Rethinking Recursion

1
Rethinking Recursion
CS1103 ????????

• Prof. Chung-Ta King
• Department of Computer Science
• National Tsing Hua University

(Contents from Dr. Jürgen Eckerle, Dr. Sameh
Elsharkawy, Dr. David Reed, www.cs.cornell.edu/cou
rses/cs211/2004fa/Lectures/Induction/induction.pdf
)
2
What Is Special about the Tree?
http//id.mind.net/zona/mmts/geometrySection/frac
tals/tree/treeFractal.html
3
How about This?
4
And These?
5
They all defined/expressed in terms of themselves

• Recursion

6
How about This?
Robot factory
7
Overview

• What is recursion?
• Why recursion?
• Recursive programming
• Recursion and iteration
• Recursion and induction

8
Recursion

• In mathematics and computer science, recursion is
  a method of defining functions in which the
  function being defined is applied within its own
  definition
• For example n! n?(n-1)!
• It is also used more generally to describe a
  process of repeating objects in a self-similar
  way

9
Recursion

• Recursion is a powerful technique for specifying
  functions, sets, and programs.
• Recursively-defined functions
• factorial
• counting combinations (choose r out of n items)
• differentiation of polynomials
• Recursively-defined sets
• language of expressions
• Recursively-defined graphs, images, puzzles,
  concepts,

10
Recursive Definitions

• Consider the following list of numbers
• 24, 88, 40, 37
• Such a list can be defined as follows
• A LIST is a number
• or a number comma LIST
• That is, a LIST is defined to be a single number,
  or a number followed by a comma followed by a
  LIST
• A more concise expression (Grammar)
• LIST ? number
• LIST ? number , LIST
• The concept of a LIST is used to define itself

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Recursive Definitions

• The recursive part of the LIST definition is used
  several times, terminating with the non-recursive
  part
• number comma LIST
• 24 , 88, 40, 37
• number comma LIST
• 88 , 40, 37
• number comma LIST
• 40 , 37
• number
• 37

12
Infinite Recursion

• All recursive definitions have to have a
  non-recursive part
• If they didn't, there would be no way to
  terminate the recursive path
• Such a definition would cause infinite recursion
• This problem is similar to an infinite loop, but
  the non-terminating "loop" is part of the
  definition itself
• The non-recursive part is often called the base
  case

13
Recursion

• Every recursive definition has 2 parts
• BASE CASE(S) case(s) so simple that they can be
  solved directly
• RECURSIVE CASE(S) make use of recursion to solve
  smaller subproblems and combine into a solution
  to the larger problem
• To verify that a recursive definition works
• Check if base case(s) are handled correctly
• ASSUME RECURSIVE CALLS WORK ON SMALLER PROBLEMS,
  then check that results from the recursive calls
  are combined to solve the whole

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Recursion

• N!, for any positive integer N, is defined to be
  the product of all integers between 1 and N
  inclusive
• This definition can be expressed recursively in a
  recurrence equation as
• 1! 1 ? base case
• N! N (N-1)! ? recursive case
• A factorial is defined in terms of another
  factorial
• Eventually, the base case of 1! is reached

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Recursion

• Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, 21,
• Fibo(1) 1
• Fibo(2) 1
• Fibo(n) Fibo(n1) Fibo(n2)
• The larger problem is a combination of two
  smaller problems

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Python Code

• def factorial(n) if n 1 return 1 result
  n factorial(n-1) return resultfor i in
  range(1,10) print factorial(i)
• def fibonacci(N)
• if N lt 2 return 1
• return fibonacci(N-1)fibonacci(N-2)
• for i in range(1,10) print fibonacci(i)

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Overview

• What is recursion?
• Why recursion?
• Recursive programming
• Recursion and iteration
• Recursion and induction

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Why Recursion?

• Concise in representation and expression
• For example any algebraic expression such as( x
  y ) ( x y ) x( x y ) x z y / ( x
  x )x z / y ( x ) ( y z ) ( y x z
  y / x )
• Can be expressed using the recursive
  definitionS ? x y z S S S S S S
  S/S (S)
• For example( x y ) ? ( S S ) ? ( S ) ? S

Recursion for composition and decomposition
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Why Recursion?

• Simplify solution ? divide and conquer
• Divide a given (complex) problem into a set of
  smaller problems and solve these and merge them
  to a complete solution
• If the smaller problems have the same structure
  as the originally problem, this problem solving
  process can be applied recursively.
• This problem solving process stops as soon as
  trivial problems are reached which can be solved
  in one step.
• Only need to focus on the smaller/simplified
  subproblems, instead of overwhelming by the
  (complex) original problem

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Divide and Conquer

• Method
• If the problem P is trivial, solve it. Otherwise
• Divide Divide P into a set of smaller problems
  P0, ..., Pn-1
• Conquer Compute a solution Si of all the
  subproblems Pi
• Merge Merge all the subsolutions Si to a
  solution S of P

21
Overview

• What is recursion?
• Why recursion?
• Recursive programming
• Recursion and iteration
• Recursion and induction

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Recursive Programming

• A procedure can call itself, perhaps indirectly
• General structure
• if stopping condition then solve base problem
• else use recursion to solve smaller problem(s)
• combine solutions from smaller problem(s)
  
• Each call to the procedure sets up a new
  execution environment (stack frame or activation
  record), with new parameters and local variables
• When the procedure completes, control returns to
  the calling procedure, which may be an earlier
  invocation of itself

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Tower of Hanoi

• Which is the base case?
• Which is the recursion case?

