is and also that you know all the the bitwise operations.

I'll use the following notation for bitwise operations in the article:

&   -  bitwise and
|   -  bitwise or
^   -  bitwise xor
~   -  bitwise not
<<  -  bitwise shift left
>>  -  bitwise shift right

The numbers in the article are 8 bit signed integers (though the operations work on arbitrary length signed integers) that are represented as two's complement and they are usually named 'x'. The result is usually 'y'. The individual bits of 'x' are named b7, b6, b5, b4, b3, b3, b2, b1 and b0. The bit b7 is the sign bit (the most significant bit), and b0 is the least significant.

I'll start with the most basic bit hacks and gradually progress to more difficult ones. I'll use examples to explain how each bithack works.

If you are intrigued by this topic I urge you to subscribe to my blog. I can share a secret that there will be the 2nd part of this article where I cover more advanced bit hacks, and I will also release a cheat sheet with all these bit tricks! It's well worth subscribing!

Here we go.

Bit Hack #1. Check if the integer is even or odd.

if ((x & 1) == 0) {
  x is even
else {
  x is odd

I am pretty sure everyone has seen this trick. The idea here is that an integer is odd if and only if the least significant bit b0 is 1. It follows from the binary representation of 'x', where bit b0 contributes to either 1 or 0. By AND-ing 'x' with 1 we eliminate all the other bits than b0. If the result after this operation is 0, then 'x' was even because bit b0 was 0. Otherwise 'x' was odd.

Let's look at some examples. Let's take integer 43, which is odd. In binary 43 is 00101011. Notice that the least significant bit b0 is 1 (in bold). Now let's AND it with 1:

&   00000001   (note: 1 is the same as 00000001)

See how AND-ing erased all the higher order bits b1-b7 but left bit b0 the same it was? The result is thus 1 which tells us that the integer was odd.

Now let's look at -43. Just as a reminder, a quick way to find negative of a given number in two's complement representation is to invert all bits and add one. So -43 is 11010101 in binary. Again notice that the last bit is 1, and the integer is odd. (Note that if we used one's complement it wouldn't be true!)

Now let's take a look at an even integer 98. In binary 98 is 1100010.

&   00000001

After AND-ing the result is 0. It means that the bit b0 of original integer 98 was 0. Thus the given integer is even.

Now the negative -98. It's 10011110. Again, bit b0 is 0, after AND-ing, the result is 0, meaning -98 is even, which indeed is true.

Bit Hack #2. Test if the n-th bit is set.

if (x & (1<<n)) {
  n-th bit is set
else {
  n-th bit is not set

In the previous bit hack we saw that (x & 1) tests if the first bit is set. This bit hack improves this result and tests if n-th bit is set. It does it by shifting that first 1-bit n positions to the left and then doing the same AND operation, which eliminates all bits but n-th.

Here is what happens if you shift 1 several positions to the left:

1         00000001    (same as 1<<0)
1<<1      00000010
1<<2      00000100
1<<3      00001000
1<<4      00010000
1<<5      00100000
1<<6      01000000
1<<7      10000000

Now if we AND 'x' with 1 shifted n positions to the left we effectively eliminate all the bits but n-th bit in 'x'. If the result after AND-ing is 0, then that bit must have been 0, otherwise that bit was set.

Let's look at some examples.

Does 122 have 3rd bit set? The operation we do to find it out is:

122 & (1<<3)

Now, 122 is 01111010 in binary. And (1<<3) is 00001000.

&   00001000

We see that the result is not 0, so yes, 122 has the 3rd bit set.

Note: In my article bit numeration starts with 0. So it's 0th bit, 1st bit, ..., 7th bit.

What about -33? Does it have the 5th bit set?

    11011111      (-33 in binary)
&   00100000     (1<<5)

Result is 0, so the 5th bit is not set.

Bit Hack #3. Set the n-th bit.

y = x | (1<<n)

This bit hack combines the same (1<<n) trick of setting n-th bit by shifting with OR operation. The result of OR-ing a variable with a value that has n-th bit set is turning that n-th bit on. It's because OR-ing any value with 0 leaves the value the same; but OR-ing it with 1 changes it to 1 (if it wasn't already). Let's see how that works in action:

Suppose we have value 120, and we wish to turn on the 2nd bit.

    01111000    (120 in binary)
|   00000100    (1<<2)

What about -120 and 6th bit?

    10001000   (-120 in binary)
|   01000000   (1<<6)

Bit Hack #4. Unset the n-th bit.

y = x & ~(1<<n)

The important part of this bithack is the ~(1<<n) trick. It turns on all the bits except n-th.

