The Lumber Room

"Consign them to dust and damp by way of preserving them"

Posts Tagged ‘programming

Visualizing product of permutations

with 4 comments

A simple pedagogical trick that may come in handy: represent a permutation \sigma using arrows (curved lines) from k to \sigma(k) for each k. Then, the product of two permutations can be represented by just putting the two corresponding figures (sets of arrows) one below the other, and following the arrows.

Representing permutations and products of permutations.

Representing permutations and products of permutations.

The figure is from an article called Symmetries by Alain Connes, found via the Wikipedia article on Morley’s trisector theorem (something entirely unrelated to permutations, but the article covers both of them and more).

I’m thinking how one might write a program to actually draw these: if we decide that the “height” of the figure is some h, then each arrow needs to go from some (k, 0) to (\sigma(k), h) (using here the usual screen convention of x coordinate increasing from left to right, and y coordinate increasing from top to bottom). Further, each curve needs to have vertical slope at its two endpoints, so that successive curves can line up smoothly. The constraint on starting point, ending point, and directions at the endpoints defines almost a quadratic Bezier curve, except that here the two directions are parallel. So it’s somewhere between a quadratic and the (usual) cubic Bezier curve, which is given by the start point, end point, and derivatives at the start and end point. (Here we only care about the direction of the derivative; we can pick some arbitrary magnitude to fix the curve: the larger we pick, the more smooth it will look at the ends, at the cost of smoothness in the interior.)

Even knowing the curve, how do we generate an image?

Written by S

Thu, 2014-03-06 at 23:15:44 +05:30

The potions puzzle by Snape in Harry Potter and the Philosopher’s Stone

with one comment

Sometime this week, I reread the first Harry Potter book (after at least 10 years… wow, has it been that long?), just for contrast after reading Rowling’s adult novel The Casual Vacancy (on which more later). Anyway, in the penultimate chapter there is a puzzle:

He pulled open the next door, both of them hardly daring to look at what came next — but there was nothing very frightening in here, just a table with seven differently shaped bottles standing on it in a line.
“Snape’s,” said Harry. “What do we have to do?”
They stepped over the threshold, and immediately a fire sprang up behind them in the doorway. It wasn’t ordinary fire either; it was purple. At the same instant, black flames shot up in the doorway leading onward. They were trapped.
“Look!” Hermione seized a roll of paper lying next to the bottles. Harry looked over her shoulder to read it:

Danger lies before you, while safety lies behind,
Two of us will help you, whichever you would find,
One among us seven will let you move ahead,
Another will transport the drinker back instead,
Two among our number hold only nettle wine,
Three of us are killers, waiting hidden in line.
Choose, unless you wish to stay here forevermore,
To help you in your choice, we give you these clues four:
First, however slyly the poison tries to hide
You will always find some on nettle wine’s left side;
Second, different are those who stand at either end,
But if you would move onward, neither is your friend;
Third, as you see clearly, all are different size,
Neither dwarf nor giant holds death in their insides;
Fourth, the second left and the second on the right
Are twins once you taste them, though different at first sight.

I became curious about whether this is just a ditty Rowling made up, or the puzzle actually makes sense and is consistent. It turns out she has constructed it well. Let’s take a look. This investigation can be carried out by hand, but we’ll be lazy and use a computer, specifically Python. The code examples below are all to be typed in an interactive Python shell, the one that you get by typing “python” in a terminal.

So what we have are seven bottles, of which one will take you forward (F), one will let you go back (B), two are just nettle wine (N), and three are poison (P).

>>> bottles = ['F', 'B', 'N', 'N', 'P', 'P', 'P']

The actual ordering of these 7 bottles is some ordering (permutation) of them. As 7 is a very small number, we can afford to be inefficient and resort to brute-force enumeration.

>>> import itertools
>>> perms = [''.join(s) for s in set(itertools.permutations(bottles))]
>>> len(perms)
420

The set is needed to remove duplicates, because otherwise itertools.permutations will print 7! “permutations”. So already the number of all possible orderings is rather small (it is \frac{7!}{2!3!} = 420). We can look at a sample to check whether things look fine.

>>> perms[:10]
['PNFNPBP', 'NPPBNFP', 'FNNPBPP', 'PPPFNBN', 'NPPNBFP', 'PFNNBPP', 'NPBPPFN', 'NBNPPFP', 'NPPFBNP', 'BNPFNPP']

Now let us try to solve the puzzle. We can start with the first clue, which says that wherever a nettle-wine bottle occurs, on its left is always a poison bottle (and in particular therefore, a nettle-wine bottle cannot be in the leftmost position). So we must restrict the orderings to just those that satisfy this condition.

