What this means is, the time taken to calculate fib (n) is equal to the sum of time taken to calculate fib (n-1) and fib (n-2). Time Complexity: In the above code “Hello World” is printed only once on the screen. What are the advantages of recursion over iteration? Recursion can reduce time complexity. Iteration produces repeated computation using for loops or while. Standard Problems on Recursion. 10. A filesystem consists of named files. There is less memory required in the case of iteration A recursive process, however, is one that takes non-constant (e. remembering the return values of the function you have already. pop() if node. Alternatively, you can start at the top with , working down to reach and . . High time complexity. However, for some recursive algorithms, this may compromise the algorithm’s time complexity and result in a more complex code. In the above algorithm, if n is less or equal to 1, we return nor make two recursive calls to calculate fib of n-1 and fib of n-2. Euclid’s Algorithm: It is an efficient method for finding the GCD (Greatest Common Divisor) of two integers. This study compares differences in students' ability to comprehend recursive and iterative programs by replicating a 1996 study, and finds a recursive version of a linked list search function easier to comprehend than an iterative version. However, as for the Fibonacci solution, the code length is not very long. In data structure and algorithms, iteration and recursion are two fundamental problem-solving approaches. Answer: In general, recursion is slow, exhausting computer’s memory resources while iteration performs on the same variables and so is efficient. A recursive algorithm can be time and space expensive because to count the value of F n we have to call our recursive function twice in every step. This is the iterative method. Consider writing a function to compute factorial. Time Complexity Analysis. You can find a more complete explanation about the time complexity of the recursive Fibonacci. Iteration is always faster than recursion if you know the amount of iterations to go through from the start. Because of this, factorial utilizing recursion has an O time complexity (N). Steps to solve recurrence relation using recursion tree method: Draw a recursive tree for given recurrence relation. If it is, the we are successful and return the index. The Iteration method would be the prefer and faster approach to solving our problem because we are storing the first two of our Fibonacci numbers in two variables (previouspreviousNumber, previousNumber) and using "CurrentNumber" to store our Fibonacci number. The second time function () runs, the interpreter creates a second namespace and assigns 10 to x there as well. Iteration is faster than recursion due to less memory usage. Recursion produces repeated computation by calling the same function recursively, on a simpler or smaller subproblem. For large or deep structures, iteration may be better to avoid stack overflow or performance issues. The second return (ie: return min(. Time complexity: O(n log n) Auxiliary Space complexity: O(n) Iterative Merge Sort: The above function is recursive, so uses function call stack to store intermediate values of l and h. It causes a stack overflow because the amount of stack space allocated to each process is limited and far lesser than the amount of heap space allocated to it. of times to find the nth Fibonacci number nothing more or less, hence time complexity is O(N), and space is constant as we use only three variables to store the last 2 Fibonacci numbers to find the next and so on. 2. While studying about Merge Sort algorithm, I was curious to know if this sorting algorithm can be further optimised. 1. Strictly speaking, recursion and iteration are both equally powerful. We prefer iteration when we have to manage the time complexity and the code size is large. If we look at the pseudo-code again, added below for convenience. The first function executes the ( O (1) complexity) statements in the while loop for every value between a larger n and 2, for an overall complexity of O (n). The function call stack stores other bookkeeping information together with parameters. Iteration: An Empirical Study of Comprehension Revisited. e. Complexity Analysis of Ternary Search: Time Complexity: Worst case: O(log 3 N) Average case: Θ(log 3 N) Best case: Ω(1) Auxiliary Space: O(1) Binary search Vs Ternary Search: The time complexity of the binary search is less than the ternary search as the number of comparisons in ternary search is much more than binary search. Iterative functions explicitly manage memory allocation for partial results. Which approach is preferable depends on the problem under consideration and the language used. Speed - It usually runs slower than iterative Space - It usually takes more space than iterative, called "call. I am studying Dynamic Programming using both iterative and recursive functions. Iteration. Tail-recursion is the intersection of a tail-call and a recursive call: it is a recursive call that also is in tail position, or a tail-call that also is a recursive call. If the shortness of the code is the issue rather than the Time Complexity 👉 better to use Recurtion. Iteration is the repetition of a block of code using control variables or a stopping criterion, typically in the form of for, while or do-while loop constructs. Let's try to find the time. An iteration happens inside one level of. Where