Matthew Hutchinson

What is ‘Computational Complexity’ ?

Basically a theory describing the scalability of algorithms. “As the size of the input to an algorithm increases, how do the running time and memory requirements of the algorithm change?” Wikipedia has a more complete description. Complexity can be considered in terms of;

• Time Complexity – the number of steps that it takes to solve an instance of the problem as a function of the size of the input, using the most efficient algorithm.
• Space Complexity – the amount of space, or memory required by the algorithm.

If you consider an array with n elements, and in solving a problem on this array, it takes n times n (or n²) steps. You could say the algorithm used had a complexity of n². Now with different programming languages there may be additional steps in the algorithm necessary to solve the problem. To describe a general level of complexity across any language, the Big Oh Notation is used. So the time-complexity for this algorithm would be Θ(n²)

‘Θ’ basically indicates that we are ignoring any language dependent factors, allowing complexity to be expressed purely in terms of the size of the input.

An Example

if you take a simple problem like, finding the ‘mode’ average on an array. I.e. the most frequently occurring element in a sequence (lets say of numbers) – You could do this an number of ways. Here is a brute force attempt (in Ruby).

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def calculate_mode(nums)
  hi_count = 0
  mode = nums[0]
  nums.each do | search_for |
    count = 0
    nums.each do |num|
      if search_for == num
        count += 1
      end
    end
    if count > hi_count
      hi_count = count
      mode = search_for
    end
  end
end

It should be obvious that this algorithm has a complexity of Θ(n²) – since every element of the array must be searched by each element to find the highest count value (and hence the mode).

Could it be done any faster? If the array of elements was sorted (ordered numerically) – the mode could be found by finding the longest continuing sequence in the array, that should only take n iterations. Employing a quick sort algorithm first; which can sort (on average) with Θ(n log n)

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def quicksort(a)
  return a if a.size <= 1
  pivot = a[0]
  quicksort(a.select {|i| i < pivot }) +
      a.select {|i| i == pivot } + 
      quicksort(a.select {|i| i > pivot })  
end

def calculate_mode_fast(nums, time_start)
  # sort array first
  sorted_nums = quicksort(nums)

  hi_count = 0
  mode = nums[0]  
  count = 1
  idx = 1

  sorted_nums.each do | search_for |
   if search_for == sorted_nums[idx]
     count += 1
     if count > hi_count
       hi_count = count
       mode = search_for
     end
   else
     count = 1
   end
   idx += 1 if idx < sorted_nums.length
  end
end

So we could say that this approach has (on average) a complexity of Θ(n+(n log n)) which is significantly less that Θ(n²)

I’ve posted this code listing which implements both algorithms running to compute the modal average on identical arrays of random integers. You can vary n (the number of elements) to see how each method performs. From the results (below) its clear that the 2nd algorithm is more time-efficient in this case.

calculate_mode
====================
mode => 3 (it occurs 44 times)
array length was 1000
num loops was 1000000
completed in 0.71737

calculate_mode_fast
=========================
mode => 3 (it occurs 44 times)
array length was 1000
num loops was 1040
completed in 0.011247

In recent on-line chatter over buying cars, the argument was put forward for more fun. Yes, Fun! with a capital ‘F’.

RWD, open-top and general ‘side-ways around corners’ was favoured over smooth driving, electric everything and ‘I feel like I’m doing 5mph when I’m really doing 80’.

Enter the (german made) Wiesmann GT, checking all the boxes. German made yes, but performs and looks every bit like a British sports car classic. With a BMW V8, BMW brakes and gearbox, Wiesmann work hard to make 500 cars every year. The lead time on ordering one is about 6 months. Another photo here, and a review at PistonHeads.com explains more.

And, here it is making some nice noises

Starting at £72,000, its a little pricey though. A killer-internet app and a quick sale to Google/Yahoo should sort that out; to the Bat mobile robin!

You might have noticed a minor style change in the most recent blog entries. I’ve decided (from now on) to use a light-box for showing off any embedded videos and (biggish) images on the page. It keeps the layout tidy and allows you to focus on the video you’re watching or image your viewing, without distraction. So I can do things like this for images;

And even stuff like this, for videos

All of this is made possible using Control.Modal, a light weight, unobtrusive JavaScript library for creating modal windows and lightboxes using content that is already on the page. So with javascript turned off, everything still works, with the links navigating to anchor tags in the page.

At the moment I am applying display:none; on the modal content to avoid a visible onLoad jump effect as the content gets hidden by Control.Modal javascript. I’ll be changing this (since doing this hides the content when javascript is off, and CSS is on) – There’s also a small bug with the tv-icon link style on IE6.

In Mephisto, I was able to create my own custom ‘Modal Macro’ filter, so I can easily apply the effect to any content in my articles. Here is the code for doing just that. Save this as modal_macro.rb in the lib/ folder for any new or existing vendor/plugin. As usual, any comments, questions or suggestions are appreciated.

I’m off to Bruges, (or the ‘Land of Chocolate’) in Belgium this Friday. Coincidently the Belgium Chocolate Festival also begins this weekend (in Bruges). I’ll get some pictures up on the Flickr

I imagine the whole trip will be something exactly like this (click to watch)