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Tower of Hanoi
25
Tower of Hanoi
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Python Code

• def hanoi(n, a'A', b'B', c'C')
• move n discs from a to c thru b
• if n 0
• return
• hanoi(n-1, a, c, b)
• print 'disc', n, '', a, '-gt', c
• hanoi(n-1, b, a, c)
• hanoi(3)

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Thinking Recursively

• 1. Find a way of breaking the given problem into
  smaller/simpler subproblems
• Tower of Hanoi largest disc remaining n-1
  discs
• 2. Relate the solution of the simpler subproblem
  with the solution of the larger problem
• Tower of Hanoi move n-1 discs to the middle peg
  move the largest disc to the destination peg
  move n-1 discs from the middle peg to the
  destination peg
• 3. Determine the smallest problem that cannot be
  decomposed any further and terminate there

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Generating Permutations

• Numerous applications require systematically
  generating permutations (orderings)
• Take some sequence of items (e.g. string of
  characters) and generate every possible
  arrangement without duplicates
• "123" ? "123", "132", "213", "231", "312",
  "321"

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Recursive Generation

• Recursive permutation generation for each letter
  in the word
• 1. remove that letter
• 2. get the permutation of the remaining letters
• 3. add the letter back at the front
• Example "123"
• "1" (1st permutation of "23") "1" "23"
  "123"
• "1" (2nd permutation of "23") "1" "32"
  "132"
• "2" (1st permutation of "13") "2" "13"
  "213"
• "2" (2nd permutation of "13") "2" "31"
  "231"
• "3" (1st permutation of "12") "3" "12"
  "312"
• "3" (2nd permutation of "12") "3" "21"
  "321"

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Tiled Pictures

• Consider the task of repeatedly displaying a set
  of images in a mosaic
• Three quadrants contain individual images
• Upper-left quadrant repeats pattern
• The base case is reached when the area for the
  images shrinks to a certain size

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Tiled Pictures
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Fractals

• A geometric shape made up of same pattern
  repeated in different sizes and orientations
• Koch curve
• A curve of order 0 is a straight line
• A curve of order n consists of 4 curve of order
  n-1

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Fractals

• Koch curve
• after five iteration steps (order 4 curve)

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Fractals

• Koch snowflake
• From 3 Koch curves of order 4

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Sierpinski Triangle

• A confined recursion of triangles to form a
  geometric lattice

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Shortest Path from s to v

• How to solve it with a recursive procedure?

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Shortest Path from s to v

• Decompose into smaller subproblems
• Combine and find the minimum

u
v
u
v
1
1
1
9
0

4


10
9
9
2
3
2
3
0
s
0
s
4
6
4
6
5


0

2
2
y
x
y
x
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Overview

• What is recursion?
• Why recursion?
• Recursive programming
• Recursion and iteration
• Recursion and induction

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Recursion vs. Iteration

• Iteration can be used in place of recursion
• (Nearly) every recursively defined problem can be
  solved iteratively
• Iterative optimization, e.g. by compiler, can be
  implemented after recursive design
• Recursive solutions are often less efficient, in
  terms of both time and space (next page)
• Recursion can simplify the solution of a problem,
  often resulting in shorter, more easily
  understood, correct source code
• What if we have multiple processors?

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Recursion and Redundancy

• Consider the recursive fibonacci method
• fib(5)
• fib(4) fib(3)
• fib(3) fib(2) fib(2) fib(1)
• fib(2) fib(1)
• SIGNIFICANT amount of redundancy in recursion
  recursive calls gt loop iterations (by an
  exponential amount!)
• Recursive version is often slower than iterative
  version

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Shortest Path from s to v

• The recursive procedure builds a search tree of
  an exponential complexity

0
s
5
10


x
u
2
1
2
3
9
v

u
y

x

v


6
v
There are more efficient algorithms e.g.
Dijkstras algorithm (O(n2))

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Divide-and-Conquer Again

• What have we done with n! in terms of
  divide-and-conquer?
• n! n (n 1)!
• We only divide one number off in each recursion
  ? the two subproblems are imbalanced
• Can we do this?
• n! (n (n-1) (n/2 1)) (n/2)! even
  n
• n! n ((n-1) ((n-1)/21)) ((n-1)/2)!
  odd n
• Very much like a binary tree
• Any difference between the two?

Tail recursion
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When Recursion?

