
'''simple docstring'''
from abc import ABC, abstractmethod
from typing import List, Optional






class    UpperCAmelCase_				(	snake_case__					):





 def __init__(    self):
     # test for the above condition
     self.test()





 def        snake_case__		(    self):
     snake_case_ :    List[str]     			=						0
     snake_case_ :    List[str]     			=						False
     while not completed:
         if counter == 1:
             self.reset()
         snake_case_ :    Optional[Any]     			=						self.advance()
         if not self.does_advance(lowercase_):
             raise Exception(
                 "Custom Constraint is not defined correctly. self.does_advance(self.advance()) must be true.")

         snake_case_			, snake_case_			, snake_case_ :    List[str]     			=						self.update(lowercase_)
         counter += 1

         if counter > 1_00_00:
             raise Exception("update() does not fulfill the constraint.")

     if self.remaining() != 0:
         raise Exception("Custom Constraint is not defined correctly.")





 @abstractmethod
 def        snake_case__		(    self):
     raise NotImplementedError(
         F'{self.__class__} is an abstract class. Only classes inheriting this class can be called.')





 @abstractmethod
 def        snake_case__		(    self ,       lowercase_):
     raise NotImplementedError(
         F'{self.__class__} is an abstract class. Only classes inheriting this class can be called.')





 @abstractmethod
 def        snake_case__		(    self ,       lowercase_):
     raise NotImplementedError(
         F'{self.__class__} is an abstract class. Only classes inheriting this class can be called.')





 @abstractmethod
 def        snake_case__		(    self):
     raise NotImplementedError(
         F'{self.__class__} is an abstract class. Only classes inheriting this class can be called.')





 @abstractmethod
 def        snake_case__		(    self):
     raise NotImplementedError(
         F'{self.__class__} is an abstract class. Only classes inheriting this class can be called.')





 @abstractmethod
 def        snake_case__		(    self ,       lowercase_=False):
     raise NotImplementedError(
         F'{self.__class__} is an abstract class. Only classes inheriting this class can be called.')






class    UpperCAmelCase_				(	snake_case__					):





 def __init__(    self ,       lowercase_):
     super(lowercase_ ,       self).__init__()

     if not isinstance(lowercase_ ,       lowercase_) or len(lowercase_) == 0:
         raise ValueError(F'`token_ids` has to be a non-empty list, but is {token_ids}.')
     if any((not isinstance(lowercase_ ,       lowercase_) or token_id < 0) for token_id in token_ids):
         raise ValueError(F'Each list in `token_ids` has to be a list of positive integers, but is {token_ids}.')

     snake_case_ :    str     			=						token_ids

     snake_case_ :    Dict     			=						len(self.token_ids)
     snake_case_ :    str     			=						-1  # the index of the currently fulfilled step
     snake_case_ :    Dict     			=						False





 def        snake_case__		(    self):
     if self.completed:
         return None
     return self.token_ids[self.fulfilled_idx + 1]





 def        snake_case__		(    self ,       lowercase_):
     if not isinstance(lowercase_ ,       lowercase_):
         raise ValueError(F'`token_id` has to be an `int`, but is {token_id} of type {type(lowercase_)}')

     if self.completed:
         return False

     return token_id == self.token_ids[self.fulfilled_idx + 1]





 def        snake_case__		(    self ,       lowercase_):
     if not isinstance(lowercase_ ,       lowercase_):
         raise ValueError(F'`token_id` has to be an `int`, but is {token_id} of type {type(lowercase_)}')

     snake_case_ :    Any     			=						False
     snake_case_ :    Union[str, Any]     			=						False
     snake_case_ :    int     			=						False

     if self.does_advance(lowercase_):
         self.fulfilled_idx += 1
         snake_case_ :    List[str]     			=						True
         if self.fulfilled_idx == (self.seqlen - 1):
             snake_case_ :    Union[str, Any]     			=						True
         snake_case_ :    str     			=						completed
     else:
         # failed to make progress.
         snake_case_ :    int     			=						True
         self.reset()
     return stepped, completed, reset





 def        snake_case__		(    self):
     snake_case_ :    int     			=						False
     snake_case_ :    Union[str, Any]     			=						0





 def        snake_case__		(    self):
     return self.seqlen - (self.fulfilled_idx + 1)





