Rethinking Recursion

Prof. Chung-Ta KingDepartment of Computer

ScienceNational Tsing Hua University

CS1103 電機資訊工程實習

(Contents from Dr. Jürgen Eckerle, Dr. Sameh Elsharkawy, Dr. David Reed,www.cs.cornell.edu/courses/cs211/2004fa/Lectures/Induction/induction.pdf)

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What Is Special about the Tree?

http://id.mind.net/~zona/mmts/geometrySection/fractals/tree/treeFractal.html

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How about This?

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And These?

They all defined/expressed in terms of themselves

Recursion

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How about This?

Robot factory

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Overview

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

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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

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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, …

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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 numberLIST 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

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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

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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 caseN! = 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) = 1Fibo(2) = 1Fibo(n) = Fibo(n–1) + Fibo(n–2)

The larger problem is a combination of two smaller problems

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Python Code

def factorial(n):if n == 1: return 1result = n * factorial(n-1)return result

for i in range(1,10): print factorial(i)

def fibonacci(N):if N <= 2: return 1return 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 definition:S 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 P[0], ..., P[n-1] [Conquer] Compute a solution S[i] of all the

subproblems P[i] [Merge] Merge all the subsolutions S[i] to a

solution S of P

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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 problemelse 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

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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, '->', 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 word1. remove that letter2. get the permutation of the remaining letters3. 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?

s

u v

yx

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4 67

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Shortest Path from s to v

Decompose into smaller subproblems

Combine and find the minimum

s

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yx

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2 394 6

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s

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yx

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2 394 6

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1 49

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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 > # 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

s

u v

yx

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2 394 6

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s

u

u y

x

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xv

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v

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v6

There are more efficient algorithms:e.g. Dijkstra’s 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 nn! = n * ((n-1) * … * ((n-1)/2+1)) * ((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 k+1.

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) => P(k+1): if property P is true for integer k, it

is true for integer k+1 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 > 0

SQ2(n) = n(n+1)(2n+1)/6

where they all calculate SQ(n) = 02+12+…+n2

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Inductive Proof

Let proposition be P(j): SQ1(j) = SQ2(j)

Two parts of proof: Prove P(0). Prove P(k+1) assuming P(k).

P(0) P(1) P(2) P(k) P(k+1)

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Inductive Proof

P(0): show SQ1(0) = SQ2(0) (easy) SQ1(0) = 0 = SQ2(0)

P(k) P(k+1): Assume SQ1(k) = SQ2(k) SQ1(k+1) = SQ1(k) + (k+1)2 (definition of SQ1)

= SQ2(k) + (k+1)2 (inductive assumption)

= k(k+1)(2k+1)/6 + (k+1)2 (definition of SQ2)=(k+1)(k+2)(2k+3)/6 (algebra)

= SQ2(k+1) (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 2n+1 x 2n+1 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 2k+1x 2k+1

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