@@ -117,7 +117,7 @@ permutateSeq existSeq (x:xs) = map (++ [x]) existSeq ++
117117 (x, y): current position
118118 (c, r): final position
119119 constrs: cells are not available (can't include the start cell)
120- -}
120+ -}
121121move :: (Nat , Nat ) -> (Nat , Nat ) -> [(Nat , Nat )]-> [Maybe Nat , Nat )]
122122move (x, y) (c, r) constrs
123123 | x+ 1 <= c && y+ 1 <= r && xCell `notElem` constrs && yCell `notElem` constrs =
@@ -157,13 +157,13 @@ allPathsHelper (Just (x, y):xs) final constrs =
157157
158158-- Return the Completed Heap
159159allPaths :: (Nat , Nat ) -> (Nat , Nat ) -> [(Nat , Nat )] -> [Maybe Nat , Nat )]
160- allPaths start end constrs =
160+ allPaths start end constrs =
161161 Just start: removeNothing (allPathsHelper [Just start] end constrs)
162162
163163-- Return all unique paths index in a Heap
164164allPaths' :: (Nat , Nat ) -> (Nat , Nat ) -> [(Nat , Nat )] -> [[Nat ]]
165165allPaths' start end constrs = findAllPaths completedHeap [] (length completedHeap- 1 )
166- where completedHeap =
166+ where completedHeap =
167167 Just start: removeNothing (allPathsHelper [Just start] end constrs)
168168
169169-- Remove Nothing on the tail
@@ -178,13 +178,87 @@ isValidPath (x, y) (c, r) constrs
178178 | x == c && y == r = True
179179 | x > c || y > r = False
180180 | (x, y) `elem` constrs = False
181- | otherwise = isValidPath (x+ 1 , y) (c, r) constrs ||
182- isValidPath (x, y+ 1 ) (c, r) constrs
181+ | otherwise = isValidPath (x+ 1 , y) (c, r) constrs ||
182+ isValidPath (x, y+ 1 ) (c, r) constrs
183183
184184findOnePath :: (Nat , Nat ) -> (Nat , Nat ) -> [(Nat , Nat )] -> [(Nat , Nat )]
185185findOnePath (x, y) (c, r) constrs
186- | isValidPath (x+ 1 , y) (c, r) constrs =
186+ | isValidPath (x+ 1 , y) (c, r) constrs =
187187 (x+ 1 , y): findOnePath(x+ 1 , y) (c,r) constrs
188- | isValidPath (x, y+ 1 ) (c, r) constrs =
188+ | isValidPath (x, y+ 1 ) (c, r) constrs =
189189 (x, y+ 1 ): findOnePath(x, y+ 1 ) (c,r) constrs
190190 | otherwise = []
191+ 192+ {-
193+ 8.3
194+ A magic index in an array A [0 ... n-1] is defined to be an index such
195+ that A[i] = i. Given a sorted array of distinct integers, write a
196+ method to find a magic index, if one exists, in array A.
197+
198+ For example:
199+ |0 |1 |2 |3 |4 |5 |6
200+ |-40 |-20 |0 |1 |4 |10 |12
201+
202+ Solution 1: brutal force
203+ Solution 2: binary search
204+ -}
205+ 206+ findMagicIndex :: [Int -> Nat -> Int
207+ findMagicIndex [] _ = error " No magic index in this array."
208+ findMagicIndex arr preInd
209+ | arr !! mid == (preInd + mid) = preInd + mid
210+ | arr !! mid > (preInd + mid) = findMagicIndex fstHalf preInd
211+ | otherwise = findMagicIndex sndHalf (preInd+ mid)
212+ where mid = length arr `div` 2
213+ (fstHalf, sndHalf) = splitAt mid arr
214+ 215+ {-
216+ 8.3 with a sorted array of non-distinct integers
217+ |0 |1 |2 |3 |4 |5 |6 |7 |8 |9 |10
218+ |-10 |-5 |2 |2 |2 |3 |4 |7 |9 |12 |13
219+ -}
220+ 221+ {-
222+ 8.4
223+ Write a method to return all subsets of a set.
224+
225+ input: a = [1,3,4,6,10]
226+ output: the power set of a
227+
228+ test case = powerSetOf [3,3,7,10,1,15]
229+ -}
230+ powerSetOf :: [a ] -> [[a ]]
231+ powerSetOf xs = map (map (xs!! )) resultInInd
232+ where resultInInd = powerSetHelper initalSeq (length xs) 0
233+ initalSeq = powerSetOfInd 0 (length xs)
234+ 235+ -- Initialize the list of index for pairing
236+ -- It is better to use index for paring because the index is a unique ID
237+ powerSetOfInd :: Nat -> Nat -> [[Nat ]]
238+ powerSetOfInd i max
239+ | i < max = [i]: powerSetOfInd (i+ 1 ) max
240+ | otherwise = []
241+ 242+ {-
243+ Copy the current set, and accumulate with new set
244+
245+ inp: give list of seq,
246+ len: length of original set,
247+ count: count number (start with 0) indicate when to stop the loop
248+ -}
249+ powerSetHelper :: [[Nat ]] -> Nat -> Nat -> [[Nat ]]
250+ powerSetHelper inp len count
251+ | count < len = inp ++ powerSetHelper newInp len (count+ 1 )
252+ | otherwise = []
253+ where newInp = newPair inp len
254+ 255+ -- Accumulate all new pair in the list
256+ newPair :: [[Nat ]] -> Nat -> [[Nat ]]
257+ newPair [] len = []
258+ newPair (x: xs) len =
259+ newPairHelper x [(last x + 1 ) .. len- 1 ] ++ newPair (xs) len
260+ 261+ -- Give a new seq append new possible pairing to it
262+ newPairHelper :: [Nat ] -> [Nat ] -> [[Nat ]]
263+ newPairHelper _ [] = []
264+ newPairHelper a (x: xs) = (a ++ [x]): newPairHelper a xs
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