Recursive Algorithms. We would like to write an algorithm that computes the factorial of a given non-negative integer number n.
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1 ITNP21 - Foundations of IT 1 Recursive Algorithms We would like to write an algorithm that computes the factorial of a given non-negative integer number n. The factorial of 0 is 1 The factorial of n is n (n-1 (n Examples factorial(0 = factorial(1 = factorial(2 = factorial(3 = factorial(4 = factorial(5 = = = = = 120 The factorial function counts the number of ways n people can form a queue. the number of anagrams of Algorithm. 2
2 ITNP21 - Foundations of IT 2 Is it a good algorithm? Algorithm: MyFactorial(n Require: A number n Return: The factorial of n 1: if n 0 then 2: return 1 3: else if n 1 then 4: return 1 5: else if n 2 then 6: return 2... The algorithm to compute factorial(n factorial(0 = 1 factorial(n = n (n 1 (n This is going nowhere! We need a new ingredient! 3 Recursion the woman is holding an object which in turn contains a smaller image of herself holding the same object, and so on Droste effect a visual form of recursion Recursion a method of defining functions, in which the function being defined is applied within its own definition. 4
3 ITNP21 - Foundations of IT 3 Recursion two clauses Your parents are your ancestors; the base case The parents of your ancestors are also your ancestors. the recursive step You need both these ingredients! 5 Defining factorial We can apply the idea of recursion to the factorial function. Base case: factorial(0 = 1 Recursive step: factorial(n = n factorial(n 1 6
4 ITNP21 - Foundations of IT 4 The Factorial algorithm Algorithm: Factorial(n Require: A number n Return: The factorial of n. 1: if n 0 then 2: return 1 3: else 4: return n Factorial(n 1 Base case: factorial(0 = 1 Recursive step: factorial(n = n factorial(n 1 7 Recursion: a bit of intuition Try to split the problem into parts. Base case: What is an easy case to solve? Recursive step: How can I get a step closer to the base case from any starting point? Use the solution to a simpler version of the same problem to help you get there. Phone a friend to ask the solution of an easier problem. 8
5 ITNP21 - Foundations of IT 5 What does this program do? Danger! Make sure you always write down the base case. Never, ever, ever leave it out! Algorithm: NoBase(n Require: A number n Return:? 1: return n + NoBase(n 1 9 Calculation NoBase(2 = 2 + NoBase(1 = NoBase(0 = NoBase(-1 =... When does it stop? Algorithm: NoBase(n Require: A number n Return:? 1: return n + NoBase(n 1 Danger! Make sure you always write down the base case. Never, ever, ever leave it out! 10
6 ITNP21 - Foundations of IT 6 What does this program do? Algorithm: Loop(n Require: A number n Return:? 1: if n 0 then 2: return 1 3: else 4: return n Loop(n Danger! When making the recursive call, you need to be a step closer to the base case. 11 Calculation Loop(0 = 1 Loop(1 = 1 Loop(1 = 1 1 Loop(1 = Loop(1 = Loop(1 = Loop(1 =... When does it stop? Algorithm: Loop(n Require: A number n Return:? 1: if n 0 then 2: return 1 3: else 4: return n Loop(n 12
![. x 1 fact(n - 1 fact(n = n x fact(n-1 if n > 0 1 if n = 1 Von Koch curve snowflake: Lists: e.g. [1, 2, 3], i.e. the list of 1 and [2, 3]! Ls ::= [] [ n, Ls ] e.g. [1, [2, [3, []]]] Recursive structures support recursive procedures [divide-and-conquer strategy]: 1.](/docs-images/88/115919373/images/7-1.jpg "decompose the problem in simpler instances of the same problem [same structure], these instances are solved by further applications of the same procedure, until recursive call 2.")
