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Chapter 16 Recursion Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. What is Recursion? A recursive function is one that solves its task by calling itself on smaller pieces of data. • Similar to recurrence function in mathematics. • Like iteration -- can be used interchangeably; sometimes recursion results in a simpler solution. n Example: Running sum (  i ) Mathematical Definition: RunningSum(1) = 1 RunningSum(n) = n + RunningSum(n-1) 1 Recursive Function: int RunningSum(int n) { if (n == 1) return 1; else return n + RunningSum(n-1); } 16-2 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Executing RunningSum res = RunningSum(4); return value = 10 RunningSum(4) return 4 + RunningSum(3); return value = 6 RunningSum(3) return 3 + RunningSum(2); RunningSum(2) return value = 3 return 2 + RunningSum(1); return value = 1 RunningSum(1) return 1; 16-3 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. High-Level Example: Binary Search Given a sorted set of exams, in alphabetical order, find the exam for a particular student. 1. Look at the exam halfway through the pile. 2. If it matches the name, we're done; if it does not match, then... 3a. If the name is greater (alphabetically), then search the upper half of the stack. 3b. If the name is less than the halfway point, then search the lower half of the stack. 16-4 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Binary Search: Pseudocode Pseudocode is a way to describe algorithms without completely coding them in C. FindExam(studentName, start, end) { halfwayPoint = (end + start)/2; if (end < start) ExamNotFound(); /* exam not in stack */ else if (studentName == NameOfExam(halfwayPoint)) ExamFound(halfwayPoint); /* found exam! */ else if (studentName < NameOfExam(halfwayPoint)) /* search lower half */ FindExam(studentName, start, halfwayPoint - 1); else /* search upper half */ FindExam(studentName, halfwayPoint + 1, end); } 16-5 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. High-Level Example: Towers of Hanoi Task: Move all disks from current post to another post. Post 1 Post 2 Post 3 Rules: (1) Can only move one disk at a time. (2) A larger disk can never be placed on top of a smaller disk. (3) May use third post for temporary storage. 16-6 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Task Decomposition Suppose disks start on Post 1, and target is Post 3. 1. Move top n-1 disks to Post 2. 1 2 3 1 2 3 1 2 3 2. Move largest disk to Post 3. 3. Move n-1 disks from Post 2 to Post 3. 16-7 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Task Decomposition (cont.) Task 1 is really the same problem, with fewer disks and a different target post. • "Move n-1 disks from Post 1 to Post 2." And Task 3 is also the same problem, with fewer disks and different starting and target posts. • "Move n-1 disks from Post 2 to Post 3." So this is a recursive algorithm. • The terminal case is moving the smallest disk -- can move directly without using third post. • Number disks from 1 (smallest) to n (largest). 16-8 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Towers of Hanoi: Pseudocode MoveDisk(diskNumber, startPost, endPost, midPost) { if (diskNumber > 1) { /* Move top n-1 disks to mid post */ MoveDisk(diskNumber-1, startPost, midPost, endPost); printf("Move disk number %d from %d to %d.\n", diskNumber, startPost, endPost); /* Move n-1 disks from mid post to end post */ MoveDisk(diskNumber-1, midPost, endPost, startPost); } else printf("Move disk number 1 from %d to %d.\n", startPost, endPost); } 16-9 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Detailed Example: Fibonacci Numbers Mathematical Definition: f (n ) f (n  1)  f (n  2) f (1) 1 f (0) 1 In other words, the n-th Fibonacci number is the sum of the previous two Fibonacci numbers. 16-10 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Fibonacci: C Code int Fibonacci(int n) { if ((n == 0) || (n == 1)) return 1; else return Fibonacci(n-1) + Fibonacci(n-2); } 16-11 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Activation Records Whenever Fibonacci is invoked, a new activation record is pushed onto the stack. main calls Fibonacci(3) Fibonacci(3) calls Fibonacci(2) Fibonacci(2) calls Fibonacci(1) main main main Fib(3) Fib(3) R6 Fib(3) R6 Fib(2) Fib(2) R6 Fib(1) 16-12 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Activation Records (cont.) Fibonacci(2) calls Fibonacci(0) Fibonacci(3) calls Fibonacci(1) Fibonacci(3) returns R6 main main Fib(3) Fib(3) main R6 Fib(2) Fib(1) R6 Fib(0) 16-13 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Tracing the Function Calls If we are debugging this program, we might want to trace all the calls of Fibonacci. • Note: A trace will also contain the arguments passed into the function. For Fibonacci(3), a trace looks like: Fibonacci(3) Fibonacci(2) Fibonacci(1) Fibonacci(0) Fibonacci(1) What would trace of Fibonacci(4) look like? 16-14 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Fibonacci: LC-2 Code Activation Record bookkeeping local return value return address dynamic link n temp arg Compiler generates temporary variable to hold result of first Fibonacci call. 16-15 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. LC-2 Code (part 1 of 3) Fibonacci STR R7, R6, #1 ; save ret addr LDR R0, R6, #3 ; load n BRz FIB_END ; check for ADD R0, R0, #-1 ; terminal cases BRz FIB_END ; temp = Fibonacci(n-1) LDR R0, R6, #3 ; calc n-1 ADD R0, R0, #-1 STR R0, R6, #8 ; store as arg STR R6, R6, #7 ; store dyn link ADD R6, R6, #5 ; push JSR Fibonacci ; call self LDR R0, R6, #5 ; store to temp STR R0, R6, #4 16-16 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. LC-2 Code (part 2 of 3) ; R0 = Fibonacci(n-2) LDR R0, R6, #3 ADD R0, R0, #-2 STR R0, R6, #8 STR R6, R6, #7 ADD R6, R6, #5 JSR Fibonacci LDR R0, R6, #5 ; return R0 + temp LDR R1, R6, #4 ADD R0, R0, R1 STR R0, R6, #0 LDR R7, R6, #1 LDR R6, R6, #2 RET ; calc n-2 ; store as arg ; store dyn link ; push ; call self ; store to return value ; restore R7, R6 16-17 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. LC-2 Code (part 3 of 3) FIB_END ; terminal: n is zero or one AND R0, R0, #0 ; set R0=1 ADD R0, R0, #1 STR R0, R6, #0 ; store to return value LDR R7, R6, #1 ; restore R7, R6 LDR R6, R6, #2 RET 16-18 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. A Final C Example: Printing an Integer Recursively converts an unsigned integer as a string of ASCII characters. • If integer <10, convert to char and print. • Else, call self on first (n-1) digits and then print last digit. void IntToAscii(int num) { int prefix, currDigit; if (num < 10) putchar(num + '0'); /* prints single char */ else { prefix = num / 10; /* shift right one digit */ IntToAscii(prefix); /* print shifted num */ /* then print shifted digit */ currDigit = num % 10; putchar(currDigit + '0'); } } 16-19 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Trace of IntToAscii Calling IntToAscii with parameter 12345: IntToAscii(12345) IntToAscii(1234) IntToAscii(123) IntToAscii(12) IntToAscii(1) putchar('1') putchar('2') putchar('3') putchar('4') putchar('5') 16-20