Class 23: Algorithm Analysis

Held Tuesday, October 3, 2000

Summary

Today we move from details of Java to consideration of the ways in which one might compare and analyze algorithms. We will ground our analysis in a few sample algorithms for finding the smallest value in an array.

Notes

• Are there questions on Homework 3?
  • A few of you asked for an extension, so it is now due next Monday. However, I won't be here on Friday or over the weekend.
• I'm putting off questions on Bailey until tomorrow (or at least the end of class today).

Overview

• Finding the smallest value
• Comparing algorithms
• Removing details
• Big O analysis

Algorithms, Revisited

• You may recall from the first day of class that the foci of CS152 are algorithms and data structures.
  • Algorithms are sets of instructions for completing tasks.
  • Data structures are mechanisms that organize information.
• We will begin with some consideration of algorithm analysis.
• To help, we'll consider two algorithms for finding the smallest integer in an array.
• Here's the first algorithm:

  Pick a value in the array .  Make that our initial guess.
  While there are values we haven't looked at
    Pick a value we haven't looked at.
    If that value is smaller than our guess, update our guess.
  Once we've looked at all the values, return our guess.
  

• We might phrase this in Java as

  public static int smallest(int[] stuff) {
    // Guess the smallest value.
    int guess = stuff[0];
    // Look at each remaining element
    for (int index = 1; index < stuff.length; index++) {
      if (stuff[index] < guess) {
        guess = stuff[index];
      }
    } // for
  } // smallest(int[])
  

• Here's a second algorithm:

  If the collection has only one element, then
    Return that element
  Otherwise
    Split the collection in half.
    Find the smallest element in each half.
    Return the minimum of those two.
  

• Here's a quick attempt to code that in Java.

  public int smallest(int[] stuff, int startIndex, int endIndex) {
    // If the collection has only one element
    if (startIndex == endIndex) {
      // Return that element
      return stuff[startIndex];
    }
    else {
      // Determine the index of the middle, so that we can split in half.
      int mid = (startIndex + endIndex) / 2;
      // Find the smallest element in each half
      int firstSmall = smallest(stuff, startIndex, mid);
      int secondSmall = smallest(stuff, mid+1, endIndex);
      // Return the smallest of thsoe two.
      return Math.min(firstSmall, secondSmall);
    }
  } // smallest(int[], int, int)
  

• Which do you prefer? Why?

Algorithm Analysis

• As you may have noted, there are often multiple algorithms one can use to solve the same problem.
  • In searching an ordered list, one can use linear search, binary search, or ``look randomly'' (as well as many others). [Don't worry if you don't know any of these.]
  • In finding the minimum element of a list, you can step through the list, keeping track of the current minimum. You could also sort the list and grab the first element.
  • You can come up with your own variants.
• How do we choose which algorithm is the best?
  • The fastest/most efficient algorithm.
  • The one that uses the fewest resources.
  • The clearest.
  • The shortest.
  • The easiest to write.
  • The most general.
  • ...
• Frequently, we look at the ``speed''. That is, we consider how long the algorithm takes to run.
  • It is therefore important for us to be able to analyze the running time of our algorithms.

Difficulties Analyzing Running Times

Is there an exact number we can provide for the running time of an algorithm? Surprisingly, no.

• Different inputs lead to different running times. For example, if there are conditionals in the algorithm (as there are in both of our smallest algorithms), different instructions will be executed depending on the input.
• Not all operations take the same time. For example, addition is typically quicker than multiplication, and integer addition is typically quicker than floating point addition.
• The same operation make take different times on different machines.
• The same operation make appear to take different times on the same machine, particularly if other things are happening on the same machine.
• Many things that affect running time happen behind the scenes and cannot be easily predicted. For example, the computer might move some frequently-used data to cache memory.

Asymptotic Analysis

• Noting problems in providing a precise analysis of the running time of programs, computer scientists developed a technique which is often called asymptotic analysis. In asymptotic analysis of algorithms, one describes the general behavior of algorithms in terms of the size of input, but without delving into precise details.
  • The analysis is ``asymptotic'' in that we look at the behavior as the input gets larger.
• There are many issues to consider in analyzing the asymptotic behavior of a program. One particularly useful metric is an upper bound on the running time of an algorithm. We call this the ``Big-O'' of an algorithm.
• Big-O is defined somewhat mathematically, as a relationship between functions.
• f(n) is in O(g(n)) iff
  • there exists a number n0
  • there exists a number d > 0
  • for all n > n0, abs(f(n)) <= abs(d*g(n))
• What does this say? It says that after a certain value, n0, f(n) is bounded above by a constant (d) times g(n).
  • The constant (d) helps accommodate the variation in the algorithm.
  • We don't usually identify the d precisely.
  • The n0 says ``for big enough n''.
• We can apply big-O to algorithms.
  • n is the "size" of the input (e.g., the number of items in a list or vector to be manipulated).
  • f(n) is the running time of the algorithm.
• Some common Big-O bounds
  • An algorithm that is O(1) takes constant time. That is, the running time is independent of the input. Getting the size of an array should be an O(1) algorithm.
  • An algorithm that is O(n) takes time linear in the size of the input. That is, we basically do constant work for each ``element'' of the input. Finding the smallest element in a list is often an O(n) algorithm.
  • An algorithm that is O(log_2(n)) takes logarithmic time. While the running time is dependent on the size of the input, it is clear that not every element of the input is processed. Many such algorithms involve the strategy of ``divide and conquer''.

Eliminating Constants

• One of the nice things about asymptotic analysis is that it makes constants ``unimportant'' because they can be ``hidden'' in the d.
• If f(n) is 100*n seconds and g(n) is 0.5*n seconds, then f(n) is in O(g(n)) [let d be 200] and g(n) is in f(n).
• If f(n) is 100*n seconds and g(n) is n*n seconds, then f(n) is O(g(n)) [let n0 be 100 and d be 1; let n0 be 1 and d be 100; ...].
• However, g(n) is not O(f(n)). Why not? Suppose there were an n0 and a d. Consider what happens for n = 101d. d*f(n) = d*100*101*d = d*d*100*101. However, g(n) = d*d*101*101, which is even larger. If n0 is greater than 101d, we'll still have this problem [proof left to reader].
• Since constants can be eliminated, we normally don't write them.
  • That is, we say that the running time of an algorithm is O(n) or O(n²), ....

Asymptotic Analysis in Practice

• We now have a theoretical grounding for asymptotic analysis. How do we do it in practice?
• At this point in your career, it's often best to ``count'' the steps in an algorithm and then add them up. After you've taken combinatorics, you can use recurrence relations.
• Over the next few days, we'll look at a number of examples. Some starting ones.
  • Finding the smallest/largest element in an array of integers.
  • Finding the average of all the elements in an array of integers.
  • Putting the largest element in an array at the end of the array. if we're only allowed to swap subsequent elements.
  • Computing the nth Fibonacci number.
  • ...

Our Smallest Algorithms

• Let us begin by considering the two algorithms we introduced earlier.
• What can we say about the first algorithm?
• What can we say about the second?

The Role of Details

• Although the previous discussion may make it seem like precise details of algorithms aren't important, some details are particularly important.
• Consider our recursive ``find the smallest value'' algorithm. Suppose we stored the values in a list.
  • How do we split the list?
  • How long should that take?
  • Is that a constant?