M.C. Escher: "Drawing Hands" (1948)

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M.C. Escher"Drawing Hands" (1948)
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Recursion

• Dave Feinberg

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Computation

• All computation consists of chugging along from
  state to state to state ...
• There is a set of rules that tell us, given the
  current state, which state to go to next.

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Arithmetic as Rewrite Rules

• 2 3 4
• 5 4
• 9
• We stop when we reach a number.

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Functions as New Rules

• def square(n)
• return n n
• When we see square(something)
• Rewrite it as something something

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Functions as Rewrite Rules

• def square(n)
• return n n
• square(3)
• 3 3
• 9

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Piecewise Functions

• f(n) 1 if n 1
• n - 1 if n gt 1
• f(4)
• 4 - 1
• 3

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In Python

• def f(n)
• if n 1
• return 1
• else
• return n - 1

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This is just math, right?

• Difference between mathematical functions and
  computation functions. Computation functions must
  be effective.
• For example, we can define the square-root
  function asvx the y such that y 0 and y²
  x
• This defines a valid mathematical function, but
  it doesn't tell us how to compute the square root
  of a given number.

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Fancier Functions

• def f(n)
• return n (n - 1)
• Find f(4)

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Fancier Functions

• def f(n)
• return n (n - 1)
• def g(n)
• return n f(n - 1)
• Find g(4)

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Fancier Functions

• def f(n)
• return n (n - 1)
• def g(n)
• return n f(n - 1)
• def h(n)
• return n h(n - 1)
• Find h(4)

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Recursion

• def h(n)
• return n h(n - 1)
• h is a recursive function,because it is defined
  in terms of itself.

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Definition

• Recursion
• See "Recursion".

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Recursion

• def h(n)
• return n h(n - 1)
• h(4)
• 4 h(3)
• 4 3 h(2)
• 4 3 2 h(1)
• 4 3 2 1 h(0)
• 4 3 2 1 0 h(-1)
• 4 3 2 1 0 -1 h(-2)
• ...
• Evaluating h leads to an infinite loop!

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What you are thinking

• "Ok, recursion is bad.What's the big deal?"

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Recursion

• def f(n)
• if n 1
• return 1
• else
• return f(n - 1)
• Find f(1)
• Find f(2)
• Find f(3)
• Find f(100)

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Recursion

• def f(n)
• if n 1
• return 1
• else
• return f(n - 1)
• f(3)
• f(3 - 1)
• f(2)
• f(2 - 1)
• f(1)
• 1

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Terminology

• def f(n)
• if n 1
• return 1
• else
• return f(n - 1)
• "Useful" recursive functions have
• at least one recursive case
• at least one base caseso that the computation
  terminates

base case
recursive case
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Recursion

• def f(n)
• if n 1
• return 1
• else
• return f(n 1)
• Find f(5)
• We have a base case and a recursive case. What's
  wrong?

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Recursion

• The recursive case
• should call the function
• on a simpler input,
• bringing us closer and closer
• to the base case.

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Recursion

• def f(n)
• if n 0
• return 0
• else
• return 1 f(n - 1)
• Find f(0)
• Find f(1)
• Find f(2)
• Find f(100)

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Recursion

• def f(n)
• if n 0
• return 0
• else
• return 1 f(n - 1)
• f(3)
• 1 f(2)
• 1 1 f(1)
• 1 1 1 f(0)
• 1 1 1 0
• 3

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Factorial

• 4! 4 3 2 1 24

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Factorial

• Does anyone know the value of 9!
• 362,880
• Does anyone know the value of 10!
• How did you know?

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Factorial

• 9! 987654321
• 10! 10 987654321
• 10! 10 9!
• n! n (n - 1)!
• That's a recursive definition!

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Factorial

• def fact(n)
• return n fact(n - 1)
• fact(3)
• 3 fact(2)
• 3 2 fact(1)
• 3 2 1 fact(0)
• 3 2 1 0 fact(-1)
• ...

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Factorial

• What did we do wrong?
• What is the base case for factorial?

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Factorial

• n! of ways to seat n people in a room with n
  chairs.
• How many ways are there to seat 3 people in a
  room with 3 chairs?

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Factorial

• There are 3! 3 2 1 6 waysto seat 3
  people in a room with 3 chairs.

A
B
C
A
C
B
B
A
C
B
C
A
C
A
B
C
B
A
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Factorial

• There are 2! 2 1 2 waysto seat 2 people in
  a room with 2 chairs.

A
B
B
A
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Factorial

• There is only 1! 1 wayto seat 1 person in a
  room with 1 chair.
• How many ways are there to seat 0 people in a
  room with 0 chairs?

A
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Factorial

• There is 1 way to seat 0 people in a room with 0
  chairs.
• This is why 0! is defined to be 1.
• That's our base case!