• When it is the most natural way of thinking about
  and implementing a solution
• can solve problem by breaking into smaller
  instances, solve, combine solutions
• When it is roughly equivalent in efficiency to an
  iterative solution, orWhen the problems to be
  solved are so small that efficiency doesn't
  matter
• think only one level deep
• make sure the recursion handles the base case(s)
  correctly
• assume recursive calls work correctly on smaller
  problems
• make sure solutions to the recursive problems are
  combined correctly
• avoid infinite recursion
• make sure there is at least one base case each
  recursive call gets closer

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Overview

• What is recursion?
• Why recursion?
• Recursive programming
• Recursion and iteration
• Recursion and induction

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Recursion and Induction

• Recursion
• A solution strategy that solves a large problem
  by breaking it up into smaller problems of same
  kind
• A concept that make something by itself, e.g.
  tools that make tools, robots that make robots,
  Droste pictures
• Induction
• A mathematical strategy for proving statements
  about integers (more generally, about sets that
  can be ordered in some fairly general ways)
• Understanding induction is useful for figuring
  out how to write recursive code.

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Prove Inductively

• Assume equally spaced dominoes, where spacing is
  less than domino length.
• How would you argue that all dominoes would fall?
• Domino 0 falls because we push it over.
• Domino 1 falls because domino 0 falls, domino 0
  is longer than inter-domino spacing, so it knocks
  over domino 1
• Domino 2 falls because
• Is there a more compact argument?

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Prove Inductively

• Better argument
• Domino 0 falls because we push it over.
• Suppose domino k falls over. Because its length
  is larger than inter-domino spacing, it will
  knock over domino k1.
• Therefore, all dominoes will fall over.
• This is an inductive argument.
• Not only is it more compact, but it works even
  for an infinite number of dominoes!

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Induction over Integers

• We want to prove that some property P holds for
  all integers.
• Inductive argument
• P(0) show that property P is true for integer 0
• P(k) gt P(k1) if property P is true for integer
  k, it is true for integer k1
• This means P(n) holds for all integers n

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Consider This

• Can we show that these two definitions of SQ(n)
  are equal?
• SQ1(0) 0 for n 0
• SQ1(n) SQ1(n-1) n2 for n gt 0
• SQ2(n) n(n1)(2n1)/6
• where they all calculate SQ(n) 0212n2

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Inductive Proof

• Let proposition be P(j) SQ1(j) SQ2(j)
• Two parts of proof
• Prove P(0).
• Prove P(k1) assuming P(k).

P(0)
P(1)
P(2)
P(k)
P(k1)
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Inductive Proof

• P(0) show SQ1(0) SQ2(0)
• (easy) SQ1(0) 0 SQ2(0)
• P(k) ? P(k1)
• Assume SQ1(k) SQ2(k)
• SQ1(k1) SQ1(k) (k1)2 (definition of SQ1)
• SQ2(k) (k1)2 (inductive assumption)
• k(k1)(2k1)/6 (k1)2 (definition of SQ2)
• (k1)(k2)(2k3)/6 (algebra)
• SQ2(k1) (definition of SQ2)

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Consider the Tiling Problem

• Problem
• A chess-board has one square cut out of it.
• Can the remaining board be tiled using tiles of
  the shape shown in the picture?
• Not obvious that we can use induction to solve
  this problem.

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Idea

• Consider boards of size 2n x 2n for n 1,2,..
• Base case show that tiling is possible for 2 x 2
  board.
• Inductive case assuming 2n x 2n board can be
  tiled, show that 2n1 x 2n1 board can be tiled.
• Chess-board (8x8) is a special case of this
  argument.

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Base Case

• For a 2x2 board, it is trivial to tile the board
  regardless of which one of the four pieces has
  been cut.

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4x4 Case

• Divide 4x4 board into four 2x2 sub-boards.
• One of the four sub-boards has the missing piece.
  That sub-board can be tiled since it is a 2x2
  board with a missing piece.
• Tile the center squares of the three remaining
  sub-boards as shown.
• This leaves 3 2x2 boards with a missing piece,
  which can be tiled.

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Inductive Proof

• Claim Any board of size 2n x 2n with one missing
  square can be tiled.
• Base case (n 1) trivial
• Inductive case assume inductive hypothesis for
  (n k) and consider board of size 2k1x 2k1
• Divide board into four equal sub-boards of size
  2k X 2k
• One of the sub-boards has the missing piece by
  inductive assumption, this can be tiled.
• Tile the central squares of the remaining three
  sub-boards
• This leaves three sub-boards with a missing
  square each, which can be tiled by inductive
  assumption.

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When Induction Fails

• Proposition any n x n board with one missing
  square can be tiled
• Problem a 3 x 3 board with one missing square
  has 8 remaining squares, but our tile has 3
  squares ? tiling is impossible
• Therefore, any attempt to give an inductive proof
  of the proposition must fail
• This does not say anything about the 2n x 2n cases