>>> def clue1(s): return all(i > 0 and s[i-1] == 'P' for i in range(len(s)) if s[i]=='N')
...
>>> len([s for s in perms if clue1(s)])
60

(In the code, the 7 positions are 0 to 6, as array indices in code generally start at 0.)
Then the second clue says that the bottles at the end are different, and neither contains the potion that lets you go forward.

>>> def clue2(s): return s[0] != s[6] and s[0] != 'F' and s[6] != 'F'
...
>>> len([s for s in perms if clue1(s) and clue2(s)])
30

The third clue says that the smallest and largest bottles don’t contain poison, and this would be of help to Harry and Hermione who can see the sizes of the bottles. But as we readers are not told the sizes of the bottles, this doesn’t seem of any help to us; let us return to this later.

The fourth clue says that the second-left and second-right bottles have the same contents.

>>> def clue4(s): return s[1] == s[5]
...
>>> len([s for s in perms if clue1(s) and clue2(s) and clue4(s)])
8

There are now just 8 possibilities, finally small enough to print them all.

>>> [s for s in perms if clue1(s) and clue2(s) and clue4(s)]
['PPNBFPN', 'BPNPFPN', 'BPFPNPN', 'BPPNFPN', 'PNPFPNB', 'BPNFPPN', 'PPNFBPN', 'PNFPPNB']

Alas, without knowing which the “dwarf” and “giant” bottles are, we cannot use the third clue, and this seems as far as we can go. We seem to have exhausted all the information available…

Almost. It is reasonable to assume that the puzzle is meant to have a solution. So even without knowing where exactly the “dwarf” and “giant” bottles are, we can say that they are in some pair of locations that ensure a unique solution.

>>> def clue3(d, g, s): return s[d]!='P' and s[g]!='P'
...
>>> for d in range(7):
...   for g in range(7):
...     if d == g: continue
...     poss = [s for s in perms if clue1(s) and clue2(s) and clue4(s) and clue3(d,g,s)]
...     if len(poss) == 1:
...       print d, g, poss[0]
... 
1 2 PNFPPNB
1 3 PNPFPNB
2 1 PNFPPNB
2 5 PNFPPNB
3 1 PNPFPNB
3 5 PNPFPNB
5 2 PNFPPNB
5 3 PNPFPNB

Aha! If you look at the possible orderings closely, you will see that we are down to just two possibilities for the ordering of the bottles.
Actually there is some scope for quibbling in what we did above: perhaps we cannot say that there is a unique solution determining the entire configuration; perhaps all we can say is that the puzzle should let us uniquely determine the positions of just the two useful bottles. Fortunately, that gives exactly the same set of possibilities, so this distinction happens to be inconsequential.

>>> for d in range(7):
...   for g in range(7):
...     if d == g: continue
...     poss = [(s.index('F'),s.index('B')) for s in perms if clue1(s) and clue2(s) and clue4(s) and clue3(d,g,s)]
...     if len(set(poss)) == 1:
...       print d, g, [s for s in perms if clue1(s) and clue2(s) and clue4(s) and clue3(d,g,s)][0]
... 
1 2 PNFPPNB
1 3 PNPFPNB
2 1 PNFPPNB
2 5 PNFPPNB
3 1 PNPFPNB
3 5 PNPFPNB
5 2 PNFPPNB
5 3 PNPFPNB

Good. Note that there are only two configurations above. So with only the clues in the poem, and the assumption that the puzzle can be solved, we can narrow down the possibilities to two configurations, and be sure of the contents of all the bottles except the third and fourth. We know that the potion that lets us go forward is in either the third or the fourth bottle.

In particular we see that the last bottle lets us go back, and indeed this is confirmed by the story later:

“Which one will get you back through the purple flames?”
Hermione pointed at a rounded bottle at the right end of the line.
[…]
She took a long drink from the round bottle at the end, and shuddered.

But we don’t know which of the two it is, as we can’t reconstruct all the relevant details of the configuration. Perhaps we can reconstruct something with the remaining piece of information from the story?

“Got it,” she said. “The smallest bottle will get us through the black fire — toward the Stone.”
Harry looked at the tiny bottle.
[…]
Harry took a deep breath and picked up the smallest bottle.

So we know that the bottle that lets one move forward is in fact in the smallest one, the “dwarf”.

