06 list processing

Kyung-Goo Doh

Hanyang University - ERICA Computer Science & Engineering

Functional Programming / Imperative ProgrammingCSE215 Fundamentals of Program Design

CSE215 Fundamentals of Program Design Fall2009

06

List Processing

Version 2.0

Hanyang University - ERICA


Computer Science & Engineering

CSE215 Fundamentals of Program Design06 List Processing

Fall 2009Kyung-Goo Doh

Predefined Polymorphic Lists# [] ;; (+ Nil +)- : 'a list = []# 1::[] ;; (+ Cons +)- : int list = [1]# true::[];;- : bool list = [true]# 1::2::3::4::5::[] ;;- : int list = [1; 2; 3; 4; 5]# [1;2;3;4;5] ;;- : int list = [1; 2; 3; 4; 5]# 1::2::[3;4;5] ;;- : int list = [1; 2; 3; 4; 5]# List.hd [1;2;3;4;5] ;; (+ head +)- : int = 1# List.tl [1;2;3;4;5] ;; (+ tail +)- : int list = [2; 3; 4; 5]# [1;2;3]@[4;5] ;; (+ append +)- : int list = [1; 2; 3; 4; 5]






Standard Library: List






Definition: Lists in OCaml

Structural Inductive Definition of Lists of type ‘a (basis) An empty list, [], is a list of type ‘a.

(induction) If xs is a list of values of type ‘a and x is a value of type ‘a, then so is x::xs.

Nothing else is a list.

Structural Induction1. is P([]) true?

2. Assume P(xs) is true, is P(x::xs) true?

3. If we can say “yes” to these two questions, then we can assure that P(xs) is true for all lists xs.






Computing List Length# let rec length xs = match xs with | [] -> 0 | _::xs -> 1 + length xs ;;val length : 'a list -> int = <fun># length [3;3;3;3;3;3;3;3;3;3;3;3;3;3;3;3;3;3] ;;- : int = 18# length [] ;;- : int = 0

# let length xs = let rec loop xs n = match xs with | [] -> n | _::xs -> loop xs (n+1) in loop xs 0 ;;val length : 'a list -> int = <fun>

Prim

itive

Tail






Computation Trace (Primitive list length)

length [1;2;3;4;5] 1 + length [2;3;4;5] 1 + (1 + length [3;4;5]) 1 + (1 + (1 + length [4;5])) 1 + (1 + (1 + (1 + length [5]))) 1 + (1 + (1 + (1 + (1 + length [])))) 1 + (1 + (1 + (1 + (1 + 0)))) 1 + (1 + (1 + (1 + 1))) 1 + (1 + (1 + 2)) 1 + (1 + 3) 1 + 4 5

Complexity- time: θ(n)- space: θ(n)

let rec length xs = match xs with | [] -> 0 | _::xs -> 1 + length xs






Computation Trace (Tail list length)

length [1;2;3;4;5] loop [1;2;3;4;5] 0 loop [2;3;4;5] 1 loop [3;4;5] 2 loop [4;5] 3 loop [5] 4 loop [] 5 5

Complexity- time: θ(n)- space: θ(1)

let length xs = let rec loop xs n = match xs with | [] -> n | _::xs -> loop xs (n+1) in loop xs 0






Verification (Primitive list length)

Theorem : The primitive recursive function call length xs evaluates to the length of xs, |xs|, for all lists xs.

Proof: We prove by structural induction on xs. (Basis) xs = [], length [] = 0 by definition of program ‘length’ = |[]| by inspection (Induction hypothesis) For some list xs of length ≥ 0, length xs evaluates to |xs| (Induction step) It suffices to show that length (x::xs) evaluates to |x::xs|

length (x::xs) = 1 + length xs by definition of program length

= 1 + |xs| by induction hypothesis = |x::xs| by inspection

let rec length xs = match xs with | [] -> 0 | _::xs -> 1 + length xs






Verification (Tail list length)

Theorem : The tail recursive function call length xs evaluates to the length of xs, |xs|, for all lists xs.

Proof: We first show that loop xs n evaluates to n+|xs| by structural induction on xs. (Basis) xs = [], loop [] n = n by definition of program ‘loop’ = n + 0 by algebra = n + |[]| by inspection (Induction hypothesis) For some list xs of length ≥ 0, loop xs n evaluates to n+|

xs| (Induction step) It suffices to show that loop (x::xs) n evaluates to n+|x::xs|

loop (x::xs) n = loop xs (n+1) by definition of program loop

= n+1+|xs| by induction hypothesis = n+|x::xs| by inspectionThus length xs = loop xs 0 evaluates to 0+|xs| = |xs|.

let length xs = let rec loop xs n = match xs with | [] -> n | _::xs -> loop xs (n+1) in loop xs 0






Computing List Append (* Concatenate two lists next to each other *)# let rec append xs ys = match xs with | [] -> ys | x::xs -> x::append xs ys ;;val append : 'a list -> 'a list -> 'a list = <fun>

Prim

itive

append [1;2;3] [4;5] 1 :: append [2;3] [4;5] 1 :: 2 :: append [3] [4;5] 1 :: 2 :: 3 :: append [] [4;5] 1 :: 2 :: 3 :: [4;5] 1 :: 2 :: [3;4;5] 1 :: [2;3;4;5] [1;2;3;4;5]

Execution Trace

Complexity- time: θ(n)- space: θ(n)

@ is the infix version of this append function






Verification (Primitive list append)

Theorem : The primitive recursive function call append xs ys evaluates to the concatenation of lists xs and ys, xs@ys, for all lists xs and ys.

Proof: We prove by structural induction on xs. (Basis) xs = [], append [] ys = ys by definition of program

‘append’ = []@ys by property of concatenation (Induction hypothesis) For some list xs of length ≥ 0, append xs ys evaluates to

xs@ys (Induction step) It suffices to show that append (x::xs) ys evaluates to (x::xs)@ys

append (x::xs) ys = x :: append xs ys by definition of program ‘append’

= x :: (xs@ys) by induction hypothesis = (x::xs)@ys by property of :: and

concatenation

let rec append xs ys = match xs with | [] -> ys | x::xs -> x::append xs ys






Computing List Reverse

(* Rearrange a list in reverse order*)# let rec rev xs = match xs with | [] -> [] | x::xs -> rev xs @ [x] ;;val rev : 'a list -> 'a list = <fun>

Prim

itive

Complexity- time: θ(n2)- space: θ(n)

rev [1;2;3;4] rev [2;3;4] @ [1] rev [3;4] @ [2] @ [1] rev [4] @ [3] @ [2] @ [1] rev[] @ [4] @ [3] @ [2] @ [1] [] @ [4] @ [3] @ [2] @ [1] [4] @ [3] @ [2] @ [1] [4;3] @ [2] @ [1] [4;3;2] @ [1] [4;3;2;1]

Execution Trace






Verification (Primitive list reverse)

Theorem : The primitive recursive function call reverse xs evaluates to the reverse of xs, ~xs, for all lists xs.

Proof: We prove by structural induction on xs. (Basis) xs = [], rev [] = [] by definition of program ‘rev’ = ~[] by inspection (Induction hypothesis) For some list xs of length ≥ 0, rev xs evaluates to ~xs (Induction step) It suffices to show that rev (x::xs) evaluates to ~(x::xs)

rev (x::xs) = rev xs @ [x] by definition of program ‘rev’ = ~xs @ [x] by induction hypothesis = ~(x::xs) by inspection

let rec rev xs = match xs with | [] -> [] | x::xs -> rev xs @ [x]






Computing List Reverse

(* Rearrange a list in reverse order*)# let rev xs = let rec loop xs ys = match xs with | [] -> ys | x::xs -> loop xs (x::ys) in loop xs [] ;;val rev : 'a list -> 'a list = <fun>

Tail


rev [1;2;3;4] loop [1;2;3;4] [] loop [2;3;4] [1] loop [3;4] [2;1] loop [4] [3;2;1] loop [] [4;3;2;1] [4;3;2;1]

Execution Trace






Verification (Tail list reverse)

Theorem : The tail recursive function call rev xs evaluates to the reverse of xs, ~xs, for all lists xs.

Proof: We first show that loop xs ys evaluates to ~xs@ys by structural induction on xs. (Basis) xs = [], loop [] ys = ys by definition of program ‘loop’ = []@ys by property of @ = ~[]@ys by inspection (Induction hypothesis) For some list xs of length ≥ 0, loop xs ys evaluates to

~xs@ys (Induction step) It suffices to show that loop (x::xs) ys evaluates to ~(x::xs)@ys

loop (x::xs) ys = loop xs (x::ys) by definition of program ‘loop’

= ~xs@(x::ys) by induction hypothesis = ~xs@[x]@ys = ~(x::xs)@ys by inspectionThus rev xs = loop xs [] evaluates to ~xs@[] = ~xs

let rev xs = let rec loop xs ys = match xs with | [] -> ys | x::xs -> loop xs (x::ys) in loop xs []






Computing List Append# let append xs ys = let rec loop xs ys = match xs with | [] -> ys | x::xs -> loop xs (x::ys) in loop (rev xs) ys ;;val append : 'a list -> 'a list -> 'a list = <fun>

