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Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013 Recursion: The Mirrors Chapter 2.

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Presentation on theme: "Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013 Recursion: The Mirrors Chapter 2."— Presentation transcript:

1 Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013 Recursion: The Mirrors Chapter 2

2 Contents Recursive Solutions Recursion That Returns a Value Recursion That Performs an Action Recursion with Arrays Organizing Data More Examples Recursion and Efficiency Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

3 Recursive Solutions Recursion breaks a problem into smaller identical problems Some recursive solutions are inefficient, impractical Complex problems can have simple recursive solutions Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

4 Recursive Solutions A recursive solution calls itself Each recursive call solves an identical, smaller problem Typically two parts:  Base case: Condition upon which does not call itself  Recursive case: Calls self with smaller problem(s) Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

5 A Recursive Valued Function The factorial of n Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

6 Characteristics of Recursive Solutions At least:  One base case and one recursive call to self Base case must be executed before recursive call Recursive functions must work with smaller problems (closer to base case) Must not skip over base case Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

7 A Recursive Valued Function FIGURE 2-2 fact(3) Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

8 The Box Trace FIGURE 2-3 A box FIGURE 2-4 The beginning of the box trace Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

9 The Box Trace FIGURE 2-5 Box trace of fact(3) Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

10 The Box Trace FIGURE 2-5 Box trace of fact(3) … continued Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

11 The Box Trace FIGURE 2-5 Box trace of fact(3) … continued Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

12 A Recursive Void Function FIGURE 2-6 A recursive solution Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

13 A Recursive Void Function Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013 The function writeBackwards

14 A Recursive Void Function FIGURE 2-7 Box trace of writeBackward("cat") Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

15 A Recursive Void Function FIGURE 2-7 writeBackward("cat") continued Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

16 A Recursive Void Function FIGURE 2-7 writeBackward("cat") continued Data Structures and Problem Solving with C++: Walls and Mirrors, Frank Carrano, © 2012

17 A Recursive Void Function FIGURE 2-8 Box trace of writeBackward("cat") in pseudocode Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

18 A Recursive Void Function FIGURE 2-8 Box trace of writeBackward("cat") in pseudocode (continued) Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013`

19 A Recursive Void Function FIGURE 2-8 Box trace of writeBackward("cat") in pseudocode (continued) Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

20 A Recursive Void Function FIGURE 2-8 Box trace of writeBackward("cat") in pseudocode (continued) Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

21 A Recursive Void Function FIGURE 2-8 Box trace of writeBackward("cat") in pseudocode (continued) Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

22 A Recursive Void Function FIGURE 2-8 Box trace of writeBackward("cat") in pseudocode (concluded) Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

23 A Recursive Void Function FIGURE 2-9 Box trace of writeBackward2("cat") in pseudocode Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

24 A Recursive Void Function FIGURE 2-9 Box trace of writeBackward2("cat") in pseudocode (continued) Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

25 A Recursive Void Function FIGURE 2-9 Box trace of writeBackward2("cat") in pseudocode (continued) Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

26 A Recursive Void Function FIGURE 2-9 Box trace of writeBackward2("cat") in pseudocode (continued) Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

27 A Recursive Void Function FIGURE 2-9 Box trace of writeBackward2("cat") in pseudocode (concluded) Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

28 Writing an Array’s Entries in Backward Order Pseudocode Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

29 Writing an Array’s Entries in Backward Order Source code Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

30 The Binary Search A high-level binary search for the array problem Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

31 The Binary Search Issues to consider 1.How to pass a half array to recursive call 2.How to determine which half of array has target value 3.What is the base case? 4.How will result of binary search be indicated? Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

32 The Binary Search FIGURE 2-10 Box traces of binarySearch with anArray = : (a) a successful search for 9 Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

33 The Binary Search FIGURE 2-10 Box traces of binarySearch with anArray = : (b) an unsuccessful search for 6 Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

34 The Binary Search FIGURE 2-11 Box trace with a reference argument Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

35 Finding the Largest Value in an Array Recursive algorithm Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

36 Finding the Largest Value in an Array FIGURE 2-12 Recursive solution to the largest-value problem Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

37 Finding the Largest Value in an Array FIGURE 2-13 The recursive calls that maxArray( ) generates Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

38 Organizing Data Towers of Hanoi FIGURE 2-16 (a) The initial state; (b) move n – 1 disks from A to C; Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

39 Towers of Hanoi FIGURE 2-16 (c) move 1 disk from A to B; (d) move n – 1 disks from C to B Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

40 Towers of Hanoi Pseudocode solution: Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

41 Towers of Hanoi FIGURE 2-17 The order of recursive calls that results from solve Towers(3, A, B, C) Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

42 Towers of Hanoi Source code for solveTowers Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

43 The Fibonacci Sequence (Multiplying Rabbits) Assumed “facts” about rabbits: Rabbits never die. A rabbit reaches sexual maturity exactly two months after birth Rabbits always born in male-female pairs. At the beginning of every month, each sexually mature male-female pair gives birth to exactly one male-female pair. Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

44 The Fibonacci Sequence (Multiplying Rabbits) Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013 MonthRabbit Population 1One pair 2One pair, still 3Two pairs 4Three pairs 5Five pairs 68 pairs

45 The Fibonacci Sequence (Multiplying Rabbits) FIGURE 2-18 Recursive solution to the rabbit problem Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

46 The Fibonacci Sequence (Multiplying Rabbits) A C++ function to compute rabbit(n) Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

47 The Fibonacci Sequence (Multiplying Rabbits) FIGURE 2-19 The recursive calls that rabbit(7) generates Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

48 Choosing k Out of n Things Recursive solution: Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

49 Choosing k Out of n Things Recursive function: Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

50 Choosing k Out of n Things FIGURE 2-20 The recursive calls that g (4, 2) generates Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

51 Recursion and Efficiency Inefficiency factors  Overhead associated with function calls  Inherent inefficiency of some recursive algorithms Fibonacci: re-compute previously computed solution Principle:  Do not use recursive solution if inefficient and clear, efficient iterative solution exists Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

52 End Chapter 2

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