Chapter 10 Recursion Dr. Jiung-yao Huang Dept. Comm. Eng. Nat. Chung Cheng Univ. TA: 鄭筱親 陳昱豪

Copyright ©2004 Pearson Addison-Wesley. All rights reserved.10-2 Outline 10.1 THE NATURE OF RECURSION 10.2 TRACING A RECURSIVE FUNCTION 10.3 RECURSIVE MATHMETICAL FUNCTIONS 10.4 RECURSIVE FUNCTIONS WITH ARRAY AND STRING PARAMETERS –CASE STUDY FINDING CAPITAL LETTERS IN A STRING –CASE STUDY RECURSIVE SELECTION SORT 10.5 PROBLEM SOLVING WITH RECURSION –CASE STUDY OPERATIONS ON SETS 10.6 A CLASSIC CASE STUDY IN RECURSION –TOWERS OF HANOI 10.7 COMMON PROGRAMMING ERRORS

Copyright ©2004 Pearson Addison-Wesley. All rights reserved THE NATURE OF RECURSION Recursive function –Function that calls itself or that is part of a cycle in the sequence of function calls Simple case –Problem case for which a straightforward solution is known Recursive solution characteristics –One or more simple cases of the problem have a straightforward, nonrecursive solution –The other cases can be redefined in terms of problems that are closer to the simple cases –By applying this redefinition process every time the recursive function is called, eventually the problem is reduced entirely to simple cases, which are relatively easy to solve

Copyright ©2004 Pearson Addison-Wesley. All rights reserved.10-4 Recursive Algorithm if this is a simple case solve it else redefine the problem using recursion

Copyright ©2004 Pearson Addison-Wesley. All rights reserved.10-5 Figure 10.1 Splitting a Problem into Smaller Problems

Copyright ©2004 Pearson Addison-Wesley. All rights reserved.10-6 EXAMPLE 10.1 Figure 10.2 implements multiplication as the recursive C function multiply that returns the product m x n of its two arguments. The body of function multiply implements the general form of a recursive algorithm. The simplest case is reached when the condition n==1 is true.

Copyright ©2004 Pearson Addison-Wesley. All rights reserved.10-7 Figure 10.2 Recursive Function multiply

Copyright ©2004 Pearson Addison-Wesley. All rights reserved.10-8 EXAMPLE 10.2 Develop a function to count the number of times a particular character appears in a string. Figure 10.3 shows thought process that fits into our generic else clause.

Copyright ©2004 Pearson Addison-Wesley. All rights reserved.10-9 Figure 10.3 Thought Process of Recursive Algorithm Developer

Copyright ©2004 Pearson Addison-Wesley. All rights reserved Figure 10.4 Recursive Function to Count a Character in a String

Copyright ©2004 Pearson Addison-Wesley. All rights reserved TRACING A RECURSIVE FUNCTION Two types of tracing the execution of a recursive function –Tracing a recursive function that returns a value –Tracing a void function that is recursive

Copyright ©2004 Pearson Addison-Wesley. All rights reserved Tracing a recursive function that returns a value Activation frame –Representation of one call to a function Figure 10.5 shows three calls to function multiply.

Copyright ©2004 Pearson Addison-Wesley. All rights reserved Figure 10.5 Trace of Function multiply

Copyright ©2004 Pearson Addison-Wesley. All rights reserved Tracing a void function that is recursive Function reverse_input_words in Fig.10.6 is a recursive module that takes n words of input and prints them in reverse order. Terminating condition –A condition that is true when a recursive algorithm is processing a simple case

Copyright ©2004 Pearson Addison-Wesley. All rights reserved Figure 10.6 Function reverse_input_words

Copyright ©2004 Pearson Addison-Wesley. All rights reserved Figure 10.7 Trace of reverse_input_words(3) When the Words Entered are "bits" "and" "bytes"

Copyright ©2004 Pearson Addison-Wesley. All rights reserved Figure 10.8 Sequence of Events for Trace of reverse_input_words(3)

Copyright ©2004 Pearson Addison-Wesley. All rights reserved Parameter And Local Variable Stacks (1/5) C uses the stack data structure to keep track of the values of n and word at any given point. After first call to reverse_input_words nword 3 ?

Copyright ©2004 Pearson Addison-Wesley. All rights reserved Parameter And Local Variable Stacks (2/5) Before the second call to reverse_input_words n word 3 bits After second call to reverse_input_words n word 3bits 2?