Here is how it looks:

~1        11111110  (same as ~(1<<0))
~(1<<1)   11111101
~(1<<2)   11111011
~(1<<3)   11110111
~(1<<4)   11101111
~(1<<5)   11011111
~(1<<6)   10111111
~(1<<7)   01111111

The effect of AND-ing variable 'x' with this quantity is eliminating n-th bit. It does not matter if the n-th bit was 0 or 1, AND-ing it with 0 sets it to 0.

Here is an example. Let's unset 4th bit in 127:

    01111111    (127 in binary)
&   11101111    (~(1<<4))

Bit Hack #5. Toggle the n-th bit.

y = x ^ (1<<n)

This bit hack also uses the wonderful "set n-th bit shift hack" but this time it XOR's it with the variable 'x'. The result of XOR-ing something with something else is that if both bits are the same, the result is 0, otherwise it's 1. How does it toggle n-th bit? Well, if n-th bit was 1, then XOR-ing it with 1 changes it to 0; conversely, if it was 0, then XOR-ing with with 1 changes it to 1. See, the bit got flipped.

Here is an example. Suppose you want to toggle 5th bit in value 01110101:

^   00100000

What about the same value but 5th bit originally 0?

^   00100000

Notice something? XOR-ing the same bit twice returned it to the same value. This nifty XOR property is used in calculating parity in RAID arrays and used in simple cryptography cyphers, but more about that in some other article.

Bit Hack #6. Turn off the rightmost 1-bit.

y = x & (x-1)

Now it finally gets more interesting!!! Bit hacks #1 - #5 were kind of boring to be honest.

This bit hack turns off the rightmost one-bit. For example, given an integer 00101010 (the rightmost 1-bit in bold) it turns it into 00101000. Or given 00010000 it turns it into 0, as there is just a single 1-bit.

Here are more examples:

    01010111    (x)
&   01010110    (x-1)

    01011000    (x)
&   01010111    (x-1)

    10000000    (x = -128)
&   01111111    (x-1 = 127 (with overflow))

    11111111    (x = all bits 1)
&   11111110    (x-1)

    00000000    (x = no rightmost 1-bits)
&   11111111    (x-1)

Why does it work?

If you look at the examples and think for a while, you'll realize that there are two possible scenarios:

1. The value has the rightmost 1 bit. In this case subtracting one from it sets all the lower bits to one and changes that rightmost bit to 0 (so that if you add one now, you get the original value back). This step has masked out the rightmost 1-bit and now AND-ing it with the original value zeroes that rightmost 1-bit out.

2. The value has no rightmost 1 bit (all 0). In this case subtracting one underflows the value (as it's signed) and sets all bits to 1. AND-ing all zeroes with all ones produces 0.

Bit Hack #7. Isolate the rightmost 1-bit.

y = x & (-x)

This bit hack finds the rightmost 1-bit and sets all the other bits to 0. The end result has only that one rightmost 1-bit set. For example, 01010100 (rightmost bit in bold) gets turned into 00000100.

Here are some more examples:

    10111100  (x)
&   01000100  (-x)

    01110000  (x)
&   10010000  (-x)

    00000001  (x)
&   11111111  (-x)

    10000000  (x = -128)
&   10000000  (-x = -128)

    11111111  (x = all bits one)
&   00000001  (-x)

    00000000  (x = all bits 0, no rightmost 1-bit)
&   00000000  (-x)

This bit hack works because of two's complement. In two's complement system -x is the same as ~x+1. Now let's examine the two possible cases:

1. There is a rightmost 1-bit bi. In this case let's pivot on this bit and divide all other bits into two flanks - bits to the right and bits to the left. Remember that all the bits to the right bi-1, bi-2 ... b0 are 0's (because bi was the rightmost 1-bit). And bits to the left are the way they are. Let's call them bi+1, ..., bn.

Now, when we calculate -x, we first do ~x which turns bit bi into 0, bits bi-1 ... b0 into 1s, and inverts bits bi+1, ..., bn, and then we add 1 to this result.

Since bits bi-1 ... b0 are all 1's, adding one makes them carry this one all the way to bit bi, which is the first zero bit.

If we put it all together, the result of calculating -x is that bits bi+1, ..., bn get inverted, bit bi stays the same, and bits bi-1, ..., b0 are all 0's.

Now, AND-ing x with -x makes bits bi+1, ..., bn all 0, leaves bit bi as is, and sets bits bi-1, ..., b0 to 0. Only one bit is left, it's the bit bi - the rightmost 1-bit.

2. There is no rightmost 1-bit. The value is 0. The negative of 0 in two's complement is also 0. 0&0 = 0. No bits get turned on.

We have proved rigorously that this bithack is correct.

Bit Hack #8. Right propagate the rightmost 1-bit.

y = x | (x-1)

This is best understood by an example. Given a value 01010000 it turns it into 01011111. All the 0-bits right to the rightmost 1-bit got turned into ones.

This is not a clean hack, tho, as it produces all 1's if x = 0.

