Archive for September 2010

SICP Section 2.3.4   Leave a comment

This section puts what we’ve learned about sets into practice implementing a method of encoding and decoding messages encoded with huffman code trees, and also a method of generating the trees.

;;Exercise 2.67
> (decode sample-message sample-tree)
(A D A B B C A)

;;Exercise 2.68
The encode-symbol procedure I wrote is straightforward, but takes a rather large number of steps to encode a symbol, because we have to search the set of symbols at each node for the correct branch to take. I’m not sure if there’s a better way to do this…

(define (encode-symbol symbol tree)
  (cond ((leaf? tree) '())
        ((element-of-set? symbol (symbols (left-branch tree)))
         (cons 0 (encode-symbol symbol (left-branch tree))))
        ((element-of-set? symbol (symbols (right-branch tree)))
         (cons 1 (encode-symbol symbol (right-branch tree))))
        (else (error "symbol not in tree - ENCODE-SYMBOL" symbol))))

;;Exercise 2.69
Successive-merge was actually fairly straightforward to code, however, when I first attempted to do this, I did not fully understand the actual tree generation procedure! Perhaps this is what the authors were warning about when they said it was tricky, I was certainly tricked initially :D

(define (successive-merge leaf-set)
  (if (= (length leaf-set) 1)
      (car leaf-set)
      (successive-merge
       (adjoin-set (make-code-tree (car leaf-set)
                                   (cadr leaf-set))
                   (cddr leaf-set)))))

;;Exercise 2.70
I never knew 50s rock was so repetitive ;)
Probably better than the junk that gets put out these days though…

(define freq-pairs (list (list 'A 2) (list 'BOOM 1) (list 'GET 2) (list 'JOB 2) (list 'NA 16) (list 'SHA 3) (list 'YIP 9) (list 'WAH 1)))

(define message '(GET A JOB
                  SHA NA NA NA NA NA NA NA NA
                  GET A JOB
                  SHA NA NA NA NA NA NA NA NA
                  WAH YIP YIP YIP YIP YIP YIP YIP YIP YIP
                  SHA BOOM))

(define rock50s-tree (generate-huffman-tree freq-pairs))

(define encoded-message (encode message rock50s-tree))
(define huffman-length (length encoded-message))

(define logbase2-of-8 3)
(define fixed-length (* (length message) logbase2-of-8))

fixed-length turns out to be 108 bits long, whereas the huffman-length is only 84 bits long. That’s a pretty fair length saving.

;;Exercise 2.71
I’m not really a fan of drawing large trees, so I’ll just give the code to generate it.

(define freq-pairs-5 (list (list '1 1) (list '2 2) (list '3 4) (list '4 8) (list '5 16)))
(define freq-pairs-tree-5 (generate-huffman-tree freq-pairs-5))

(define freq-pairs-10 (list (list '1 1) (list '2 2) (list '3 4) (list '4 8) (list '5 16) (list '6 32) (list '7 64) (list '8 128) (list '9 256) (list '10 512)))
(define freq-pairs-tree-10 (generate-huffman-tree freq-pairs-10))

The number of bits required for n=5 code are
Largest frequency: 1
Lowest frequency: 4
For n=10
Largest frequency: 1
Lowest frequency 9

It’s fairly obvious that the largest frequency will always have 1 bit, and the lowest frequency will have (n-1) bits.

;;Exercise 2.72
As we descend the tree we have to search the list of symbols at the node we are on, which is of the order O(n). We will need to descend n levels worst case, therefore n searches n deep is O(n2)

SICP Section 2.3.3   Leave a comment

The focus of this section is on representing mathematical sets with Scheme’s built-in data structures, and methods of combining these data structures to enable us to write faster algorithms for performing set operations. We go from a basic list based representation, to binary trees. I found this section to be quite a test of my abstract thinking skills; visualizing the operations on the representations was quite difficult for me.

;;Exercise 2.59
Implementing union-set is fairly straightforward for the unordered-list representation.

(define (union-set set1 set2)
  (cond ((null? set1) set2)
        ((null? set2) set1)
        ((element-of-set? (car set1) set2)
         (union-set (cdr set1) set2))
        (else (cons (car set1) (union-set (cdr set1) set2)))))

;;Exercise 2.60
This exercise was pretty interesting, it makes you think about situations in which what would seem to be an inefficient algorithm is actually the best for a job. Here’s my implementation of the set operations for an unordered set which allows duplicates.