Copyright ©2004 Pearson Addison-Wesley. All rights reserved Parameter And Local Variable Stacks (3/5) Before the third call to reverse_input_words n word bits 3 2and After third call to reverse_input_words n word 3bits 2 and 1?

Copyright ©2004 Pearson Addison-Wesley. All rights reserved Parameter And Local Variable Stacks (4/5) During this execution of the function, the word “bytes” is scanned and stored in word, and “bytes” is echo printed immediately because n is 1 (a simple case) n word 3bits 2 and 1bytes The function return pops both stacks After first return nword 3bits 2and

Copyright ©2004 Pearson Addison-Wesley. All rights reserved Parameter And Local Variable Stacks (5/5) Because control is returned to a printf call, the value of word at the top of the stack is then displayed. Another return occurs, poping the stacks again. After second return nword 3bits

Copyright ©2004 Pearson Addison-Wesley. All rights reserved Implementation of Parameter Stacks in C System stack –Area of memory where parameters and local variables are allocated when a function is called and deallocated when the function returns When and how to trace recursive functions –During algorithm development, it is best to trace a specific case simply by trusting any recursive call to return a correct value based on the function purpose. –Figure 10.9 shows a self-tracing version of function multiply as well as output generated by the call.

Copyright ©2004 Pearson Addison-Wesley. All rights reserved Figure 10.9 Recursive Function multiply with Print Statements to Create Trace and Output from multiply(8, 3)

Copyright ©2004 Pearson Addison-Wesley. All rights reserved Figure 10.9 Recursive Function multiply with Print Statements to Create Trace and Output from multiply(8, 3) (cont’d)

Copyright ©2004 Pearson Addison-Wesley. All rights reserved RECURSIVE MATHMETICAL FUNCTIONS Many mathmatical functions can be defined recursively. For example –The factorial of a number n (n!)

Copyright ©2004 Pearson Addison-Wesley. All rights reserved Figure Recursive factorial Function

Copyright ©2004 Pearson Addison-Wesley. All rights reserved EXAMPLE 10.4 Figure shows the trace of fact=factorial(3) Figure uses iterative version to solve the factorial of the number n

Copyright ©2004 Pearson Addison-Wesley. All rights reserved Figure Trace of fact = factorial(3);

Copyright ©2004 Pearson Addison-Wesley. All rights reserved Figure Iterative Function factorial

Copyright ©2004 Pearson Addison-Wesley. All rights reserved EXAMPLE 10.5 The Fibonacci numbers are a sequence of numbers that have many varied uses. The Fibonacci sequence is defined as –Fibonacci 1 is1 –Fibonacci 2 is 1 –Fibonacci n is Fibonacci n-2 +Fibonacci n-1, for n>2

Copyright ©2004 Pearson Addison-Wesley. All rights reserved Figure Recursive Function fibonacci

Copyright ©2004 Pearson Addison-Wesley. All rights reserved EXAMPLE 10.6 Euclid’s algorithm for finding the gcd can be defined recursively –gcd(m,n) is n if n divides m evently –gcd(m,n) is gcd(n, remainder of m divided by n) otherwise

Copyright ©2004 Pearson Addison-Wesley. All rights reserved Figure Program Using Recursive Function gcd

Copyright ©2004 Pearson Addison-Wesley. All rights reserved Figure Program Using Recursive Function gcd (cont’d)

Copyright ©2004 Pearson Addison-Wesley. All rights reserved RECURSIVE FUNCTIONS WITH ARRAY AND STRING PARAMETERS CASE STUDY: FINDING CAPITAL LETTERS IN A STRING(1/4) Step 1: Problem –Form a string containing all the capital letters found in another string.

Copyright ©2004 Pearson Addison-Wesley. All rights reserved CASE STUDY: FINDING CAPITAL LETTERS IN A STRING(2/4) Step 2: Analysis –Problem Input char *str –Problem Output char *caps

Copyright ©2004 Pearson Addison-Wesley. All rights reserved CASE STUDY: FINDING CAPITAL LETTERS IN A STRING(3/4) Step 3: Design –Algorithm 1. if str is the empty string 2. Store empty string in caps (a string with no letters certainly has no caps) else 3. if initial letter of str is a capital letter 4. Store in caps this letter and the capital letters from the rest of str else 5. Store in caps the capital letters from the rest of str

Copyright ©2004 Pearson Addison-Wesley. All rights reserved CASE STUDY: FINDING CAPITAL LETTERS IN A STRING(4/4) Step 4: Implementation (Figure10.15) Step 5: Testing (Figure 、 Figure 10.17)

Copyright ©2004 Pearson Addison-Wesley. All rights reserved Figure Recursive Function to Extract Capital Letters from a String

Copyright ©2004 Pearson Addison-Wesley. All rights reserved Figure Trace of Call to Recursive Function find_caps

Copyright ©2004 Pearson Addison-Wesley. All rights reserved Figure Sequence of Events for Trace of Call to find_caps from printf Statements

Copyright ©2004 Pearson Addison-Wesley. All rights reserved CASE STUDY RECURSIVE SELECTION SORT(1/4) Step 1: Problem –Sort an array in ascending order using a selection sort.