>>> for d in range(7):
...   for g in range(7):
...     poss = [s for s in perms if clue1(s) and clue2(s) and clue4(s) and clue3(d,g,s)]
...     if len(poss) == 1 and poss[0][d] == 'F':
...       print d, g, poss[0]
... 
2 1 PNFPPNB
2 5 PNFPPNB
3 1 PNPFPNB
3 5 PNPFPNB

This narrows the possible positions of the smallest and largest bottles (note that it says the largest bottle is one that contains nettle wine), but still leaves the same two possibilities for the complete configuration. So we can stop here.

Conclusion
What we can conclude is the following: apart from the clues mentioned in the poem, the “dwarf” (the smallest bottle) was in either position 2 (third from the left) or 3 (fourth from the left). The biggest bottle was in either position 1 (second from the left) or 5 (sixth from the left). With this information about the location of the smallest bottle (and without necessarily assuming the puzzle has a unique solution!), Hermione could determine the contents of all the bottles. In particular she could determine the location of the two useful bottles: namely that the bottle that lets you go back was the last one, and that the one that lets you go forward was the smallest bottle.

>>> for (d,g) in [(2,1), (2,5), (3,1), (3,5)]:
...   poss = [s for s in perms if clue1(s) and clue2(s) and clue4(s) and clue3(d, g, s)]
...   assert len(poss) == 1
...   s = poss[0]
...   assert s.index('B') == 6
...   assert s.index('F') == d
...   print (d,g), s
... 
(2, 1) PNFPPNB
(2, 5) PNFPPNB
(3, 1) PNPFPNB
(3, 5) PNPFPNB

It is not clear why she went to the effort to create a meaningful puzzle, then withheld details that would let the reader solve it fully. Perhaps some details were removed during editing. As far as making up stuff for the sake of a story goes, though, this is nothing; consider for instance the language created for Avatar which viewers have learned.

Added:
See also http://www.zhasea.com/logic/snape.html which does it by hand, and has a perfectly clear exposition (it doesn’t try the trick of guessing that solution is unique before reaching for the additional information from the story).

Written by S

Mon, 2012-10-22 at 02:00:12 +05:30

How does Tupper’s self-referential formula work?

with 16 comments

[I write this post with a certain degree of embarrassment, because in the end it turns out (1) to be more simple than I anticipated, and (2) already done before, as I could have found if I had internet access when I did this. :-)]

The so-called “Tupper’s self-referential formula” is the following, due to Jeff Tupper.

Graph the set of all points {(x,y)} such that

\displaystyle  \frac12 < \left\lfloor \mathrm{mod} \left( \left\lfloor{\frac{y}{17}}\right\rfloor 2^{-17\lfloor x \rfloor - \mathrm{mod}(\lfloor y \rfloor, 17)}, 2 \right) \right\rfloor

in the region

\displaystyle  0 < x < 106

\displaystyle  N < y < N+17

where N is the following 544-digit integer:
48584506361897134235820959624942020445814005879832445494830930850619
34704708809928450644769865524364849997247024915119110411605739177407
85691975432657185544205721044573588368182982375413963433822519945219
16512843483329051311931999535024137587652392648746133949068701305622
95813219481113685339535565290850023875092856892694555974281546386510
73004910672305893358605254409666435126534936364395712556569593681518
43348576052669401612512669514215505395545191537854575257565907405401
57929001765967965480064427829131488548259914721248506352686630476300

The result is the following graph:

Figure 1: The graph of the formula, in some obscure region, is a picture of the formula itself.

Whoa. How does this work?

At first sight this is rather too incredible for words.

But after a few moments we can begin to guess what is going on, and see that—while clever—this is perhaps not so extraordinary after all. So let us calmly try to reverse-engineer this feat.

Read the rest of this entry »

Written by S

Tue, 2011-04-12 at 13:05:20 +05:30

Posted in mathematics

Tagged with , , ,

Firebug “console is undefined”

with 2 comments

If you’re using Firebug and don’t want to bother removing or commenting out all the console.log debug messages for users who aren’t, put this at the top:

if(typeof(console) === "undefined" || typeof(console.log) === "undefined") 
    var console = { log: function() { } };

This reminds me of the trick I use for C/C++ of putting debugging statements inside a macro:

D(cerr<<"The number is "<<n<<endl;);

where the macro D is defined as

#define D(A) A

when you want the debugging code to run, and

#define D(A) 

when you don’t. :-)

(The final semicolon after the D() is for Emacs to indent it reasonably.)

Update: Of course, the above are just hacky hacks to save typing. The “right” way for conditional C code is usually to use #ifdef DEBUG and the like, and the right way around the Firebug problem is to use something more general like this code at metamalcolm.

