MATHEMATICAL INDUCTION
Mathematical induction is an important proof technique used in mathematics. Mathematical induction proof consists of two steps and these are the base step and the inductive step.
In this article, we will look at some example problems and solve it recursively and iteratively. Sometimes it is possible to define an algorithm in terms of itself. This process is called recursion. It is similar to mathematical induction.
The article [Recursion and Mathematical Induction]({% post_url 20200213recursionandmathematicalinduction %}) explains how they are related to each other in more detail.
Recursion is a way to design solutions to problems by [Divide and Conquer]({% post_url 20190404divideandconquer %}) or decreaseandconquer. Reduce a problem to simpler versions of the same problem.
Recursion is a programming technique where a function calls itself. The solutions has one or more base cases that are easy to solve. Must solve the same problem on some other input with the goal of simplifying the larger problem input.
Recursion can also be single, when the method calls itself once or multiple, when the method calls itself multiple times.
Notice that the method does not return a value in the inductive step, it returns a function instead. Each time a procedure is activated recursively, a new set of local, bound variables is created. Although they have the same names as their corresponding elements in the set local to the previous instance of the procedure, their values are distinct.
Normally, a computer allocates two areas of memory for a program: the stack and the heap.
The stack is used to store information about method calls. When a piece of code calls a method, information about the call is placed on the stack. When the method returns, that information is popped off the stack, so the program can resume execution just after the point where it called the method.
The list of methods that were called to get to a particular point of execution is called the call stack.
The heap is another piece of memory that the program can use to create variables and perform calculations.
Typically, the stack is much smaller than the heap. The stack usually is large enough for normal programs because your code typically doesn’t include methods calling other methods to a very great depth. However, recursive algorithms can sometimes create extremely deep call stacks and exhaust the stack space, causing the program to crash.
The article [Recursion Basics]({% post_url 20190328recursionbasics %}) has a section on Removing Recursion that explains how you can prevent this kind of deep recursion from exhausting the stack space and crashing the program.
For this reason, it’s important to evaluate the maximum depth of recursion that a recursive algorithm requires in addition to studying its run time and memory requirements.
A recursive function is a function that is defined by itself or that calls itself. The sum function is given as an example, which is defined as follows:
sum(n) = 1 + 2 + ... + n
That means, the first n natural numbers are added. So, for n = 4 we get:
sum(4) = 1 + 2 + 3 + 4 = 10
If we want to calculate the result of the sum function for a certain n and we already know the result for n  1, n just has to be added to this result:
sum(n) = sum(n1) + n
Such a definition is called a recursive step. In order to calculate the sum function for some n in this way, we still need the base case for the smallest n:
sum(1) = 1
Using these definitions, we are now able to calculate the sum function for some n:
sum(4) = sum(3) + 4
= (sum(2) + 3) + 4
= ((sum(1) + 2) + 3) + 4
= ((1 + 2) + 3) + 4
= 10
We can implement this in code. Here is the C language function to compute sum of first n positive numbers:


You can see the last line that makes the recursive call is using the same form we saw earlier:
sum(n) = sum(n1) + n
Multiply two numbers using multiplication operator. For instance:
5 * 3 = 5 + 5 + 5
We add 5, 3 times. This is the same as:
5 * 3 = 5 + 5 * (3  1)
We keep going until we multipy by 1. This is the base case.
5 * 3 = 5 + 5 + 5 * (2  1)
5 * 3 = 5 + 5 + 5 * 1
We can generalize this as below:
a * b = (a + a + a + a + a + a)
Add a to itself b times.
a * b = (a + a + a + a + a + a)
b times
= a + (a + a + a + a + a)
b  1 times
= a + a * (b1)
The last line:
a + a * (b1)
is called the Recursive Reduction. Notice that the original problem is reduced to a smaller problem that is similar to the original problem.


Compute the nth power of a number Xn.
The mathematical function is defined as follows:
For n = 0, Xn = 1
Otherwise if n > 0 Xn = X * Xn1
The C implementation of exponentiation:


In situations where both iterative and recursive solutions are equally easy to code, choose iterative solution. The advantage of recursion is that sometimes a solution that is very complex to understand can be easily visualized recursively. We just need to solve the problem for the base case and leave the rest of the problem to be solved by recursion. Tower of Hanoi problem is an example.
If the recursive call is made before the function performs its own task, then it is called Head Recursion.
Tail recursion occurs when the last thing a singly recursive algorithm does before returning is call itself.
For example, consider the following implementation of the Factorial algorithm:
Integer: Factorial(Integer: n)
If (n == 0) Then Return 1
Integer: result = n * Factorial(n  1)
Return result
End Factorial
The algorithm starts by checking to see whether it needs to call itself recur sively or whether it can simply return the value 1. If the algorithm must call itself, it does so, multiplies the returned result by n, and returns the result.
You can convert this recursive version of the algorithm into a nonrecursive version by using a loop. Within the loop, the algorithm performs whatever tasks the original algorithm did.
Before the end of the loop, the algorithm should set its parameters to the values they had during the recursive call. If the algorithm returns a value, as the Factorial algorithm does, you need to create a variable to keep track of the return value.
When the loop repeats, the parameters are set for the recursive call, so the algorithm does whatever the recursive call did.
The loop should end when the condition occurs that originally ended the recursion.
For the Factorial algorithm, the stopping condition is:
n == 0
so that condition controls the loop. When the algorithm calls itself recursively, it decreases its parameter n by 1, so the nonrecursive version should also decrease n by 1 before the end of the loop.
The following pseudocode shows the new nonrecursive version of the Factorial algorithm:
Integer: Factorial(Integer: n)
// Make a variable to keep track of the returned value.
// Initialize it to 1 so we can multiply it by returned results.
// (The result is 1 if we do not enter the loop at all.)
Integer: result = 1
// Start a loop controlled by the recursion stopping condition.
While (n != 0)
// Save the result from this "recursive" call.
result = result * n
// Prepare for "recursion."
n=n 1 Loop
// Return the accumulated result.
Return result
End Factorial
This algorithm looks a lot longer than it really is because of all the comments.
Removing tail recursion is straightforward enough that some compilers can do it automatically to reduce stack space requirements.
Of course, the problem with the Factorial algorithm isn’t the depth of recursion, it’s the fact that the results become too big to store in data types of fixed size.
Tailrecursive functions can always be translated directly into iterative functions. The reason is that when we do the tail call, we are effectively replacing the current copy of the function with a new copy with new arguments. We can reuse the parent copy by assigning new values to its arguments and jumping back to the top of the function.
Recursive algorithms are particularly appropriate when the underlying problem or the data to be treated are defined in recursive terms. This does not mean that such recursive definitions guarantee that a recursive algorithm is the best way to solve the problem.
Every recursive program can be transformed into an iterative one. This involves the explicit handling of a recursion stack and these operations will often obscure the essence of a program to such an extent that it becomes difficult to comprehend.
Algorithms which by their nature are recursive rather than iterative should be formulated as recursive procedures. A good example is QuickSort.
We can use recursion in cases the underlying data structures let the choice of recursive solutions appear obvious and natural.
For example, trees are naturally recursive because branches divide into smaller branches that divide into still smaller branches and so on. For that reason, algorithms that build, draw, and search trees are often recursive.
Some problems are naturally recursive. They have a structure that allows a recursive algorithm to easily keep track of its progress and find a solution.
How to approach problems in computer science using recursion, a technique in which a problem is solved by breaking it down into simpler, smaller instances of the same problem.
The post starts with an analogy to mathematical induction, a proof technique in mathematics. Mathematical induction, much like recursion, breaks down a larger problem into a base case (an easily solved instance of the problem) and an inductive step (a rule that allows the solution of larger instances to be derived from smaller ones).
Algorithmically, recursion is an application of the “Divide and Conquer” strategy. The problem at hand is repeatedly divided into simpler versions of the same problem until it can be solved directly (base case).
Semantically, recursion in programming refers to a function that calls itself to solve a problem. Such a function has one or more base cases that can be easily solved, and it also must solve the same problem on some smaller input with the goal of simplifying the larger problem. The function could call itself either once (single recursion) or multiple times (multiple recursion).
The post also introduces the concept of stack depth, which refers to the amount of memory allocated to storing the information related to recursive calls of a function. The memory for a program is typically divided into a stack and a heap. The stack keeps track of method calls (known as the call stack), while the heap is used for other calculations. Recursive algorithms can sometimes exhaust the stack space due to their deep call stacks, leading to a crash. Therefore, one needs to consider the maximum depth of recursion that a recursive algorithm requires, along with its time and memory requirements.
The post then illustrates recursion through examples, such as computing the sum of first ’n’ natural numbers and multiplication of two numbers. In each of these examples, the problems are solved both recursively and iteratively to highlight the differences between the two methods.
The recursive computation of the sum of first ’n’ natural numbers uses a base case of ’n = 1’ and recursively calls the function with ’n1’ to compute the sum. The multiplication of two numbers is also achieved similarly using recursive and iterative methods.
Next, the post discusses how to compute the ’n’th power of a number using recursion, with the base case being ’n = 0’ (which yields a result of 1), and the recursive case being the multiplication of the number with the ’n1’th power of the number.
Recursion can be further divided into two types based on where the recursive call is made: Head Recursion and Tail Recursion. In head recursion, the recursive call is made before the function performs its task, whereas in tail recursion, the recursive call is the last operation in the function. Tail recursive functions are especially amenable to conversion into iterative versions.
The post also emphasizes that while every recursive program can be converted to an iterative one, it’s not always beneficial. There are situations where recursion is more intuitive and straightforward than iteration, particularly when the problem or data structures have a recursive nature. For instance, tree traversal algorithms are typically recursive due to the inherent recursive structure of trees.
In summary, this article gives you the introduction to recursion, explaining its concepts, implementation, and use cases with various examples. It also highlights important considerations when using recursion, such as the possibility of stack overflow, the tradeoff between time and memory in recursive solutions, and the conditions under which recursion might be preferred over iteration.