have I gone wrong and why is recursion different from iteration when analyzing for Big-O? recursion; iteration; big-o; computer-science; Share. After every iteration ‘m', the search space will change to a size of N/2m. the last step of the function is a call to the. As such, you pretty much have the complexities backwards. The first recursive computation of the Fibonacci numbers took long, its cost is exponential. The recursive function runs much faster than the iterative one. Recursive algorithm's time complexity can be better estimated by drawing recursion tree, In this case the recurrence relation for drawing recursion tree would be T(n)=T(n-1)+T(n-2)+O(1) note that each step takes O(1) meaning constant time,since it does only one comparison to check value of n in if block. Processes generally need a lot more heap space than stack space. To estimate the time complexity, we need to consider the cost of each fundamental instruction and the number of times the instruction is executed. Which is better: Iteration or Recursion? Sometime finding the time complexity of recursive code is more difficult than that of Iterative code. Recursion, broadly speaking, has the following disadvantages: A recursive program has greater space requirements than an iterative program as each function call will remain in the stack until the base case is reached. For medium to large. Similarly, Space complexity of an algorithm quantifies the amount of space or memory taken by an algorithm to run as a function of the length of the input. But it is stack based and stack is always a finite resource. 2. Iterative codes often have polynomial time complexity and are simpler to optimize. Improve this answer. Follow. Iteration terminates when the condition in the loop fails. , it runs in O(n). With your iterative code, you're allocating one variable (O (1) space) plus a single stack frame for the call (O (1) space). This is usually done by analyzing the loop control variables and the loop termination condition. Also remember that every recursive method must make progress towards its base case (rule #2). The Java library represents the file system using java. There is more memory required in the case of recursion. In a recursive step, we compute the result with the help of one or more recursive calls to this same function, but with the inputs somehow reduced in size or complexity, closer to a base case. Remember that every recursive method must have a base case (rule #1). Generally speaking, iteration and dynamic programming are the most efficient algorithms in terms of time and space complexity, while matrix exponentiation is the most efficient in terms of time complexity for larger values of n. Time complexity is relatively on the lower side. For example, MergeSort - it splits the array into two halves and calls itself on these two halves. This involves a larger size of code, but the time complexity is generally lesser than it is for recursion. Iterative codes often have polynomial time complexity and are simpler to optimize. Iteration is preferred for loops, while recursion is used for functions. As for the recursive solution, the time complexity is the number of nodes in the recursive call tree. N logarithm N (N * log N) N*logN complexity refers to product of N and log of N to the base 2. It's a equation or a inequality that describes a functions in terms of its values and smaller inputs. One uses loops; the other uses recursion. It is faster because an iteration does not use the stack, Time complexity. Recursion: The time complexity of recursion can be found by finding the value of the nth recursive call in terms of the previous calls. There are factors ignored, like the overhead of function calls. Time Complexity calculation of iterative programs. Space Complexity. In the recursive implementation on the right, the base case is n = 0, where we compute and return the result immediately: 0! is defined to be 1. It may vary for another example. Note: To prevent integer overflow we use M=L+(H-L)/2, formula to calculate the middle element, instead M=(H+L)/2. T (n) = θ. And to emphasize a point in the previous answer, a tree is a recursive data structure. " Recursion is also much slower usually, and when iteration is applicable it's almost always prefered. So whenever the number of steps is limited to a small. The complexity is only valid in a particular. Time Complexity: Intuition for Recursive Algorithm. Recursion terminates when the base case is met. Recursion is more natural in a functional style, iteration is more natural in an imperative style. I'm a little confused. The difference comes in terms of space complexity and how programming language, in your case C++, handles recursion. In your example: the time complexity of this code can be described with the formula: T(n) = C*n/2 + T(n-2) ^ ^ assuming "do something is constant Recursive call. Space complexity of iterative vs recursive - Binary Search Tree. It is faster than recursion. Btw, if you want to remember or review the time complexity of different sorting algorithms e. Obviously, the time and space complexity of both. If time complexity is the point of focus, and number of recursive calls would be large, it is better to use iteration. Our iterative technique has an O(N) time complexity due to the loop's O(N) iterations (N). An algorithm that uses a single variable has a constant space complexity of O (1). This can include both arithmetic operations and data. Space Complexity : O(2^N) This is due to the stack size. However, just as one can talk about time complexity, one can also talk about space complexity. Iteration terminates when the condition in the loop fails. Practice. , at what rate does the time taken by the program increase or decrease is its time complexity. Control - Recursive call (i. Recursion is when a statement in a function calls itself repeatedly. 