Written by S

Wed, 2009-08-05 at 14:32:15 +05:30

Reverse-engineering Gmail: Initial remarks

with 11 comments

For the last week and a bit, I have been trying to do a particular something with Gmail. (Specifically, get at the Message-ID headers of messages.) This has been mostly a failure, but that’s not so surprising, as I had little experience with “all this web stuff”: JavaScript, AJAX, DOM, browser incompatibilities, Firebug, Greasemonkey… round up the usual buzzwords. I have learnt a bit, though, and thought it might help others starting in a similar situation. (And there’s also the hope that someone might actually find this and help me!)

The story so far
Gmail was launched in April 2004. Since then, it has been through many changes, the latest around October 2007 when there came to our inboxes a “Newer version”, also sometimes called “Gmail 2″. (Note that officially Gmail is still in Beta; it hasn’t even released a 1.0!)
When Gmail was released the set of practices that go by the name of “AJAX” was still new and unfamiliar; it has been refined and better-understood since. (And it turns out to require neither asynchrony nor JavaScript nor XML.)

Johnvey Hwang reverse-engineered much of Gmail’s original version, and even made a “Gmail API” out of it. It no longer works of course, and the site is often down too, but it’s available on the Wayback Machine and the section documenting “the Gmail engine and protocol” is still worth a read, if only for its glimpse into the labyrinthine ways in which Ajax applications can work. He turned it (in May 2005) into a SourceForge project (“Gmail API”), last updated June 2005, and the associated Google Group (” Gmail Agent API”) is also largely defunct and indicates that the API, or whatever came of it, has not been working since the changes in October 2007, at any rate.

My goal
At this point, I might as well reveal what I want to do: I want to make it easy to get the “Message-ID:” header of messages in Gmail. (I like to read email in Gmail but not to send, so one way to reply to a specific message would be to get the Message-ID and ask my other mail client to reply to the message with that message-ID.) In the current interface, the (only) way of getting it is to click on the pulldown menu next to “Reply”, and click on “Show original”. This will open up a page that contains the raw text of the message with all its headers, and “Message-ID:” is always one of them. Since I use Firefox, I’ve been trying to make this easier with a Greasemonkey script.

Trap-patching the P() function
As Greasemonkey scripts for Gmail go, much useful information comes from Mihai Parparita, who wrote many Greasemonkey scripts for Gmail. Quoting from here:

As others have documented, Gmail receives data from the server in form of JavaScript snippets. Looking at the top of any conversation list’s source, we can see that the D() function that receives data in turns calls a function P() in the frame where all the JavaScript resides. Since all data must pass through this global P() function, we can use Greasemonkey to hook into it. This is similar to the trap patching way of extending Classic Mac OS. Specifically, the Greasemonkey script gets a hold of the current P() function and replaces it with a version that first records relevant data in an internal array, and then calls the original function (so that Gmail operations are not affected).

Clever. This same information is also documented at Greasespot wiki, with a few remarks on what different parameters to P() mean. Alas, it no longer works, because Gmail changed their functions around and renamed all of them, so there is no P() function anymore, and I can’t find what the new equivalent is, or if there is one.

Changes of October 2007
Gmail made certain changes in October 2007, including introducing a “newer version”, but also changing the “older version” that is still available: so it’s not really the older version. As far as Greasemonkey scripts go, another change was in January 2008, where they made all the Javascript load in a separate iframe. So “unsafeWindow” in a Greasemonkey script now refers to this iframe (which is the first frame, frame[0], in the window, and can also be got as top.js). So any scripts written in September 2007 or earlier are certainly useless now.

A lesson from all this is that Gmail will always be a moving target, and one must consider whether it’s worth chasing it.

Gmail’s Greasemonkey “API”:
Sometime in November 2007 or so, after the latest changes, Google even released a basic Greasemonkey API for Gmail, which lets you do a few things, like adding things to the pane at the left. It is too limited for what I need, but it works very well for what is meant for, and is also very well-documented, by Mark Pilgrim with his usual “Dive Into” excellence. It is comprehensive, accurate, well-illustrated and to-the-point, and great as documentation goes; it just happens that the API doesn’t provide what I need.

