Recursion

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Recursion
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Overview

• Learn about recursive definitions
• Explore the base case and the general case of a
  recursive definition
• Discover recursive algorithms
• Learn about recursive functions
• Explore how to use recursive functions to
  implement recursive algorithms
• Learn about recursion and backtracking

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Recursive Definitions

• Recursion
• The process of solving a problem by reducing it
  to a smaller versions of itself
• Recursive definition
• Definition in which a problem is expressed in
  terms of a smaller version of itself
• Has one or more base cases

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Recursive Definitions

• Recursive algorithm
• Algorithm that finds the solution to a given
  problem by reducing the problem to smaller
  versions of itself
• Has one or more base cases
• Can be implemented using recursive functions
• Recursive function
• function that calls itself
• Base case
• Case in recursive definition (or function) in
  which the solution is obtained directly (without
  making a recursive function call)
• Halts the recursive function calling
• Without a base case, recursion would continue
  endlessly

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Recursive Definitions

• General solution
• Breaks problem into smaller versions of itself
• General case
• Case in recursive definition in which a smaller
  version of itself is called
• Must eventually be reduced to a base case

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Tracing a Recursive Function

• Every recursive call has
• its own code
• own set of parameters
• own set of local variables
• After completing recursive call
• Control goes back to calling environment
• Recursive call must execute completely before
  control goes back to previous call
• Execution in previous call begins from point
  immediately following recursive call

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Recursive Definitions

• Directly recursive a function that calls itself
• Indirectly recursive a function that calls
  another function and eventually results in the
  original function call
• Tail recursive function recursive function in
  which the last statement executed is the
  recursive call
• Infinite recursion the case where every
  recursive call results in another recursive call

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Designing Recursive Functions

• Understand problem requirements
• Determine limiting conditions
• Identify base cases
• Provide direct solution to each base case
• Identify general case(s)
• Provide solutions to general cases in terms of
  smaller versions of itself

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Recursive Factorial Function

• int fact(int num)
• if(num lt 0)
• return 1
• else
• return num fact(num 1)

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Largest Value in Array

• int largest(const int list, //in
• int lowerIndex, //in
• int upperIndex //in
• )
• int max
• if(lowerIndex upperIndex)
• return listlowerIndex
• else
• max largest(list, lowerIndex 1,
  upperIndex)
• if(listlowerIndex gt max)
• return listlowerIndex
• else
• return max

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Towers of Hanoi Problem with 3 Disks
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Towers of Hanoi Three Disk Solution
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Towers of Hanoi Three Disk Solution
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Towers of Hanoi Recursive Algorithm

• void moveDisks(int count, int source, int target,
  int spare)
• if(count gt 0)
• moveDisks(count - 1, source, spare,
  target)
• coutltlt"Move disk "ltltcountltlt from "ltltsource
• ltlt to "ltlttargetltlt"."ltltendl
• moveDisks(count - 1, spare, target,
  source)

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Decimal to Binary

• void decToBin(int num, int base)
• if(num gt 0)
• decToBin(num/base, base)
• coutltltnum base

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Recursion or Iteration?

• Two ways to solve particular problem
• Iteration
• Recursion
• Iterative control structures uses looping to
  repeat a set of statements
• Tradeoffs between two options
• Sometimes recursive solution is easier
• Recursive solution is often slower

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Backtracking Algorithm

• Attempts to find solutions to a problem by
  constructing partial solutions
• Makes sure that any partial solution does not
  violate the problem requirements
• Tries to extend partial solution towards
  completion

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Backtracking Algorithm

• If it is determined that partial solution would
  not lead to solution
• partial solution would end in dead end
• algorithm backs up by removing the most recently
  added part and then tries other possibilities

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Solution to 8-Queens Puzzle
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4-Queens Puzzle
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4-Queens Tree