1 Predefined List Loops. This is the essence of recursion – solving a larger problem by breaking it down into smaller instances of the. Iteration is generally going to be more efficient. For each node the work is constant. Recursion can sometimes be slower than iteration because in addition to the loop content, it has to deal with the recursive call stack frame. Generally, it has lower time complexity. We. No. I assume that solution is O(N), not interesting how implemented is multiplication. Its time complexity anal-ysis is similar to that of num pow iter. This complexity is defined with respect to the distribution of the values in the input data. With recursion, you repeatedly call the same function until that stopping condition, and then return values up the call stack. Sometimes it’s more work. In a recursive step, we compute the result with the help of one or more recursive calls to this same function, but with the inputs somehow reduced in size or complexity, closer to a base case. If it's true that recursion is always more costly than iteration, and that it can always be replaced with an iterative algorithm (in languages that allow it) - than I think that the two remaining reasons to use. What are the benefits of recursion? Recursion can reduce time complexity. In the recursive implementation on the right, the base case is n = 0, where we compute and return the result immediately: 0! is defined to be 1. The time complexity of recursion is higher than Iteration due to the overhead of maintaining the function call stack. Recursion can be replaced using iteration with stack, and iteration can also be replaced with recursion. Utilization of Stack. Time complexity is O(n) here as for 3 factorial calls you are doing n,k and n-k multiplication . It is fast as compared to recursion. So does recursive BFS. 0. In contrast, the iterative function runs in the same frame. When you're k levels deep, you've got k lots of stack frame, so the space complexity ends up being proportional to the depth you have to search. Any recursive solution can be implemented as an iterative solution with a stack. Line 6-8: 3 operations inside the for-loop. The reason is because in the latter, for each item, a CALL to the function st_push is needed and then another to st_pop. No. So a filesystem is recursive: folders contain other folders which contain other folders, until finally at the bottom of the recursion are plain (non-folder) files. How many nodes are. geeksforgeeks. Consider for example insert into binary search tree. Recursion allows us flexibility in printing out a list forwards or in reverse (by exchanging the order of the. Recursion is a way of writing complex codes. Things get way more complex when there are multiple recursive calls. If i use iteration , i will have to use N spaces in an explicit stack. Because of this, factorial utilizing recursion has. "use a recursion tree to determine a good asymptotic upper bound on the recurrence T (n)=T (n/2)+n^2. Time & Space Complexity of Iterative Approach. Some tasks can be executed by recursion simpler than iteration due to repeatedly calling the same function. It's all a matter of understanding how to frame the problem. io. Therefore the time complexity is O(N). Readability: Straightforward and easier to understand for most programmers. This is a simple algorithm, and good place to start in showing the simplicity and complexity of of recursion. Moving on to slicing, although binary search is one of the rare cases where recursion is acceptable, slices are absolutely not appropriate here. Because of this, factorial utilizing recursion has. iterations, layers, nodes in each layer, training examples, and maybe more factors. Courses Practice What is Recursion? The process in which a function calls itself directly or indirectly is called recursion and the corresponding function is called a recursive function. It consists of three poles and a number of disks of different sizes which can slide onto any pole. In this video, I will show you how to visualize and understand the time complexity of recursive fibonacci. Naive sorts like Bubble Sort and Insertion Sort are inefficient and hence we use more efficient algorithms such as Quicksort and Merge Sort. Recursive functions provide a natural and direct way to express these problems, making the code more closely aligned with the underlying mathematical or algorithmic concepts. High time complexity. If a k-dimensional array is used, where each dimension is n, then the algorithm has a space. Consider writing a function to compute factorial. In the above recursion tree diagram where we calculated the fibonacci series in c using the recursion method, we. The definition of a recursive function is a function that calls itself. In the illustration above, there are two branches with a depth of 4. e. By examining the structure of the tree, we can determine the number of recursive calls made and the work. Introduction. Since this is the first value of the list, it would be found in the first iteration. If the Time Complexity is important and the number of recursive calls would be large 👉 better to use Iteration. A loop looks like this in assembly. The order in which the recursive factorial functions are calculated becomes: 1*2*3*4*5. But at times can lead to difficult to understand algorithms which can be easily done via recursion. Let a ≥ 1 and b > 1 be constants, let f ( n) be a function, and let T ( n) be a function over the positive numbers defined by the recurrence. It also covers Recursion Vs Iteration: From our earlier tutorials in Java, we have seen the iterative approach wherein we declare a loop and then traverse through a data structure in an iterative manner by taking one element at a time. In our recursive technique, each call consumes O(1) operations, and there are O(N) recursive calls overall. 