Copyright ©2004 Pearson Addison-Wesley. All rights reserved CASE STUDY RECURSIVE SELECTION SORT(2/4) Step 2: Analysis (Figure 10.18)

Copyright ©2004 Pearson Addison-Wesley. All rights reserved Figure Trace of Selection Sort

Copyright ©2004 Pearson Addison-Wesley. All rights reserved CASE STUDY RECURSIVE SELECTION SORT(3/4) Step 3: Design –Recursive algorithm for selection sort 1. if n is 1 2. The array is sorted. else 3. Place the largest array value in last array element 4. Sort the subarray which excludes the last array element (array[0]..array[n-2])

Copyright ©2004 Pearson Addison-Wesley. All rights reserved CASE STUDY RECURSIVE SELECTION SORT(4/4) Step 4: Implementation (Figure10.19) Step 5: Testing

Copyright ©2004 Pearson Addison-Wesley. All rights reserved Figure Recursive Selection Sort

Copyright ©2004 Pearson Addison-Wesley. All rights reserved Figure Recursive Selection Sort (cont’d)

Copyright ©2004 Pearson Addison-Wesley. All rights reserved PROBLEM SOLVING WITH RECURSION CASE STUDY OPERATIONS ON SETS(1/4) Step 1: Problem –Develop a group of functions to perform the E (is an element of), (is a subset of) and ∪ (union) operations on sets of characters. –Also develop functions to check that a certain set is valid (that is, that it contains no duplicate characters), to check for the empty set, and to print a set in standard set notation.

Copyright ©2004 Pearson Addison-Wesley. All rights reserved CASE STUDY OPERATIONS ON SETS(2/4) Step 2: Analysis –Character strings provide a fairly natural representation of sets of characters. –Like sets, strings can be of varying sizes and can be empty. –If a character array that is to hold a set is declared to have one more than the number of characters in the universal set (to allow room for the null character), then set operations should never produce a string that will overflow the array.

Copyright ©2004 Pearson Addison-Wesley. All rights reserved CASE STUDY OPERATIONS ON SETS(3/4) Step 3: Design –Algorithm for is_empty(set) 1. Is initial character ‘\0’ ? –Algorithm for is_element(ele, set) 1.if is_empty(set) 2. Answer is false else if initial character of set matches ele 3. Answer is true else 4. Answer depends on whether ele is in the rest of set

Copyright ©2004 Pearson Addison-Wesley. All rights reserved Step 3: Design (cont’s) –Algorithm for is_set(set) 1.if is_empty(set) 2. Answer is true else if is_element(initial set character, rest of set) 3. Answer is false else 4. Answer depends on whether rest of set is valid set –Algorithm for is_subset(sub, set) 1.if is_empty(sub) 2. Answer is true else if initial character of sub is not an element of set 3. Answer is false else 4. Answer depends on whether rest of sub is a subset of set

Copyright ©2004 Pearson Addison-Wesley. All rights reserved Step 3: Design (cont’s) –Algorithm for union of set1 and set2 1.if is_empty(set1) 2. Result is set2 else if initial character of sset1 is also an element of set2 3. Result is union of the rest of set1 with set2 else 4. Result includes initial character of set1 and the union of the rest of set1 with set2

Copyright ©2004 Pearson Addison-Wesley. All rights reserved Step 3: Design (cont’s) –Algorithm for print_set(set) 1.Output a {. 2.if set is not empty, print elements seperated by commas. 3.Output a }. –Algorithm for print_with_commas(set) 1.If set has exactly one element 2. Print it else 3. Print initial element and a comma 4. print_with_commas the rest of set

Copyright ©2004 Pearson Addison-Wesley. All rights reserved CASE STUDY OPERATIONS ON SETS(4/4) Step 4: Implementation (Figure10.20) Step 5: Testing

Copyright ©2004 Pearson Addison-Wesley. All rights reserved Figure Recursive Set Operations on Sets Represented as Character Strings

