Brace expansion II¶
Time: O(PxLxLog(PxL)); Space: O(PxL); hard
Under a grammar given below, strings can represent a set of lowercase words. Let’s use R(expr) to denote the set of words the expression represents.
Grammar can best be understood through simple examples:
Single letters represent a singleton set containing that word.
R(“a”) = {“a”}
R(“w”) = {“w”}
When we take a comma delimited list of 2 or more expressions, we take the union of possibilities.
R(“{a,b,c}”) = {“a”,“b”,“c”}
R(“{{a,b},{b,c}}”) = {“a”,“b”,“c”} (notice the final set only contains each word at most once)
When we concatenate two expressions, we take the set of possible concatenations between two words where the first word comes from the first expression and the second word comes from the second expression.
R(“{a,b}{c,d}”) = {“ac”,“ad”,“bc”,“bd”}
R(“a{b,c}{d,e}f{g,h}”) = {“abdfg”, “abdfh”, “abefg”, “abefh”, “acdfg”, “acdfh”, “acefg”, “acefh”}
Formally, the 3 rules for our grammar: 1. For every lowercase letter x, we have R(x) = {x} 2. For expressions e_1, e_2, … , e_k with k >= 2, we have R({e_1,e_2,…}) = R(e_1) ∪ R(e_2) ∪ … 3. For expressions e_1 and e_2, we have R(e_1 + e_2) = {a + b for (a, b) in R(e_1) × R(e_2)}, where + denotes concatenation, and × denotes the cartesian product.
Given an expression representing a set of words under the given grammar, return the sorted list of words that the expression represents.
Example 1:
Input: expression = “{a,b}{c,{d,e}}”
Output: [“ac”,“ad”,“ae”,“bc”,“bd”,“be”]
Example 2:
Input: expression = “{{a,z},a{b,c},{ab,z}}”
Output: [“a”,“ab”,“ac”,“z”]
Explanation:
Each distinct word is written only once in the final answer.
Constraints:
1 <= len(expression) <= 60
expression[i] consists of ‘{’, ‘}’, ’,’or lowercase English letters.
The given expression represents a set of words based on the grammar given in the description.
Hints:
You can write helper methods to parse the next “chunk” of the expression. If you see eg. “a”, the answer is just the set {a}. If you see “{”, you parse until you complete the “}” (the number of { and } seen are equal) and that becomes a chunk that you find where the appropriate commas are, and parse each individual expression between the commas.
[3]:
import itertools
class Solution1(object):
"""
Time: O(P*L*Log(P*L)), P is the production of all number of options, L is the length of a word
Space: O(P*L)
"""
def braceExpansionII(self, expression):
"""
:type expression: str
:rtype: List[str]
"""
def form_words(options):
words = list(map(''.join, itertools.product(*options)))
words.sort()
return words
def generate_option(expr, i):
option_set = set()
while i[0] != len(expr) and expr[i[0]] != "}":
i[0] += 1 # { or ,
for option in generate_words(expr, i):
option_set.add(option)
i[0] += 1 # }
option = list(option_set)
option.sort()
return option
def generate_words(expr, i):
options = []
while i[0] != len(expr) and expr[i[0]] not in ",}":
tmp = []
if expr[i[0]] not in "{,}":
tmp.append(expr[i[0]])
i[0] += 1 # a-z
elif expr[i[0]] == "{":
tmp = generate_option(expr, i)
options.append(tmp)
return form_words(options)
return generate_words(expression, [0])
[4]:
s = Solution1()
expression = "{a,b}{c,{d,e}}"
assert s.braceExpansionII(expression) == ["ac","ad","ae","bc","bd","be"]
expression = "{{a,z},a{b,c},{ab,z}}"
assert s.braceExpansionII(expression) == ["a","ab","ac","z"]
[5]:
class Solution2(object):
def braceExpansionII(self, expression):
"""
:type expression: str
:rtype: List[str]
"""
def form_words(options):
words = []
total = 1
for opt in options:
total *= len(opt)
for i in range(total):
tmp = []
for opt in reversed(options):
i, c = divmod(i, len(opt))
tmp.append(opt[c])
tmp.reverse()
words.append(''.join(tmp))
words.sort()
return words
def generate_option(expr, i):
option_set = set()
while i[0] != len(expr) and expr[i[0]] != "}":
i[0] += 1 # { or ,
for option in generate_words(expr, i):
option_set.add(option)
i[0] += 1 # }
option = list(option_set)
option.sort()
return option
def generate_words(expr, i):
options = []
while i[0] != len(expr) and expr[i[0]] not in ",}":
tmp = []
if expr[i[0]] not in "{,}":
tmp.append(expr[i[0]])
i[0] += 1 # a-z
elif expr[i[0]] == "{":
tmp = generate_option(expr, i)
options.append(tmp)
return form_words(options)
return generate_words(expression, [0])
[6]:
s = Solution2()
expression = "{a,b}{c,{d,e}}"
assert s.braceExpansionII(expression) == ["ac","ad","ae","bc","bd","be"]
expression = "{{a,z},a{b,c},{ab,z}}"
assert s.braceExpansionII(expression) == ["a","ab","ac","z"]