(define (element-of-set? x set)
  (cond ((null? set) false)
        ((equal? x (car set)) true)
        (else (element-of-set? x (cdr set)))))

(define (adjoin-set x set)
  (cons x set))

(define (intersection-set set1 set2)
  (cond ((or (null? set1) (null? set2)) '())
        ((element-of-set? (car set1) set2)
         (cons (car set1)
               (intersection-set (cdr set1) set2)))
        (else (intersection-set (cdr set1) set2))))

(define (union-set set1 set2)
  (append set2 set1))

From the code we can see that, when duplicates are allowed,

  • element-of-set? is still O(n)
  • adjoin-set is O(1) from O(n)
  • intersection-set is still O(n^2)
  • union-set is O(n) from O(n^2)

By allowing duplicates, we have sped things up quite a bit, even if the algorithms are going to use up a lot more memory. In choosing which algorithm to use in a certain situation, we need to take into account the environment it will be used in. If we are, for example, on a machine with limited CPU speed, but a lot of memory (not uncommon in machines that have been upgraded), the duplicate representation would be best. On the other hand, if we were dealing with huge numbers, it would be a big drain on memory to use the duplicate representation, and the non-duplicate representation would be better for keeping overhead down.

;;Exercise 2.61
Now we’re starting to optimize the representations a bit more, it’s really neat to see how simple ideas like ordering the sets can produce fairly large performance gains :)

(define (adjoin-set x S)
  (cond ((null? S) (list x))
        ((< x (car S)) (cons x S))
        ((> x (car S)) (cons (car S) (adjoin-set x (cdr S))))
        ((= x (car S)) S)))

The ordered-list representation will take less time to adjoin, on the average, because we will not necessarily have to process the whole list in order to know if a duplicate exists.

;;Exercise 2.62
This particular exercise required a fair bit of thought. Here’s my explanation.
If the first element of set1 is less than the first element of set2, we make that element the car of a new list, and the cdr of the new list is the union of the cdr of set1 and set2…

(define (union-set set1 set2)
  (cond ((null? set1) set2)
        ((null? set2) set1)
        ((< (car set1) (car set2))
         (cons (car set1) (union-set (cdr set1) set2)))
        ((> (car set1) (car set2))
         (cons (car set2) (union-set set1 (cdr set2))))
        ((= (car set1) (car set2))
         (cons (car set1)
               (union-set (cdr set1)
                          (cdr set2))))))

;;Exercise 2.63
;;A
I think this exercise is designed to see how well we can transform a diagram into the actual representation of a tree as the most difficult part was writing down the trees :D

(define tree1 (make-tree 7
                         (make-tree 3
                                    (make-tree 1 '() '())
                                    (make-tree 5 '() '()))
                         (make-tree 9
                                    '()
                                    (make-tree 11 '() '()))))

(define tree2 (make-tree 3
                         (make-tree 1 '() '())
                         (make-tree 7
                                    (make-tree 5 '() '())
                                    (make-tree 9
                                               '()
                                               (make-tree 11 '() '())))))

(define tree3 (make-tree 5
                         (make-tree 3
                                    '()
                                    (make-tree 1 '() '()))
                         (make-tree 9
                                    (make-tree 7 '() '())
                                    (make-tree 11 '() '()))))

After running these through both functions, you’ll see that both the recursive and iterative procedures produce the same result, which is expected if they are to do the same job…

;B
Yes, they both have the same order of growth, however tree->list-2 uses less memory than tree->list-3, and thus could be faster in certain situations.

;;Exercise 2.64
Partial-tree works in a very recursive fashion :)
The best way I can explain is to work through an example.
Let’s say we have the list '(1 2 3 4 5) to turn into a tree.
Partial-tree first assigns equal, or almost equal sizes for the left and right branch of the new tree. So for our tree left-size will be 2, and right-size will be 3 (because the size of the list is not even). Then, partial tree conses together the result of partial-tree on the left and right halves of the given list. It’s quite a bit for me to wrap my head around, to be sure!

The list looks like so:

;;Exercise 2.65
Still a work in progress for me…

;;Exercise 2.66
The lookup procedure is basically element-of-set?, except instead of returning true or false, we return the item requested by key, or false if it’s not found.