 def        snake_case__		(    self ,       lowercase_=False):
     snake_case_ :    str     			=						PhrasalConstraint(self.token_ids)

     if stateful:
         snake_case_ :    int     			=						self.seqlen
         snake_case_ :    int     			=						self.fulfilled_idx
         snake_case_ :    List[str]     			=						self.completed

     return new_constraint






class    UpperCAmelCase_				:





 def __init__(    self ,       lowercase_ ,       lowercase_=True):
     snake_case_ :    List[str]     			=						max([len(lowercase_) for one in nested_token_ids])

     snake_case_ :    List[str]     			=						{}
     for token_ids in nested_token_ids:
         snake_case_ :    Optional[Any]     			=						root
         for tidx, token_id in enumerate(lowercase_):
             if token_id not in level:
                 snake_case_ :    Dict     			=						{}

             snake_case_ :    List[Any]     			=						level[token_id]

     if no_subsets and self.has_subsets(lowercase_ ,       lowercase_):
         raise ValueError(
             "Each list in `nested_token_ids` can't be a complete subset of another list, but is"
             F' {nested_token_ids}.')

     snake_case_ :    Optional[int]     			=						root





 def        snake_case__		(    self ,       lowercase_):
     snake_case_ :    str     			=						self.trie

     for current_token in current_seq:
         snake_case_ :    Union[str, Any]     			=						start[current_token]

     snake_case_ :    str     			=						list(start.keys())

     return next_tokens





 def        snake_case__		(    self ,       lowercase_):
     snake_case_ :    List[str]     			=						self.next_tokens(lowercase_)

     return len(lowercase_) == 0





 def        snake_case__		(    self ,       lowercase_):
     snake_case_ :    Optional[int]     			=						list(root.values())
     if len(lowercase_) == 0:
         return 1
     else:
         return sum([self.count_leaves(lowercase_) for nn in next_nodes])





 def        snake_case__		(    self ,       lowercase_ ,       lowercase_):
     snake_case_ :    str     			=						self.count_leaves(lowercase_)
     return len(lowercase_) != leaf_count






class    UpperCAmelCase_				(	snake_case__					):





 def __init__(    self ,       lowercase_):
     super(lowercase_ ,       self).__init__()

     if not isinstance(lowercase_ ,       lowercase_) or len(lowercase_) == 0:
         raise ValueError(F'`nested_token_ids` has to be a non-empty list, but is {nested_token_ids}.')
     if any(not isinstance(lowercase_ ,       lowercase_) for token_ids in nested_token_ids):
         raise ValueError(F'`nested_token_ids` has to be a list of lists, but is {nested_token_ids}.')
     if any(
         any((not isinstance(lowercase_ ,       lowercase_) or token_id < 0) for token_id in token_ids)
         for token_ids in nested_token_ids):
         raise ValueError(
             F'Each list in `nested_token_ids` has to be a list of positive integers, but is {nested_token_ids}.')

     snake_case_ :    List[str]     			=						DisjunctiveTrie(lowercase_)
     snake_case_ :    Union[str, Any]     			=						nested_token_ids

     snake_case_ :    Union[str, Any]     			=						self.trie.max_height
     snake_case_ :    int     			=						[]
     snake_case_ :    int     			=						False





 def        snake_case__		(    self):
     snake_case_ :    List[Any]     			=						self.trie.next_tokens(self.current_seq)

     if len(lowercase_) == 0:
         return None
     else:
         return token_list





 def        snake_case__		(    self ,       lowercase_):
     if not isinstance(lowercase_ ,       lowercase_):
         raise ValueError(F'`token_id` is supposed to be type `int`, but is {token_id} of type {type(lowercase_)}')

     snake_case_ :    Optional[int]     			=						self.trie.next_tokens(self.current_seq)

     return token_id in next_tokens





 def        snake_case__		(    self ,       lowercase_):
     if not isinstance(lowercase_ ,       lowercase_):
         raise ValueError(F'`token_id` is supposed to be type `int`, but is {token_id} of type {type(lowercase_)}')

     snake_case_ :    Union[str, Any]     			=						False
     snake_case_ :    List[str]     			=						False
     snake_case_ :    Optional[int]     			=						False

     if self.does_advance(lowercase_):
         self.current_seq.append(lowercase_)
         snake_case_ :    List[Any]     			=						True
     else:
         snake_case_ :    str     			=						True
         self.reset()