Tail

append [1;2;3] [4;5] loop (rev [1;2;3]) [4;5] loop (loop [1;2;3] []) [4;5] loop (loop [2;3] [1]) [4;5] loop (loop [3] [2;1]) [4;5] loop (loop [] [3;2;1]) [4;5] loop [3;2;1] [4;5] loop [2;1] [3;4;5] loop [1] [2;3;4;5] loop [] [1;2;3;4;5] [1;2;3;4;5]

Execution Trace







Verification (Tail list append)

Theorem : The tail recursive function call append xs ys evaluates to the concatenation of xs and ys, xs@ys, for all lists xs and ys.

Proof: The ‘loop’ function here is identical to the ‘loop’ function in the tail recursive version of ‘rev’. Thus we already proved that loop xs ys evaluates to ~xs@ys [1] . append xs ys = loop (rev xs) ys by definition of ‘append’

= loop ~xs ys by previous theorem = ~(~xs)@ys by [1] = xs@ys by property of reverse

let append xs ys = let rec loop xs ys = match xs with | [] -> ys | x::xs -> loop xs (x::ys) in loop (rev xs) ys






Sorting

sort : ‘a list -> ‘a list input: the list of values of type output: the rearranged list of values in ascending order.






Selection Sort

Algorithm1. Initialization

unsorted list = input list

sorted list = empty

2. Find the minimum value in the unsorted list.

3. Move it to the back of the sorted list.

4. Repeat the steps 2~3 until the unsorted list is empty.






Animation Selection Sort






Program Selection Sort

(* find_min : ‘a list -> ‘a * ‘a list *)let rec find_min xs = match xs with | y::[] -> (y,[]) | y::ys -> let (m,zs) = find_min ys in if y<m then (y,ys) else (m,y::zs) | [] -> failwith "This message will never be displayed"

(* selection_sort : 'a list -> 'a list *)let rec selection_sort xs = match xs with | [] -> [] | _ -> let (m,ys) = find_min xs in m :: selection_sort ys






Insertion Sort

Algorithm1. Initialization

unsorted list = input list

sorted list = empty

2. Remove a value from the unsorted list.

3. Insert it at the correct position in the sorted list.

4. Repeat the steps 2~3 until the unsorted list is empty.






Animation Insertion Sort






Program Insertion Sort

(* insert : 'a list -> 'a -> 'a list *)let rec insert xs x = match xs with | [] -> [x] | y::ys -> if y<x then y :: insert ys x else x :: xs

(* insertion_sort : 'a list -> 'a list *)let insertion_sort xs = let rec loop sorted unsorted = match unsorted with | [] -> sorted | y::ys -> loop (insert sorted y) ys in loop [] xs






Merge Sort

Algorithm1. If the list is of length 0 or 1, then it is already sorted.

2. Otherwise:

Split the unsorted list into two sublists of about half the size.

Sort each sublist recursively by re-applying merge sort.

Merge the two sorted sublist back into one sorted list.






Animation Merge Sort






Program Merge Sort ~

(* split : 'a list -> 'a list * 'a list *)let rec split xs = match xs with | [] -> ([],[]) | [x] -> ([x],[]) | x::xs' -> let (ls,rs) = split xs' in if List.length ls <= List.length rs then (x::ls,rs) else (ls,x::rs)

(* merge : 'a list -> 'a list -> 'a list *)let rec merge xs ys = match (xs,ys) with | ([],_) -> ys | (_,[]) -> xs | (x::xs',y::ys') -> if x<y then x :: merge xs' ys else y :: merge xs ys'






Program Merge Sort

(* merge_sort : 'a list -> 'a list *)let rec merge_sort xs = match xs with | [] -> [] | [x] -> [x] | _ -> let (ls,rs) = split xs in merge (merge_sort ls) (merge_sort rs)






Quicksort

Algorithm1. Pick an element, called a pivot, from the list.

2. [Partition] Reorder the list so that all elements which are less than the pivot come

before the pivot and

so that all elements greater than the pivot come after it

3. Recursively sort the sublist of lesser elements and the sublist of greater elements.






Animation Quicksort






Program Quicksort

(* quicksort : 'a list -> 'a list *)let rec quicksort xs = match xs with | [] -> [] | pivot::xs' -> let (ls,rs) = partition pivot xs' in (quicksort ls) @ [pivot] @ (quicksort rs)

(* partition : 'a -> 'a list -> 'a list * 'a list *)let partition pivot xs = (List.filter (fun x -> x < pivot) xs, List.filter (fun x -> pivot <= x) xs)

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