(define (lookup key records)
  (cond ((null? records) false)
        ((= key (entry records)) (record-value (entry records)))
        ((< key (entry records))
         (lookup key (left-branch records)))
        ((> key (entry records))
         (lookup key (right-branch records)))))

;;Where a record structure looks like
(define (make-record key value)
  (cons key value))

Posted 23/09/2010 by Emmanuel Jacyna in Scheme, SICP

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Walk Through The Desert   Leave a comment

As Promised

I spent my entire day today working on this awesome song, which I’ve titled “Walk Through The Desert”. It’s a sort of rock balladish thing, and I’m quite pleased with what I managed to knock out in ten hours. Yesterday I randomly decided to listen to some “Ramble On Rose” by The Grateful Dead, and jammed along with it for a little while. I got an interesting chord progression out of that jam, and I was in a bit of an interesting mood, so I sung along to the riff and came up with some random lyrics for it. Today I decided to flesh it out a bit, so I got the guitar out, plugged along and came up with some lyrics that actually made sense. Because I’m not wonderful at keeping time, I made up a little drum beat using Hydrogen. To record my guitar playing, I used a rather old webcam microphone wrapped in a (clean) sock. This produced surprisingly good results!

My Mic Setup

I then spent a good few hours coming up with an interesting guitar accompaniment, which I am very pleased with. When I developed this accompaniment, I basically had an idea of what it would sound like in my head, but I didn’t bother to figure out what scale (if any ;) it was actually in. I find that this approach actually lets me experiment more, as I’m not constrained  by what I “should” be playing. While I was figuring out my guitar part, I made sure to record all of what I was doing, using chord charts, and tabulation. I used audacity for all my recording, and it took plenty of takes for me to actually hit it right, so it was good to have a good method of recording in pieces.

Audacity

It snowballed from there, and after spending a good few hours coming up with some guitar accompaniment, I can now present to the general public, “Walk Through The Desert”.

Walk Through The Desert

Before I forget, I also need to note the command that I used to turn it into a video

ffmpeg -ab 192k -i logo.jpg -i walk_through_the_desert.mp3 desert.avi

-ab is the bitrate I want the music to be, which I’ve set to 192k, a fairly typical rate
-i logo.jpg specifies the image to use as the background (I’m pretty sure it needs to be jpeg)
-i walk_through_the_desert.mp3 is the mp3 file to use as audio backing.

Also, for those who are interested, here’s my studio :)

My "Studio" :D

Posted 21/09/2010 by Emmanuel Jacyna in Music

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Information Dump   Leave a comment

Excuses

It’s been a while. I purchased an electric guitar about two months ago, and from the moment it arrived, I’ve been pretty much playing guitar in my free time. So yes, not much work has been done on SICP in that interval, but I’ve certainly learned a lot about guitar in that time.

Reasons

It seems a bit spontaneous, to me, to decide to pick up guitar, but I’ve had this ache since I got into listening to rock music (~age thirteen) to learn to play guitar. Of course, when I was thirteen I lacked two things. The perseverance to actually keep going with it no matter the boredom, pain, and whatnot, and also the money to actually buy the equipment I needed. That ache has been on the backburner for a while (read 3 years :), but recently, when I began to listen to some really great guitar it was reawakened. Hearing Joe Satriani smash out a solo, or just some meaty power chords from deep purple made me wish I could experience that. So when I decided to get a guitar a couple months ago, I had a clear goal in mind. I knew that I wouldn’t be JS in a month, but what I could do, was have fun! I decided for both cost purposes and the interest of fun to forgo getting professional lessons. I believe this is the right approach, at least for me, because I’m not burdened by the boredom of running through the same exercise fifty times ’til my fingers seize up; I can just grab a tab or two and practise.

The Equipment

It was also in the interest of “fun” that I decided to purchase a Squier Stratocaster electric. There’s just so much more you can do with an electric, and after hearing a lot of my friends recommend it as a good beginners guitar I was set. It was also cheap, only ~$150. I also needed an amp, and after a fair amount of research I decided to get an orange crush 10. Orange make beautiful amps, and my ~$200 solid state amp is testament to that. Of course, the best way to save money is to use eBay, so I bought my gear from there, saved about $50, including postage, of course.

My Amp

My Amp

My Guitar

My Guitar

So, yes, I’ve managed to learn quite a bit in the last two months, and I’ve had a heck of a lot of fun doing it! Guitar is a really great instrument to play; it’s very expressive. It’s quite easy to bend strings, add a bit of tremolo, and you can apply numerous effects to modify your sound. I personally have also got an MXR Carbon Copy delay pedal, for stretching out rhythm pieces, and giving a bit of sustain to reggae sounds.

I’ll leave you with that, my next post will feature a song I’ve composed, and the method I used to record it.

Happy Hacking!

Posted 21/09/2010 by Emmanuel Jacyna in Uncategorized

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