     snake_case_ :    str     			=						self.trie.reached_leaf(self.current_seq)
     snake_case_ :    Dict     			=						completed

     return stepped, completed, reset





 def        snake_case__		(    self):
     snake_case_ :    Optional[int]     			=						False
     snake_case_ :    Dict     			=						[]





 def        snake_case__		(    self):
     if self.completed:
         # since this can be completed without reaching max height
         return 0
     else:
         return self.seqlen - len(self.current_seq)





 def        snake_case__		(    self ,       lowercase_=False):
     snake_case_ :    List[str]     			=						DisjunctiveConstraint(self.token_ids)

     if stateful:
         snake_case_ :    Optional[int]     			=						self.seqlen
         snake_case_ :    Union[str, Any]     			=						self.current_seq
         snake_case_ :    Tuple     			=						self.completed

     return new_constraint






class    UpperCAmelCase_				:





 def __init__(    self ,       lowercase_):
     snake_case_ :    Dict     			=						constraints

     # max # of steps required to fulfill a given constraint
     snake_case_ :    int     			=						max([c.seqlen for c in constraints])
     snake_case_ :    int     			=						len(lowercase_)
     snake_case_ :    Union[str, Any]     			=						False

     self.init_state()





 def        snake_case__		(    self):
     snake_case_ :    Optional[int]     			=						[]
     snake_case_ :    Dict     			=						None
     snake_case_ :    List[Any]     			=						[constraint.copy(stateful=lowercase_) for constraint in self.constraints]





 def        snake_case__		(    self):
     snake_case_ :    Any     			=						0
     if self.inprogress_constraint:
         # extra points for having a constraint mid-fulfilled
         add += self.max_seqlen - self.inprogress_constraint.remaining()

     return (len(self.complete_constraints) * self.max_seqlen) + add





 def        snake_case__		(    self):
     snake_case_ :    List[Any]     			=						[]
     if self.inprogress_constraint is None:
         for constraint in self.pending_constraints:  # "pending" == "unfulfilled yet"
             snake_case_ :    Any     			=						constraint.advance()
             if isinstance(lowercase_ ,       lowercase_):
                 token_list.append(lowercase_)
             elif isinstance(lowercase_ ,       lowercase_):
                 token_list.extend(lowercase_)
     else:
         snake_case_ :    Tuple     			=						self.inprogress_constraint.advance()
         if isinstance(lowercase_ ,       lowercase_):
             token_list.append(lowercase_)
         elif isinstance(lowercase_ ,       lowercase_):
             token_list.extend(lowercase_)

     if len(lowercase_) == 0:
         return None
     else:
         return token_list





 def        snake_case__		(    self ,       lowercase_):
     self.init_state()

     if token_ids is not None:
         for token in token_ids:
             # completes or steps **one** constraint
             snake_case_			, snake_case_ :    Dict     			=						self.add(lowercase_)

             # the entire list of constraints are fulfilled
             if self.completed:
                 break





 def        snake_case__		(    self ,       lowercase_):
     if not isinstance(lowercase_ ,       lowercase_):
         raise ValueError(F'`token_id` should be an `int`, but is `{token_id}`.')

     snake_case_			, snake_case_ :    Optional[Any]     			=						False, False

     if self.completed:
         snake_case_ :    int     			=						True
         snake_case_ :    List[str]     			=						False
         return complete, stepped

     if self.inprogress_constraint is not None:
         # In the middle of fulfilling a constraint. If the `token_id` *does* makes an incremental progress to current
         # job, simply update the state

         snake_case_			, snake_case_			, snake_case_ :    List[Any]     			=						self.inprogress_constraint.update(lowercase_)
         if reset:
             # 1. If the next token breaks the progress, then we must restart.
             #     e.g. constraint = "I love pies" and sequence so far is "I love" but `token_id` == "books".