1. If you're unsure about the iteration / recursion mechanics, insert a couple of strategic print statements to show you the data and control flow. e. 1. In the above implementation, the gap is reduced by half in every iteration. That's a trick we've seen before. g. Recursion is not intrinsically better or worse than loops - each has advantages and disadvantages, and those even depend on the programming language (and implementation). In contrast, the iterative function runs in the same frame. def tri(n: Int): Int = { var result = 0 for (count <- 0 to n) result = result + count result} Note that the runtime complexity of this algorithm is still O(n) because we will be required to iterate n times. 1. Time complexity = O(n*m), Space complexity = O(1). Recursion vs. |. Disadvantages of Recursion. Thus fib(5) will be calculated instantly but fib(40) will show up after a slight delay. The major difference between the iterative and recursive version of Binary Search is that the recursive version has a space complexity of O(log N) while the iterative version has a space complexity of O(1). One of the best ways I find for approximating the complexity of the recursive algorithm is drawing the recursion tree. Tower of Hanoi is a mathematical puzzle where we have three rods and n disks. The iterative version uses a queue to maintain the current nodes, while the recursive version may use any structure to persist the nodes. The towers of Hanoi problem is hard no matter what algorithm is used, because its complexity is exponential. Sorted by: 4. 3: An algorithm to compute mn of a 2x2 matrix mrecursively using repeated squaring. What is the time complexity to train this NN using back-propagation? I have a basic idea about how they find the time complexity of algorithms, but here there are 4 different factors to consider here i. When deciding whether to. The iterative solution has three nested loops and hence has a complexity of O(n^3) . Add a comment. There is less memory required in the case of. Iteration and recursion are key Computer Science techniques used in creating algorithms and developing software. Same with the time complexity, the time which the program takes to compute the 8th Fibonacci number vs 80th vs 800th Fibonacci number i. Backtracking at every step eliminates those choices that cannot give us the. Here are the 5 facts to understand the difference between recursion and iteration. ; It also has greater time requirements because each time the function is called, the stack grows. 1 Answer. See your article appearing on the GeeksforGeeks main page. So when recursion is doing constant operation at each recursive call, we just count the total number of recursive calls. A filesystem consists of named files. In that sense, it's a matter of how a language processes the code also, as I've mentioned, some compilers transformers a recursion into a loop on its binary depending on its computation on that code. Recursive Sorts. Time complexity. m) => O(n 2), when n == m. Time complexity. In Java, there is one situation where a recursive solution is better than a. To my understanding, the recursive and iterative version differ only in the usage of the stack. Yes. Iterative vs recursive factorial. But at times can lead to difficult to understand algorithms which can be easily done via recursion. Where branches are the number of recursive calls made in the function definition and depth is the value passed to the first call. Computations using a matrix of size m*n have a space complexity of O (m*n). Time Complexity. In our recursive technique, each call consumes O(1) operations, and there are O(N) recursive calls overall. The idea is to use one more argument and accumulate the factorial value in the second argument. The inverse transformation can be trickier, but most trivial is just passing the state down through the call chain. In the first partitioning pass, you split into two partitions. Comparing the above two approaches, time complexity of iterative approach is O(n) whereas that of recursive approach is O(2^n). Improve this question. In the Fibonacci example, it’s O(n) for the storage of the Fibonacci sequence. The result is 120. While tail-recursive calls are usually faster for list reductions—like the example we’ve seen before—body-recursive functions can be faster in some situations. File. Moreover, the recursive function is of exponential time complexity, whereas the iterative one is linear. The reason why recursion is faster than iteration is that if you use an STL container as a stack, it would be allocated in heap space. Loops are generally faster than recursion, unless the recursion is part of an algorithm like divide and conquer (which