Copyright ©2004 Pearson Addison-Wesley. All rights reserved Figure Recursive Set Operations on Sets Represented as Character Strings (cont’d)

Copyright ©2004 Pearson Addison-Wesley. All rights reserved Figure Recursive Set Operations on Sets Represented as Character Strings (cont’d)

Copyright ©2004 Pearson Addison-Wesley. All rights reserved Figure Recursive Set Operations on Sets Represented as Character Strings (cont’d)

Copyright ©2004 Pearson Addison-Wesley. All rights reserved Figure Recursive Set Operations on Sets Represented as Character Strings (cont’d)

Copyright ©2004 Pearson Addison-Wesley. All rights reserved Figure Recursive Set Operations on Sets Represented as Character Strings (cont’d)

Copyright ©2004 Pearson Addison-Wesley. All rights reserved A CLASSIC CASE STUDY IN RECURSION TOWERS OF HANOI (1/) Step 1: Problem –Move n disks from peg A to peg C using peg B as needed. –The following conditions apply 1. Only one disk at a time may be moved, and this disk must be the top disk on a peg 2. A larger disk can never be placed on top of a smaller disk

Copyright ©2004 Pearson Addison-Wesley. All rights reserved TOWERS OF HANOI (2/) Step 2: Analysis –Problem inputs int n char from_peg char to_peg char aux_peg –Problem output A list of individual disk moves

Copyright ©2004 Pearson Addison-Wesley. All rights reserved Figure Towers of Hanoi

Copyright ©2004 Pearson Addison-Wesley. All rights reserved Figure Towers of Hanoi After Steps 1 and 2

Copyright ©2004 Pearson Addison-Wesley. All rights reserved Figure Towers of Hanoi After Steps 1, 2, 3.1, and 3.2

Copyright ©2004 Pearson Addison-Wesley. All rights reserved TOWERS OF HANOI (3/) Step 3: Design –Algorithm 1.if n is 1 then 2. Move disk 1 from the from peg to the to peg else 3. Move n-1 disks from the from peg to the auxiliary peg using the to peg 4. Move disk n from the from peg to the to peg 5. Move n-1 disks from the auxiliary peg to the to peg using the from peg

Copyright ©2004 Pearson Addison-Wesley. All rights reserved TOWERS OF HANOI (4/4) Step 4: Implementation (Figure10.24) Step 5: Testing

Copyright ©2004 Pearson Addison-Wesley. All rights reserved Figure Recursive Function tower

Copyright ©2004 Pearson Addison-Wesley. All rights reserved Figure Trace of tower ('A', 'C', 'B', 3);

Copyright ©2004 Pearson Addison-Wesley. All rights reserved Figure Output Generated by tower ('A', 'C', 'B', 3);

Copyright ©2004 Pearson Addison-Wesley. All rights reserved Comparison of Iterative and Recursive Functions Recursive –Requires more time and space because of extra function calls –Is much easier to read and understand –To researchers developing solutions to the complex problems that are at the frontiers of their research areas, the benefits gained from increased clarity far outweigh the extra cost in time and memory of running a recursive program

Copyright ©2004 Pearson Addison-Wesley. All rights reserved COMMON PROGRAMMING ERRORS A recursive function may not be terminate properly. –A run-time error message noting stack overflow or an access violation is an indicator that a recursive function is not terminating Be aware that it is critical that every path through a nonvoid function leads to a return statement The recopying of large arrays or other data structures can quickly consume all available memory Introduce a nonrecursive function to handle preliminaries and call the recursive function when there is error checking

Copyright ©2004 Pearson Addison-Wesley. All rights reserved Chapter Review A recursive function either calls itself or initiates a sequence of function calls in which it may be called again Designing a recursive solution involves identifying simple cases that have straightforward solutions and then redefining more complex cases in terms of problems that are closer to simple cases Recursive functions depend on the fact that for each cal to a function, space is allocated on the stack for the function’s parameters and local variables

Copyright ©2004 Pearson Addison-Wesley. All rights reserved Question?

Copyright ©2004 Pearson Addison-Wesley. All rights reserved Programming Projects 1 Develop a program to count pixels (picture elements) belonging to an object in a photograph. The data are in a two-dimensional grid of cells, each of which may be empty (value 0) or filled (value 1). The filled cells that are connected form a blob (an object). Figure shows a grid with three blobs. Include in your program a function blob_check that takes as parameters the grid and the x-y coordinates of a cell and returns as its value the number of cells in the blob to which the indicated cell belongs.

Copyright ©2004 Pearson Addison-Wesley. All rights reserved Figure Grid with Three Blobs