             #     But that doesn't mean we self.init_state(), since we only reset the state for this particular
             #     constraint, not the full list of constraints.

             self.pending_constraints.append(self.inprogress_constraint.copy(stateful=lowercase_))
             snake_case_ :    List[str]     			=						None

         if complete:
             # 2. If the next token completes the constraint, move it to completed list, set
             #     inprogress to None. If there are no pending constraints either, then this full list of constraints
             #     is complete.

             self.complete_constraints.append(self.inprogress_constraint)
             snake_case_ :    Tuple     			=						None

             if len(self.pending_constraints) == 0:
                 # we're done!
                 snake_case_ :    Any     			=						True

     else:
         # Not in the middle of fulfilling a constraint. So does this `token_id` helps us step towards any of our list
         # of constraints?

         for cidx, pending_constraint in enumerate(self.pending_constraints):
             if pending_constraint.does_advance(lowercase_):
                 snake_case_			, snake_case_			, snake_case_ :    str     			=						pending_constraint.update(lowercase_)

                 if not stepped:
                     raise Exception(
                         "`constraint.update(token_id)` is not yielding incremental progress, "
                         "even though `constraint.does_advance(token_id)` is true.")

                 if complete:
                     self.complete_constraints.append(lowercase_)
                     snake_case_ :    Dict     			=						None

                 if not complete and stepped:
                     snake_case_ :    List[Any]     			=						pending_constraint

                 if complete or stepped:
                     # If we made any progress at all, then it's at least not a "pending constraint".

                     snake_case_ :    int     			=						(
                         self.pending_constraints[:cidx] + self.pending_constraints[cidx + 1 :]
                     )

                     if len(self.pending_constraints) == 0 and self.inprogress_constraint is None:
                         # If there's no longer any pending after this and no inprogress either, then we must be
                         # complete.

                         snake_case_ :    Union[str, Any]     			=						True

                     break  # prevent accidentally stepping through multiple constraints with just one token.

     return complete, stepped





 def        snake_case__		(    self ,       lowercase_=True):
     snake_case_ :    Union[str, Any]     			=						ConstraintListState(self.constraints)  # we actually never though self.constraints objects
     # throughout this process. So it's at initialization state.

     if stateful:
         snake_case_ :    Any     			=						[
             constraint.copy(stateful=lowercase_) for constraint in self.complete_constraints
         ]
         if self.inprogress_constraint is not None:
             snake_case_ :    str     			=						self.inprogress_constraint.copy(stateful=lowercase_)
         snake_case_ :    Any     			=						[constraint.copy() for constraint in self.pending_constraints]

     return new_state






'''simple docstring'''
from __future__ import annotations

import math

def 						UpperCamelCase_     (	__SCREAMING_SNAKE_CASE,		__SCREAMING_SNAKE_CASE,		__SCREAMING_SNAKE_CASE,		__SCREAMING_SNAKE_CASE,		__SCREAMING_SNAKE_CASE       ):
    """simple docstring"""






    if depth < 0:
        raise ValueError("Depth cannot be less than 0"       )

    if len(__SCREAMING_SNAKE_CASE       ) == 0:
        raise ValueError("Scores cannot be empty"       )

    if depth == height:
        return scores[node_index]

    if is_max:
        return max(
            minimax(depth + 1,		node_index * 2,		__SCREAMING_SNAKE_CASE,		__SCREAMING_SNAKE_CASE,		__SCREAMING_SNAKE_CASE       ),		minimax(depth + 1,		node_index * 2 + 1,		__SCREAMING_SNAKE_CASE,		__SCREAMING_SNAKE_CASE,		__SCREAMING_SNAKE_CASE       ),		)

    return min(
        minimax(depth + 1,		node_index * 2,		__SCREAMING_SNAKE_CASE,		__SCREAMING_SNAKE_CASE,		__SCREAMING_SNAKE_CASE       ),		minimax(depth + 1,		node_index * 2 + 1,		__SCREAMING_SNAKE_CASE,		__SCREAMING_SNAKE_CASE,		__SCREAMING_SNAKE_CASE       ),		)

def 						UpperCamelCase_     (	):
    """simple docstring"""






    snake_case_ :    int     			=						[9_0, 2_3, 6, 3_3, 2_1, 6_5, 1_2_3, 3_4_4_2_3]
    snake_case_ :    Union[str, Any]     			=						math.log(len(__SCREAMING_SNAKE_CASE       ),		2       )
    print("Optimal value : ",		end=""       )
    print(minimax(0,		0,		__SCREAMING_SNAKE_CASE,		__SCREAMING_SNAKE_CASE,		__SCREAMING_SNAKE_CASE       )       )


if __name__ == "__main__":
  import doctest

  doctest.testmod()
  main()