7 Recursive Algorithms Recursion, a pervasive concept: definitions/structures/procedures... defined in terms of themselves Factorial: fact(n = n x (n-1 x... x 1 fact(n - 1 fact(n = n x fact(n-1 if n > 0 1 if n = 1 Von Koch curve snowflake Recursive Algorithms Recursion, a pervasive concept: definitions/structures/procedures... defined in terms of themselves Factorial: fact(n = n x (n-1 x... x 1 fact(n - 1 fact(n = n x fact(n-1 if n > 0 1 if n = 1 Von Koch curve snowflake: Lists: e.g. [1, 2, 3], i.e. the list of 1 and [2, 3]! Ls ::= [] [ n, Ls ] e.g. [1, [2, [3, []]]] Recursive structures support recursive procedures [divide-and-conquer strategy]: 1. decompose the problem in simpler instances of the same problem [same structure], these instances are solved by further applications of the same procedure, until recursive call 2. solutions for the simplest instances are provided, then base case 3. recombine the results produced by recursive calls at each level, if needed. 14 ITNP21 - Foundations of IT 7
8 ITNP21 - Foundations of IT 8 Binary Search An example of a recursive algorithm (divide and conquer The problem: find a particular target item in an (ascending ordered list. The algorithm (phonebook approach: if the list is empty the target item is not in the list [trivially], otherwise compare the target item with the middle item: o the target item is the middle item: found!, or o the target item is less than (alphabetically before the middle item, then revert searching the target item in the first half of the ordered list, or o the target item is larger than (alphabetically after the middle item, then revert searching the target item in the second half of the ordered list. base case base case recursive call recursive call a single comparison with the middle item cuts off half of the possible candidates [by transitivity: all are bigger/lower than the middle item that is in turn bigger/lower than the target item] 15 Algorithm: Binary_Search (List, Target_Value if (List is empty then (Return: search failure recursive call recursive call ( := The middle entry in List if (Target_Value == then (Return: search successful if (Target_Value > Binary Search then Binary_Search(Second_half_of_List, Target_Value else Binary_Search(First_half_of_List, Target_Value base case Note: middle may need rounding base case Note: Recursion needs a name to be referenced in the recursive calls 16
9 ITNP21 - Foundations of IT 9 Target_Value: 32 [let s see if our binary search is quicker...] Algorithm: Binary_Search (List, Target_Value if (List is empty then (Return: search failure ( := The middle entry in List if (Target_Value == then (Return: search successful if (Target_Value > then Binary_Search(Second_half_of_List, Target_Value else Binary_Search(First_half_of_List, Target_Value! middle = (13+1/2 = 7 Comparisons so far: 1 17 Target_Value: 32 x x x x x x x! middle = (13+8/2 = > 11 Binary_Search ([32,68,75,79,82,91], 32 Comparisons so far: 2 a new recursive call of our algorithm! [since there is not a recombination phase of results, you can forget the previous ones] 18
10 ITNP21 - Foundations of IT 10 Target_Value: 32! middle = (10+8/2 = 9 Comparisons so far: 3 19 Target_Value: 32 Algorithm: Binary_Search ([32], 32 if (List is empty then (Return: search failure ( := The middle entry in List if (Target_Value == then (Return: search successful if (Target_Value > then Binary_Search(Second_half_of_List, Target_Value else Binary_Search(First_half_of_List, Target_Value 4 vs.8 comparisons w.r.t. sequential search no recombination phase! 20
11 ITNP21 - Foundations of IT 11 Target_Value: Target_Value: 33 22
12 ITNP21 - Foundations of IT 12 Target_Value: Target_Value: 33 Algorithm: Binary_Search ([32], 33 if (List is empty then (Return: search failure ( := The middle entry in List if (Target_Value == then (Return: search successful Binary_Search (Second_half_of_[32], 33 if (Target_Value > then Binary_Search(Second_half_of_List, Target_Value else Binary_Search(First_half_of_List, Target_Value 24
13 ITNP21 - Foundations of IT 13 Target_Value: 33 Algorithm: Binary_Search ([], 33 if (List is empty then (Return: search failure Algorithm design: considerations [ incomplete! ]... RECURSION: neat and effective conceptual schema, algorithm structure and programming technique if the problem has a recursive nature!!! Understand how the problem can be recursively decomposed in "smaller" problems of the same kind [understand "the dimension" along which recursion can be defined]. Clearly define all the base cases [and check that one base case can always be reached]. Assume recursive calls solve the recursive instances of the problem, how these results can be recombined into a unique solution of the "bigger" instance of the problem?... 26
![ITNP21 - Foundations of IT 14 Binary Search: JAVA CODE If you want to see the implementation, java code is provided. You will get details further on in the programme [e.g. ITNP11].](/docs-images/88/115919373/images/14-0.jpg "Still, you can appreciate the correspondence with pesudocode, run, test and modify it. The binary_search (int target, int[] a, int i, int j method is defined in the class Searching.")
14 ITNP21 - Foundations of IT 14 Binary Search: JAVA CODE If you want to see the implementation, java code is provided. You will get details further on in the programme [e.g. ITNP11]. Still, you can appreciate the correspondence with pesudocode, run, test and modify it. The binary_search (int target, int[] a, int i, int j method is defined in the class Searching. Lists are implemented on top of arrays. The search method is used by the Search_manager that defines some lists/arrays and performs some searches. If you want to modify the search, you can define an iterative algorithms for searching for the maximum element in an unordered list, expressed in pseudocode. starting from the binary search schema, define a recursive algorithm for the same problem, expressed in pseudocode. It may require a recombination phase of the results provided by the recursive calls. Try to implement the algorithms, modifying the searching code. Find errors, if any. 27 Summary 1. The concept of algorithm, i.e. "a description of a computable process" 2. Iterative algorithms 3. Recursive algorithms 4. Computational complexity, i.e. measuring algorithms efficiency (and problem complexity 5. Computability, i.e. defining what can and what cannot be computed 28
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