your example is not). To visualize the execution of a recursive function, it is. Iterative Backtracking vs Recursive Backtracking; Time and Space Complexity; Introduction to Iteration. Utilization of Stack. Recursion vs Iteration: You can reduce time complexity of program with Recursion. Recursion does not always need backtracking. Graph Search. Recursion involves creating and destroying stack frames, which has high costs. To visualize the execution of a recursive function, it is. Sorted by: 1. def function(): x = 10 function() When function () executes the first time, Python creates a namespace and assigns x the value 10 in that namespace. The Recursion and Iteration both repeatedly execute the set of instructions. There is a lot to learn, Keep in mind “ Mnn bhot karega k chor yrr a. 1. The memory usage is O (log n) in both. Recursion has a large amount of Overhead as compared to Iteration. In both cases (recursion or iteration) there will be some 'load' on the system when the value of n i. Both approaches create repeated patterns of computation. 12. This approach of converting recursion into iteration is known as Dynamic programming(DP). If not, the loop will probably be better understood by anyone else working on the project. It has relatively lower time. Recursion is a repetitive process in which a function calls itself. 1. personally, I find it much harder to debug typical "procedural" code, there is a lot of book keeping going on as the evolution of all the variables has to be kept in mind. An iteration happens inside one level of function/method call and. Recursive implementation uses O (h) memory (where h is the depth of the tree). Analyzing recursion is different from analyzing iteration because: n (and other local variable) change each time, and it might be hard to catch this behavior. In this traversal, we first create links to Inorder successor and print the data using these links, and finally revert the changes to restore original tree. This is the main part of all memoization algorithms. e. Space The Fibonacci sequence is de ned: Fib n = 8 >< >: 1 n == 0Efficiency---The Time Complexity of an Algorithm In the bubble sort algorithm, there are two kinds of tasks. Your example illustrates exactly that. when recursion exceeds a particular limit we use shell sort. An example of using the findR function is shown below. Introduction. Now, we can consider countBinarySubstrings (), which calls isValid () n times. Here are some scenarios where using loops might be a more suitable choice: Performance Concerns : Loops are generally more efficient than recursion regarding time and space complexity. This means that a tail-recursive call can be optimized the same way as a tail-call. Even now, if you are getting hard time to understand the logic, i would suggest you to make a tree-like (not the graph which i have shown here) representation for xstr = "ABC" and ystr. O ( n ), O ( n² ) and O ( n ). Iteration & Recursion. The auxiliary space required by the program is O(1) for iterative implementation and O(log 2 n) for. There is more memory required in the case of recursion. Tail recursion optimization essentially eliminates any noticeable difference because it turns the whole call sequence to a jump. Sorted by: 1. e execution of the same set of instructions again and again. Recursion happens when a method or function calls itself on a subset of its original argument. The problem is converted into a series of steps that are finished one at a time, one after another. As for the recursive solution, the time complexity is the number of nodes in the recursive call tree. The reason for this is that the slowest. However, having been working in the Software industry for over a year now, I can say that I have used the concept of recursion to solve several problems. functions are defined by recursion, so implementing the exact definition by recursion yields a program that is correct "by defintion". quicksort, merge sort, insertion sort, radix sort, shell sort, or bubble sort, here is a nice slide you can print and use:The Iteration Method, is also known as the Iterative Method, Backwards Substitution, Substitution Method, and Iterative Substitution. High time complexity. Comparing the above two approaches, time complexity of iterative approach is O(n) whereas that of recursive approach is O(2^n). Application of Recursion: Finding the Fibonacci sequenceThe master theorem is a recipe that gives asymptotic estimates for a class of recurrence relations that often show up when analyzing recursive algorithms. With regard to time complexity, recursive and iterative methods both will give you O(log n) time complexity, with regard to input size, provided you implement correct binary search logic. Evaluate the time complexity on the paper in terms of O(something). The primary difference between recursion and iteration is that recursion is a process, always. Backtracking always uses recursion to solve problems. Generally, it has lower time complexity. To visualize the execution of a recursive function, it is. When evaluating the space complexity of the problem, I keep seeing that time O() = space O(). This reading examines recursion more closely by comparing and contrasting it with iteration. Recursion versus iteration. It is fast as compared to recursion. It is the time needed for the completion of an algorithm. For mathematical examples, the Fibonacci numbers are defined recursively: Sigma notation is analogous to iteration: as is Pi notation. Iteration and recursion are two essential approaches in Algorithm Design and Computer Programming. The Tower of Hanoi is a mathematical puzzle. Both approaches create repeated patterns of computation. Strengths: Without the overhead of function calls or the utilization of stack memory, iteration can be used to repeatedly run a group of statements. Often writing recursive functions is more natural than writing iterative functions, especially for a rst draft of a problem implementation. It is. But then, these two sorts are recursive in nature, and recursion takes up much more stack memory than iteration (which is used in naive sorts) unless. In general, we have a graph with a possibly infinite set of nodes and a set of edges. But it has lot of overhead. mov loopcounter,i dowork:/do work dec loopcounter jmp_if_not_zero dowork. The first is to find the maximum number in a set. 1. Recursion shines in scenarios where the problem is recursive, such as traversing a DOM tree or a file directory. The time complexity for the recursive solution will also be O(N) as the recurrence is T(N) = T(N-1) + O(1), assuming that multiplication takes constant time. Both recursion and ‘while’ loops in iteration may result in the dangerous infinite calls situation. Big O Notation of Time vs. A recursive function solves a particular problem by calling a copy of itself and solving smaller subproblems of the original problems. Program for Tower of Hanoi Algorithm; Time Complexity Analysis | Tower Of Hanoi (Recursion) Find the value of a number raised to its reverse; Recursively remove all adjacent duplicates; Print 1 to n without using loops; Print N to 1 without loop; Sort the Queue using Recursion; Reversing a queue using. There is no difference in the sequence of steps itself (if suitable tie-breaking rules. With recursion, the trick of using Memoization the cache results will often dramatically improve the time complexity of the problem. Quoting from the linked post: Because you can build a Turing complete language using strictly iterative structures and a Turning complete language using only recursive structures, then the two are therefore equivalent. The computation of the (n)-th Fibonacci numbers requires (n-1) additions, so its complexity is linear. 1Review: Iteration vs. Condition - Exit Condition (i. – However, I'm uncertain about how the recursion might affect the time complexity calculation. But when you do it iteratively, you do not have such overhead. 1 Answer. It takes O (n/2) to partition each of those. But there are significant differences between recursion and iteration in terms of thought processes, implementation approaches, analysis techniques, code complexity, and code performance. Analysis of the recursive Fibonacci program: We know that the recursive equation for Fibonacci is = + +. Instead of many repeated recursive calls we can save the results, already obtained by previous steps of algorithm. It is fast as compared to recursion. As such, the time complexity is O(M(lga)) where a= max(r). e. And the space complexity of iterative BFS is O (|V|). Iteration is the process of repeatedly executing a set of instructions until the condition controlling the loop becomes false. We. Introduction This reading examines recursion more closely by comparing and contrasting it with iteration. Applicable To: The problem can be partially solved, with the remaining problem will be solved in the same form. Singly linked list iteration complexity. However -these are constant number of ops, while not changing the number of "iterations". It is faster than recursion. Removing recursion decreases the time complexity of recursion due to recalculating the same values. Knowing the time complexity of a method involves examining whether you have implemented an iteration algorithm or. This paper describes a powerful and systematic method, based on incrementalization, for transforming general recursion into iteration: identify an input increment, derive an incremental version under the input. For example, using a dict in Python (which has (amortized) O (1) insert/update/delete times), using memoization will have the same order ( O (n)) for calculating a factorial as the basic iterative solution. Looping may be a bit more complex (depending on how you view complexity) and code. functions are defined by recursion, so implementing the exact definition by recursion yields a program that is correct "by defintion". Now, one of your friend suggested a book that you don’t have. Memoization¶. For example, using a dict in Python (which has (amortized) O (1) insert/update/delete times), using memoization will have the same order ( O (n)) for calculating a factorial as the basic iterative solution. In Java, there is one situation where a recursive solution is better than a. It consists of initialization, comparison, statement execution within the iteration, and updating the control variable. There are possible exceptions such as tail recursion optimization. )Time complexity is very useful measure in algorithm analysis. 5: We mostly prefer recursion when there is no concern about time complexity and the size of code is small. base case) Update - It gradually approaches to base case. In simple terms, an iterative function is one that loops to repeat some part of the code, and a recursive function is one that calls itself again to repeat the code. The complexity analysis does not change with respect to the recursive version.