Some observations
Back to what I’m trying to do. Currently, the actions in the menu next to “Reply”, namely “Reply to all”, “Forward”, “Filter messages like this”, … “Show original” etc., do not actually appear in the DOM multiple times once attached to each message. Instead each of these actions corresponds to exactly one node (each) in the DOM, like these:

<div act="27" style="padding-left: 19px;" class="SAQJzb" id=":t6">Filter messages like this</div>
<div id=":t8" class="R10Zdd" act="29" style="padding-left: 19px;">Add to Contacts list</div>
<div id=":tc" class="SAQJzb" act="32" style="padding-left: 19px;">Show original</div>

etc. The IDs change, and the class name also seems to randomly change between “SAQJzb” and “R10Zdd”; the only constant between the action and the node is the “act” attribute. “Show original” is always act=32. So when you click on the down-arrow button next to Reply, this menu comes up, and when you click on something in the menu, it somehow uses the information about where this menu came up and what you clicked, to find out which message to act on.

This means that simply simulating a click on the node (initMouseEvent, etc…) does not work; we also have to somehow give it the information on what message to act on. How to do this is one thing I’m trying to find out.

The other way involves the fact that Gmail also has its own “ID” for each message. When you are looking at a thread (“conversation”) that contains a single message, it is the same as what is in the URL, e.g. if the URL is something like https://mail.google.com/mail/#inbox/11c177beaf88ffe6, Gmail’s ID of the message is 11c177beaf88ffe6. But when you’re looking at a thread containing more than one message, the ID in the URL is just that of any of the messages in the thread (usually the first one, but you can use the ID of a different message in the URL and it will show the same thread). And when you click on the “Show original” link, the URL is something like https://mail.google.com/mail/?ui=2&ik=1234567890&view=om&th=11c177beaf88ffe6 where 1234567890 is a constant (probably depending on the user) and “om” probably stands for “original message”, and the “th” parameter is the ID of the message. So if I can somehow find a way of getting the ID of messages (like the trap-patching P() method, except that it should work for the current version), then it is possible to get the Message-ID headers of messages too.

Neither has worked out yet, but I’m trying…
(And I have more to say, but will post when things actually work.)

Written by S

Sun, 2008-08-31 at 18:45:57 +05:30

Where Have I Seen…

with 9 comments

I have, for a long time, dreamed of a Firefox extension that would do this. Now, finally, empowered by Greasemonkey, I wrote one myself. (Technically, a Greasemonkey user script is not a Firefox extension, but it can be easily converted into one, and a Greasemonkey script is the Right Thing anyway.)

What it does: on any IMDB cast listing, show for each actor what other movies you have seen him/her in. Here’s a screenshot:

Clearly there is a lot of work left to do, but it took less than half a day to learn Greasemonkey (and parts of JavaScript) and write it, and it’s already usable! By me, that is.
[The following was true at the time of writing but is no longer true.] I don’t know if anyone else would be interested in this, but it currently won’t work anyway except when running on my laptop. This is its crazy “design”: on any IMDB page with a cast listing, it first looks for each actor on the page, and extracts their ID. (Reasonable so far.) To find their other movies, it then makes a xmlhttp request to a PHP script running on my laptop, which then calls a Python script and returns its raw output inside ‘pre’ tags. Now you know. The reason for this nonsense is that there was no JavaScript API/library for IMDB while there was one for Python, so it was really easier to just use the latter, and the only way available of interacting with the “outside world” from a Greasemonkey script is through xmlhttp requests, and…
Anyway it’s not all that hard to parse each actor’s “other movies” through JavaScript myself, if that’s all I’m doing, so I might get to that eventually. (I also considered keeping IMDB’s data locally and parsing the text files, but they’re huge and not very well-formatted: No IDs, for example.)

It’s currently named “WHIS”, can you think of a better name? :)

Update: It’s now a full-fledged Greasemonkey script, and is up on userscripts.org.

Written by S

Sat, 2008-08-09 at 06:41:02 +05:30

Posted in Uncategorized

Tagged with , ,

JavaScript as a programming language

with 5 comments

JavaScript has a bad reputation. It does have a few warts, but at its core it’s an okay language, and much of the ire ought to be directed at its users instead: amateur programmers with poor taste. JavaScript can’t be blamed for making it easy for them to implement their ill-considered ideas, but it often is. It is also significant that JavaScript has a somewhat unique relationship with its “users”, in the following sense. With most programming languages, one doesn’t know one is using the language unless one is programming in it. But think about what it means to have JavaScript enabled on your browser: you are basically inviting every random website to run code on your browser, and while your browser happily gives up control over all its aspects to the random code, it gives you no control over what code runs! Is it any surprise that a lot of people simply hate JavaScript for what it makes possible? The only surprising thing is that it took so long for control to be returned to the user of the browser, in the form of Greasemonkey. “Client-side” is a misnomer when only the computation happens on the client side but everything is decided by the server.
Read the rest of this entry »

Written by S

Fri, 2008-08-08 at 21:59:11 +05:30

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