# Time:  O(nloga)
# Space: O(n)

# Range Maximum Query
class SegmentTree(object):  # 0-based index
    def __init__(self, N,
                 build_fn=lambda x, y: [y]*(2*x),
                 query_fn=lambda x, y: y if x is None else max(x, y),  # (lambda x, y: y if x is None else min(x, y))
                 update_fn=lambda x, y: y,
                 default_val=0):
        self.N = N
        self.H = (N-1).bit_length()
        self.query_fn = query_fn
        self.update_fn = update_fn
        self.default_val = default_val
        self.tree = build_fn(N, default_val)
        self.lazy = [None]*N

    def __apply(self, x, val):
        self.tree[x] = self.update_fn(self.tree[x], val)
        if x < self.N:
            self.lazy[x] = self.update_fn(self.lazy[x], val)

    def update(self, L, R, h):  # Time: O(logN), Space: O(N)
        def pull(x):
            while x > 1:
                x //= 2
                self.tree[x] = self.query_fn(self.tree[x*2], self.tree[x*2+1])
                if self.lazy[x] is not None:
                    self.tree[x] = self.update_fn(self.tree[x], self.lazy[x])

        L += self.N
        R += self.N
        L0, R0 = L, R
        while L <= R:
            if L & 1:  # is right child
                self.__apply(L, h) 
                L += 1
            if R & 1 == 0:  # is left child
                self.__apply(R, h)
                R -= 1
            L //= 2
            R //= 2
        pull(L0)
        pull(R0)

    def query(self, L, R):  # Time: O(logN), Space: O(N)
        def push(x):
            n = 2**self.H
            while n != 1:
                y = x // n
                if self.lazy[y] is not None:
                    self.__apply(y*2, self.lazy[y])
                    self.__apply(y*2 + 1, self.lazy[y])
                    self.lazy[y] = None
                n //= 2

        result = None
        if L > R:
            return result

        L += self.N
        R += self.N
        push(L)
        push(R)
        while L <= R:
            if L & 1:  # is right child
                result = self.query_fn(result, self.tree[L])
                L += 1
            if R & 1 == 0:  # is left child
                result = self.query_fn(result, self.tree[R])
                R -= 1
            L //= 2
            R //= 2
        return result
    
    def __str__(self):
        showList = []
        for i in xrange(self.N):
            showList.append(self.query(i, i))
        return ",".join(map(str, showList))


# optimized from Solution3
class Solution(object):
    def bestTeamScore(self, scores, ages):
        """
        :type scores: List[int]
        :type ages: List[int]
        :rtype: int
        """
        players = sorted(zip(scores, ages))
        sorted_ages = sorted(set(ages))
        lookup = {age:i for i, age in enumerate(sorted_ages)}  # coordinate compression
        segment_tree = SegmentTree(len(lookup))
        result = 0
        for score, age in players:
            segment_tree.update(lookup[age], lookup[age], segment_tree.query(0, lookup[age])+score)
        return segment_tree.query(0, len(lookup)-1)


# Time:  O(nlogs)
# Space: O(n)
# optimized from Solution4
class Solution2(object):
    def bestTeamScore(self, scores, ages):
        """
        :type scores: List[int]
        :type ages: List[int]
        :rtype: int
        """
        players = sorted(zip(ages, scores))
        sorted_scores = sorted(set(scores))
        lookup = {score:i for i, score in enumerate(sorted_scores)}  # coordinate compression
        segment_tree = SegmentTree(len(lookup))
        result = 0
        for age, score in players:
            segment_tree.update(lookup[score], lookup[score], segment_tree.query(0, lookup[score])+score)
        return segment_tree.query(0, len(lookup)-1)
 

# Time:  O(n * a)
# Space: O(n)
import collections


# optimized from Solution5
class Solution3(object):
    def bestTeamScore(self, scores, ages):
        """
        :type scores: List[int]
        :type ages: List[int]
        :rtype: int
        """
        players = sorted(zip(scores, ages))
        sorted_ages = sorted(set(ages))
        dp = collections.defaultdict(int)
        result = 0
        for score, age in players:
            dp[age] = max(dp[a] for a in sorted_ages if a <= age) + score
        return max(dp.itervalues())


# Time:  O(n * s)
# Space: O(n)
import collections


# optimized from Solution6
class Solution4(object):
    def bestTeamScore(self, scores, ages):
        """
        :type scores: List[int]
        :type ages: List[int]
        :rtype: int
        """
        players = sorted(zip(ages, scores))
        sorted_scores = sorted(set(scores))
        dp = collections.defaultdict(int)
        result = 0
        for age, score in players:
            dp[score] = max(dp[s] for s in sorted_scores if s <= score) + score
        return max(dp.itervalues())


# Time:  O(n^2)
# Space: O(n)
# longest_increasing_subsequence like dp solution
class Solution5(object):
    def bestTeamScore(self, scores, ages):
        """
        :type scores: List[int]
        :type ages: List[int]
        :rtype: int
        """
        players = sorted(zip(scores, ages))
        dp = [0]*len(players)
        result = 0
        for i in xrange(len(players)):
            dp[i] = players[i][0]
            for j in xrange(i):
                if players[j][1] <= players[i][1]:
                    dp[i] = max(dp[i], dp[j] + players[i][0])
            result = max(result, dp[i])
        return result


# Time:  O(n^2)
# Space: O(n)
# longest_increasing_subsequence like dp solution
class Solution6(object):
    def bestTeamScore(self, scores, ages):
        """
        :type scores: List[int]
        :type ages: List[int]
        :rtype: int
        """
        players = sorted(zip(ages, scores))
        dp = [0]*len(players)
        result = 0
        for i in xrange(len(players)):
            dp[i] = players[i][1]
            for j in xrange(i):
                if players[j][1] <= players[i][1]:
                    dp[i] = max(dp[i], dp[j] + players[i][1])
            result = max(result, dp[i])
        return result