Let's look at more examples:

    10111100  (x)
|   10111011  (x-1)

    01110111  (x)
|   01110110  (x-1)

    00000001  (x)
|   00000000  (x-1)

    10000000  (x = -128)
|   01111111  (x-1 = 127)

    11111111  (x = -1)
|   11111110  (x-1 = -2)

    00000000  (x)
|   11111111  (x-1)

Let's prove it, though not as rigorously as in the previous bithack (as it's too time consuming and this is not a scientific publication). There are two cases again. Let's start with easiest first.

1. There is no rightmost 1-bit. In that case x = 0 and x-1 is -1. -1 in two's complement is 11111111. OR-ing 0 with 11111111 produces the same 11111111. (Not the desired result, but that's the way it is.)

2. There is the rightmost 1-bit bi. Let's divide all the bits in two groups again (like in the previous example). Calculating x-1 modifies only bits to the right, turning bi into 0, and all the lower bits to 1's. Now OR-ing x with x-1 leaves all the higher bits (to the left) the same, leaves bit bi as it was 1, and since lower bits are all low 1's it also turns them on. The result is that the rightmost 1-bit got propagated to lower order bits.

Bit Hack #9. Isolate the rightmost 0-bit.

y = ~x & (x+1)

This bithack does the opposite of #7. It finds the rightmost 0-bit, turns off all bits, and sets this bit to 1 in the result. For example, it finds the zero in bold in this number 10101011, producing 00000100.

More examples:

    10111100  (x)
    01000011  (~x)
&   10111101  (x+1)

    01110111  (x)
    10001000  (~x)
&   01111000  (x+1)

    00000001  (x)
    11111110  (~x)
&   00000010  (x+1)

    10000000  (x = -128)
    01111111  (~x)
&   10000001  (x+1)

    11111111  (x = no rightmost 0-bit)
    00000000  (~x)
&   00000000  (x+1)

    00000000  (x)
    11111111  (~x)
&   00000001  (x+1)

Proof: Suppose there is a rightmost 0-bit. Then ~x turns this rightmost 0 bit into 1 bit. And so does x+1 (because bits more right to the rightmost 0 bit are 1's). Now AND-ing ~x with x+1 evaporates all the bits up to this rightmost 0 bit. This is the highest order bit set in the result. Now what about lower order bits to the right of rightmost 0 bit? They also got evaporated because because x+1 turned them into 0's (they were 1's) and ~x turned them into 0's. They got AND-ed with 0 and evaporated.

Bit Hack #10. Turn on the rightmost 0-bit.

y = x | (x+1)

This hack changes the rightmost 0-bit into 1. For example, given an integer 10100011 it turns it into 10100111.

More examples:

    10111100  (x)
|   10111101  (x+1)

    01110111  (x)
|   01111000  (x+1)

    00000001  (x)
|   00000010  (x+1)

    10000000  (x = -128)
|   10000001  (x+1)

    11111111  (x = no rightmost 0-bit)
|   00000000  (x+1)

    00000000  (x)
|   00000001  (x+1)

Here is the proof as a bunch of true statements. OR-ing x with x+1 does not lose any information. Adding 1 to x fills the first rightmost 0. The result is max{x, x+1}. If x+1 overflows it's x and there were no 0 bits. If it doesn't, it's x+1 which just got rightmost bit filled with 1.

Bonus stuff.

If you decide to play more with these hacks, here are a few utility functions to print binary values of 8 bit signed integers in Perl, Python and C.

Print binary representation in Perl:

sub int_to_bin {
  my $num = shift;
  print unpack "B8", pack "c", $num;

Or you can print it from command line right away:

perl -wle 'print unpack "B8", pack "c", shift' <integer>

# For example:
perl -wle 'print unpack "B8", pack "c", shift' 113

perl -wle 'print unpack "B8", pack "c", shift' -- -128

Print binary number in Python:

def int_to_bin(num, bits=8):
 r = ''
 while bits:
  r = ('1' if num&1 else '0') + r
  bits = bits - 1
  num = num >> 1
 print r

Print binary representation in C:

void int_to_bin(int num) {
  char str[9] = {0};
  int i;
  for (i=7; i>=0; i--) {
    str[i] = (num&1)?'1':'0';
    num >>= 1;
  printf("%s\n", str);

Have fun with these! I'll write about advanced bit hacks some time soon. If you are really intrigued by this topic I encourage you to subscribe to my blog. Thanks! :)

Ps. Let me know in the comments what you think about this article, and let me know if you do not know what two's complement, or the basic binary operations are. If there are a few people who would like me to explain these concepts, I'll be glad to write another article just about these fundamental topics.

Pps. There is a book entirely on bit hacks like these. It's called "Hacker's Delight". It may be worth getting if you are into this stuff: