This is an index to the problems from THE book being solved in haskell (and counting). Some of the trivial ones will only be attempted on demand!
Implement functions to run length encode a string and decode the RLE value of an encoded string.
For example “aaaabcccaa” should be encoded to “4a1b3c2a”, while “3e4f2e” would be decoded to “eeeffffee”.
Encoding is straight forward:
run_length_encode :: [Char] -> [Char] run_length_encode xs = run_length_encode' 0 '|' xs where run_length_encode' 0 _  =  run_length_encode' 0 _ (x:xs) = run_length_encode' 1 x xs run_length_encode' count curr_ch  = (show count) ++ [curr_ch] run_length_encode' count curr_ch (x:xs) | curr_ch == x = run_length_encode' (count + 1) curr_ch xs | otherwise = (show count) ++ [curr_ch] ++ run_length_encode' 1 x xs
Decoding is also fairly straightforward, except we just need to accumulate the “count” before that many characters can be output:
run_length_decode :: [Char] -> [Char] run_length_decode xs = run_length_decode' 0 xs where run_length_decode' _  =  run_length_decode' count (x:xs) | isDigit x = run_length_decode' ((count * 10) + (digitToInt x)) xs | otherwise = (replicate count x) ++ run_length_decode' 0 xs
Given an nxm 2D array of integers, print it in spiral order.
Eg for a 3×3 matrix (with the following implicit values):
Would be printed as “1 2 3 6 9 8 7 4 5″
The straight forward solution is to have 4 iterations, and in each iteration to print one of the two directions (horizontal and vertical) alternatively.
However a more intuitive solution is to use vectors to create a tour starting from the index (0,0) and incrementing the next point based on the current direction and position.
First we create the concept of directions along with a couple of helper functions:
data Direction = North | East | South | West -- Return the clockwise direction of a given direction clockwise_of North = East clockwise_of East = South clockwise_of South = West clockwise_of West = North -- Returns the vector representing the direction vector_of North = (0, -1) vector_of South = (0, 1) vector_of East = (1, 0) vector_of West = (-1, 0) -- The primary coordinate that will be updated when traversing in a particular direction primary_coord North = 1 primary_coord South = 1 primary_coord East = 0 primary_coord West = 0 -- Given a direction and point, returns the next and previous points in the direction. next_pt dir (x,y) = ((x + fst (vector_of dir)),(y + snd (vector_of dir))) prev_pt dir (x,y) = ((x - fst (vector_of dir)),(y - snd (vector_of dir)))
Now we simply use the above helpers to start and iterate through a tour:
print_spirally width height = print_spirally' East (0,0) 0 [width, height] where print_spirally' dir (x,y) t [w,h] | w == 0 || h == 0 =  | t < [w,h] !! curr_coord = (y * width + x + 1) : print_spirally' dir (next_pt dir (x,y)) (t+1) [w,h] | otherwise = print_spirally' next_dir (next_pt next_dir (prev_pt dir (x,y))) 0 [next_w,next_h] where curr_coord = primary_coord dir dir_vector = vector_of dir next_dir = clockwise_of dir next_dir_vector = vector_of next_dir next_w = w - (abs (fst next_dir_vector)) next_h = h - (abs (snd next_dir_vector))
Couple of things to note:
* w,h represent the “remaining” width and height in the spiral as each time there is a change in direction, the available coordinate size reduces by one (height if beginning vertically, width if beginning horizontally).
* t is the number of values printed in a given direction (when this value reaches “w” or “h” depending on the direction, direction is rotated and t is reset to 0).
* When the direction needs to change (in the otherwise) the “current” point is one beyond the last point in the direction. For this reason the next point is evaluted from the previous direction in the previous point.
Given an Array of n elements, design an algorithm for rotating an Array right by i positions. Only O(1) additional storage is allowed.
The natural solution is to start from position the value at index k to k + i, repeatedly n times. This will work well if GCD(n,i) is != 1. However a general solution is to perform m jumps of size i, l times, but each time starting from the next index.
The first helper function is to perform m jumps of “size” each starting from a given index, wrapping around if necessary. For example if A = [1,2,3,4,5,6,7,8,9,10],
m_rotations A 0 3 3
would cause the following jumps: 1 -> 4 -> 7, with A resulting in:
[1, 2, 3, 1, 5, 6, 4, 8, 9, 10]
m_rotations xs index size m = elems (m_rotations' (arrayFromList xs 0) index (xs!!index) size m) where len = length xs m_rotations' arr curr_index curr_value size numleft | curr_index < 0 || size <= 0 || numleft <= 0 = arr | otherwise = m_rotations' (arr // [(next_index, curr_value)]) next_index next_value size (numleft - 1) where next_index = mod (curr_index + size) len next_value = arr!next_index
Now we create the actual rotator method that calls m_rotations k times, where k = gcd(|A|, j).
rotate_array xs j = rotate_array' xs 0 where lxs = length xs j' = mod j lxs gcd_lxs_j = greatest_common_divisor lxs j' numtimes = div lxs gcd_lxs_j rotate_array' xs start_index | start_index >= gcd_lxs_j = xs | otherwise = m_rotations ys (j' - (start_index + 1)) j' numtimes where ys = rotate_array' xs (start_index + 1)
A simpler algorithm to perform this is very similar to reversing words in a sentence:
rotate_array_simple xs j = reverse (take j rxs) ++ reverse (drop j rxs) where rxs = reverse xs
Design an algorithm for computing the GCD of two numbers without using multiplication, division or the modulus operator.
The GCD of two numbers can be computed by recursively subtracting the smaller number from the larger until one of the numbers is 0, at which point the non-zero value is the GCD.
However this can be improved by “quickly” eliminating factors of two (by inspecting the least significant bits) and doubling and halving values (via left and right shifting by 1 respectively).
greatest_common_divisor x 0 = x greatest_common_divisor 0 y = y greatest_common_divisor x y | x_is_even && y_is_even = 2 * greatest_common_divisor (x `shiftR` 1) (y `shiftR` 1) | x_is_odd && y_is_even = greatest_common_divisor x (y `shiftR` 1) | x_is_even && y_is_odd = greatest_common_divisor (x `shiftR` 1) y | x y = greatest_common_divisor (x - y) x | otherwise = x where x_is_even = (x .&. 1) == 0 y_is_even = (y .&. 1) == 0 x_is_odd = not x_is_even y_is_odd = not y_is_even
An array is increasing if each element is less than its succeeding element except for the last element.
Given an array A of n elements return the beginning and ending indices of a longest increasing subarray of A.
Let S[i] be the longest increasing subarray between the indexes (0,i – 1).
S[i] = a,b if A[i] > A[i - 1],
where a,b = S[i - 1]
In haskell this would be:
longest_contig_inc_subarray :: (Ord a) => [a] -> (Int, Int) longest_contig_inc_subarray  = (-1, -1) longest_contig_inc_subarray (x:xs) = longest_contig_inc_subarray' (0, x, 0, x) xs where longest_contig_inc_subarray' (i,ai,j,aj)  = (i,j) longest_contig_inc_subarray' (i,ai,j,aj) (x:xs) | x >= aj = longest_contig_inc_subarray' (i,ai,j + 1,x) xs | otherwise = longest_contig_inc_subarray' (j + 1,x,j + 1,x) xs
A heuristic to improve the best case complexity (but does nothing in the worst case) is to realise that if the length of the longest subarray till i is L (and A[i + 1] < A[i] – indicating an end of the longest subarray), then a larger increasing subarray must contain *atleast* L elements. So we only need to start with L items in front and check backwards.
The code for this is (here the start index and the length of the subarray are returned instead):
-- Returns the size of the largest increasing "prefix" in an array largest_inc_prefix  = 0 largest_inc_prefix (x:) = 1 largest_inc_prefix (x:y:xs) | x = y = 1 + largest_dec_prefix (y:xs) | otherwise = 1 -- Returns the size of the largest decreasing "prefix" in an array largest_dec_prefix  = 0 largest_dec_prefix (x:) = 1 largest_dec_prefix (x:y:xs) | x >= y = 1 + largest_dec_prefix (y:xs) | otherwise = 1 lcisa :: (Ord a) => [a] -> (Int, Int) lcisa  = (-1,-1) lcisa xs = lcisa' (0,1) 0 xs where lcisa' (start,maxlen) i  = (start,maxlen) lcisa' (start,maxlen) i xs | nextlen > maxlen = lcisa' nextbest (i + maxlen + inc_prefix) (drop inc_prefix rest) | otherwise = lcisa' (start,maxlen) (i + maxlen) rest where first_l = take maxlen xs rest = drop maxlen xs dec_prefix = largest_dec_prefix (reverse first_l) inc_prefix = largest_inc_prefix rest nextlen = inc_prefix + dec_prefix nextbest = (i + maxlen - dec_prefix, nextlen)
Reverse the words in a sentence.
We can define a word as any substring separated by one or more spaces. To do this on constant space, simply reverse the entire sentence, and then reverse each word within.
In haskell, the recursive solution would be:
reverse_words :: [Char] -> [Char] reverse_words  =  reverse_words xs = initial_spaces ++ (reverse after_spaces) ++ reverse_words remaining where initial_spaces = takeWhile isSpace xs spaces_dropped = dropWhile isSpace xs after_spaces = takeWhile (\x -> not (isSpace x)) spaces_dropped remaining = drop (length after_spaces) spaces_dropped
This is clearly not O(1) in space usage. The constant space solution would require maintaining array structures and swapping entries within the array.