# E-99: 21-30

The next installment of the 99 Lisp problems. 27 and 28 are incomplete as I have not sat down to actually work through them yet. More research is needed to do multinomial coefficients. This batch of problems posed some challenge and required a bit of research and dusting off math skills, as well as getting familiar with Erlang's List Comprehension syntax. For your viewing pleasure:

%% Insert an element at a given position into a list. %% Example: (insert-at 'alfa '(a b c d) 2) -> (A ALFA B C D) p21(Elem, L, Pos) -> p21insert(Elem, L, Pos, []). p21insert(Elem, L, 1, NewL) -> lists:reverse([Elem | NewL]) ++ L; p21insert(Elem, [H | L], Pos, NewL) -> p21insert(Elem, L, Pos-1, [H | NewL]). %% Create a list containing all integers within a given range. %% If first argument is smaller than second, produce a list in decreasing order. %% Example: (range 4

- -> (4 5 6 7 8 9) p22(Start, End) when Start < End ->
p22range\

_{asc}(End-Start, [Start]); p22(Start, End) -> p22range\_{desc}(Start-End, [Start]). p22range\_{asc}(0, L) -> lists:reverse(L); p22range\_{asc}(Count, [H | L]) -> p22range\_{asc}(Count-1, [H+1, H | L]). p22range\_{desc}(0, L) -> lists:reverse(L); p22range\_{desc}(Count, [H | L]) -> p22range\_{desc}(Count-1, [H-1, H | L]). %% Extract a given number of randomly selected elements from a list. %% The selected items shall be returned in a list. %% Example: (rnd-select '(a b c d e f g h) 3) -> (E D A) %% Hint: Use the built-in random number generator and the result of problem P20. p23(L, Count) when length(L) < Count -> p23rnd\_{select}(L, length(L), []); p23(L, Count) -> p23rnd\_{select}(L, Count, []). p23rnd\_{select}(\_{L}, 0, RandL) -> RandL; p23rnd\_{select}(L, Count, RandL) -> Rnd = random:uniform(length(L)), V = lists:nth(Rnd, L), p23rnd\_{select}(p20(L, Rnd), Count-1, [V | RandL]). %% Lotto: Draw N different random numbers from the set 1..M. %% The selected numbers shall be returned in a list. %% Example: (lotto-select 6 49) -> (23 1 17 33 21 37) %% Hint: Combine the solutions of problems P22 and P23. p24(Num, High) -> p23(p22(1, High), Num). %% Generate a random permutation of the elements of a list. %% Example: (rnd-permu '(a b c d e f)) -> (B A D C E F) %% Hint: Use the solution of problem P23. p25(L) -> p23(L, length(L)). %% Generate the combinations of K distinct objects chosen from the N elements of a list %% In how many ways can a committee of 3 be chosen from a group of 12 people? We all know that there are C(12,3) = 220 possibilities (C(N,K) denotes the well-known binomial coefficients). %% For pure mathematicians, this result may be great. But we want to really generate all the possibilities in a list. %% Example: (combination 3 '(a b c d e f)) -> ((A B C) (A B D) (A B E) … ) % Not the most efficient way, but generate the power set for the list and remove all but the sub-lists of length K powerset([]) -> [[]]; powerset([H | T]) -> SubPS = powerset(T), lists:append(SubPS, lists:map(fun(E) -> [H | E] end, SubPS)). p26(K, L) -> [X || X < - powerset(L), length(X) == K]. %% Group the elements of a set into disjoint subsets. %% a) In how many ways can a group of 9 people work in 3 disjoint subgroups of 2, 3 and 4 persons? Write a function that generates all the possibilities and returns them in a list. %% Example: (group3 '(aldo beat carla david evi flip gary hugo ida)) -> ( ( (ALDO BEAT) (CARLA DAVID EVI) (FLIP GARY HUGO IDA) ) … ) p27([]) -> []. %% b) Generalize the above predicate in a way that we can specify a list of group sizes and the predicate will return a list of groups. %% Example: (group '(aldo beat carla david evi flip gary hugo ida) '(2 2 5)) -> ( ( (ALDO BEAT) (CARLA DAVID) (EVI FLIP GARY HUGO IDA) ) … ) %% Note that we do not want permutations of the group members; i.e. ((ALDO BEAT) …) is the same solution as ((BEAT ALDO) …). %% However, we make a difference between ((ALDO BEAT) (CARLA DAVID) …) and ((CARLA DAVID) (ALDO BEAT) …). %% You may find more about this combinatorial problem in a good book on discrete mathematics under the term "multinomial coefficients". p28() -> []. %% Sorting a list of lists according to length of sublists %% a) We suppose that a list contains elements that are lists themselves. The objective is to sort the elements of this list according to their length. E.g. short lists first, longer lists later, or vice versa. %% Example: (lsort '((a b c) (d e) (f g h) (d e) (i j k l) (m n) (o))) -> ((O) (D E) (D E) (M N) (A B C) (F G H) (I J K L)) p29(L) -> p29lsort(L). %% code taken from Erlang Programming Examples -> List Comprehensions -> QuickSort and adapted p29lsort([]) -> []; p29lsort([Pivot | L]) -> p29lsort([ X || X < - L, length(X) < length(Pivot)]) ++ [Pivot | p29lsort([ X || X = length(Pivot)])]. %% b) Again, we suppose that a list contains elements that are lists themselves. But this time the objective is to sort the elements of this list according to their length frequency; %% i.e., in the default, where sorting is done ascendingly, lists with rare lengths are placed first, others with a more frequent length come later. %% Example: (lfsort '((a b c) (d e) (f g h) (d e) (i j k l) (m n) (o))) -> ((i j k l) (o) (a b c) (f g h) (d e) (d e) (m n)) %% Note that in the above example, the first two lists in the result have length 4 and 1, both lengths appear just once. The third and forth list have length 3 which appears twice (there are two list of this length). %% And finally, the last three lists have length 2. This is the most frequent length. p30(L) -> p30lfsort(L). p30lfsort([]) -> []; p30lfsort([H | L]) -> Grouped = p30group(L,H), Sorted = p29(Grouped), p30ungroup(Sorted, []). p30group([], Grouped) -> Grouped; p30group([H

L], Grouped) -> p30group(L, p30insert(H, Grouped, [])). %% insert Elem in to a group with the same length, or create a new group for the length of Elem p30insert(Elem, [], PrevGroups) -> lists:reverse([[Elem] | PrevGroups]); p30insert(Elem, [ [GroupH | GroupT] | RestGrouped], PrevGroups) when length(GroupH) == length(Elem) -> PrevGroups ++ [ [GroupH, Elem | GroupT] | RestGrouped]; p30insert(Elem, [ [GroupH | GroupT] | RestGrouped], PrevGroups) -> p30insert(Elem, RestGrouped, [[GroupH | GroupT] | PrevGroups]). %% remove length-specific groupings p30ungroup([], Ungrouped) -> lists:reverse(Ungrouped); p30ungroup([H | L], Ungrouped) when not is\

_{list}(H) -> [[H | L] | Ungrouped]; p30ungroup([Group | L], Ungrouped) -> p30ungroup(Group, []) ++ p30ungroup(L, Ungrouped).