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Factorial

• def fact(n)
• if n 0
• return 1
• else
• return n fact(n - 1)
• fact(3)
• 3 fact(2)
• 3 2 fact(1)
• 3 2 1 fact(0)
• 3 2 1 1
• 6

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Writing Recursive Functions

• Write if. Why?
• Base case Test for and handle simplest case.
• Recursive case Call function on simpler input.
• Assume the recursive call works. How does it
  help us?

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Honey Bees

• Drone son of queen
• Queen daughter of queen drone
• (Worker daughter of queen drone)

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Honey Bees

• 1. D individual
• 2. Q parent
• 3. QD grandparents
• 4. QDQ great grandparents
• Rewrite rules D ? Q
• Q ? QD

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Honey Bees

• 1. D individual
• 2. Q parent
• 3. QD grandparents
• 4. QDQ great grandparents
• 5. QDQQD
• 6. QDQQDQDQ
• 7. QDQQDQDQQDQQD
• How many in each generation?

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Honey Bees

• 1. D 1
• 2. Q 1
• 3. QD 2
• 4. QDQ 3
• 5. QDQQD 5
• 6. QDQQDQDQ 8
• 7. QDQQDQDQQDQQD 13
• It's the Fibonacci sequence!

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Fibonacci

• In the Fibonacci sequence,each term sum of
  previous 2 terms
• Let fib(n) be the nth term.
• Then, fib(n) fib(n - 1) fib(n - 2)
• The sequence is defined recursively!

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Fibonacci

• def fib(n)
• return fib(n - 1) fib(n - 2)
• Find fib(1)
• We need a base case!

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Fibonacci

• def fib(n)
• if n 1
• return 1
• else
• return fib(n - 1) fib(n - 2)
• Find fib(1)
• Find fib(2)
• How do we fix our function?

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Fibonacci

• def fib(n)
• if n lt 2
• return 1
• else
• return fib(n - 1) fib(n - 2)
• Find fib(1)
• Find fib(2)
• Find fib(3)

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Tiling Squares

• Rewrite rule Add square to long side.

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Tiling Squares
What is the side length of each square?
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Tiling Squares
5
8
21
1
1
3
2
13
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Spiral
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Fibonacci

• 1 1 1
• 2 1 2
• 3 2 1.5
• 5 3 1.666...
• 8 5 1.6
• 13 8 1.625
• 21 13 1.615...
• 34 21 1.619...

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Limit
fib(n) fib(n - 1)

• What is the limit ofas n approaches infinity?
• 1.6180339887498948482...
• What's that called?

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The Golden Ratio

• The proportions of a rectangle that,when a
  square is added to itresults in a rectanglewith
  the same proportions.

Square
Square


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The Golden Ratio
f2 - f - 1 0
f 1.618...
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Fibonacci

• fib(n) 1 n 1, 2
• fib(n-1) fib(n-2) n gt 2
• fib(n)

fn - (1 - f)n v5
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Top-Down Computation
fib(5)
fib(4)
fib(3)
fib(3)
fib(2)
fib(2)
fib(1)
fib(2)
fib(1)
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Overlapping Subproblems
fib(5)
fib(4)
fib(3)
fib(3)
fib(2)
fib(2)
fib(1)
fib(2)
fib(1)
fib(5)
1 time
fib(4)
1 time
fib(3)
2 times
fib(2)
3 times
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Bottom-Up Computation

• def fib(n)
• if n lt 2
• return 1
• else
• return fib(n - 1) fib(n - 2)
• To find fib(5), first find fib(1) ...

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Bottom-Up Computation

• def fib(n)
• if n lt 2
• return 1
• else
• return fib(n - 1) fib(n - 2)

n
fib(n)
1
1
2
1
3
2
4
3
5
5
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Worksheet!

• Yay.

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Why Recursion?

• Why not just use for/while loops, instead of
  recursion?
• 1. It's good to know that you don't need
  for/while to program loops.
• 2. Many problems are easier to solve recursively
  than with for/while loops.

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Merge Sort

• If only 1 element
• Do nothing.
• Otherwise
• Split the list in half.
• Merge sort each half.
• Merge the sorted halves.

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Forestry

• (Demo)

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Forestry

• Problem Identify contiguous green squares on
  the grid, and change their color.
• Story We're determining which trees get burned
  down if a given tree catches fire.

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Related Problems

• using the "paint" tool to fill a region
• clearing out the 0's in Minesweeper

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Forest Fire

• Let's Code!

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Solving Recursively

• Key Idea Change color of current location.
  Then recursively burn the 4 neighboring trees.
• Base Case location is outside grid, or not green

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The Code

• def burn(grid, row, col)
• if isvalid(grid, row, col)
• if getcolor(grid, row, col) "green"
• setcolor(grid, row, col, "red")
• burn(grid, row - 1, col)
• burn(grid, row, col - 1)
• burn(grid, row, col 1)
• burn(grid, row 1, col)

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Big Ideas

• Comp Sci focuses on how to compute.
• Computation involves chugging along from state to
  state.
• Rewrite rules tell us what state to go to next.
• Recursive functions are defined in terms of
  themselves.
• Useful recursive functions have a base case(s)
  and a recursive case(s).