Problem Solving

Algorithms:
Efficiency, Analysis and Order

Chapter 1 Foundations of Algorithms

Problem Solving Techniques

• So far, in this course we have been talking about data structures

• We now turn our attention from the data to the problems to be solved

• There are often several fundamentally different techniques that can be used to solve the same problem

• Applying a technique to a problem results in a step-by-step procedure to solve the problem

• A good programmer is capable of analyzing the steps the mind takes in solving a problem

The step-by-step solution

• This step-by-step procedure is called an algorithm

• The purpose of studying these problem-solving techniques is so that you have a repertoire of techniques to consider when trying to solve a problem

• A given problem can often be solved using several techniques, but one may be much faster than the others

• So we will be concerned with analyzing how efficient a technique is in solving a problem in terms of time and space

Communicating algorithms

• One way to write an algorithm is in English

– This is fairly hard to translate into a programming language.

• Another way is in a programming language like Java

– We often get tangled in the syntax, and lose sight of the logic (steps) of the algorithm.

• Another way is to use pseudocode

– The text uses kind of a Java shorthand, but avoids Java syntax when it is cumbersome

Examples of the text pseudocode

• Keytype is much like Comparable; it means the items are from any ordered set

• Array indexing can be from ranges other than 0..n; in this pseudocode, the range is specified

• If mathematical expressions or English is more succinct or clear than Java, it is used:

Examples: If (low< x < high)
exchange x and y

• Look at overhead of binary search in psedocode

Importance of Developing Efficient Algorithms

• Problem: Searching a sorted array

– We have two algorithms Binary and Sequential Search

• Suppose the item we are looking for is not in the array and the list has 16 items

– How many comparisons will a sequential search make?

– How many comparisons will a binary search make?

• If we double the number of items in the list, how many comparisons will each make then?

• Every time we double the size of the list, we add just one more comparison to a binary search

Algorithms to for Fibonacci sequence

• There are at least two algorithms to find the n^th term of the Fibonacci sequence

• Here is the Fibonacci sequence; how do we find the next term?
1 1 2 3 5 8 13 ……

• Two algorithmic techniques

– Recursive

– Iterative

Recursive algorithm to find nth term

• public static int fib(int n)
{ if (n<=1) return n;
else return fib(n-1) +
fib(n-2);
}

• How efficient is this algorithm? Look at the number of terms computed

Efficiency of recursive solution.

• Notice that the number of terms more than doubles every time n increases by 2.

• Call T(n) the number of terms.

• Then T(n) > 2*T(n-2)

• We know that T(0) =1

^• How did you book derive T(n) > 2^n/2

• To see this, take a concrete case n=6 (work on board)

Efficiency of the iterative solution

• The recursive solution required that the same value be computed over and over

• Suppose we save each value as it is computed

• Look at iterative code on overhead

– Once the values are computed, they are saved in an array.

The moral of this story

• The technique used to solve problems makes a lot of difference!!