from typing import Callable
import operator


class SegmentTree:
    def __init__(self, capacity: int,
                 operation: Callable, neutral_element: int) -> None:
        """
        Build a Segment Tree data structure.

        https://en.wikipedia.org/wiki/Segment_tree

        Can be used as regular array, but with two
        important differences:

            a) setting item's value is slightly slower.
               It is O(lg capacity) instead of O(1).
            b) user has access to an efficient ( O(log segment size) )
               `reduce` operation which reduces `operation` over
               a contiguous subsequence of items in the array.

        :param capacity: (int) Total size of the array - must be a power of two
        :param operation: (lambda (Any, Any): Any) operation for combining
            elements (eg. sum, max) must form a
            mathematical group together with the set of possible values for
            array elements (i.e. be associative)
        :param neutral_element: (Any) neutral element for the operation above.
            eg. float('-inf') for max and 0 for sum.
        """
        assert capacity > 0 and capacity & (
            capacity - 1) == 0, "capacity must be positive and a power of 2."
        self._capacity = capacity
        self._operation = operation
        self._value = [neutral_element for _ in range(2 * capacity)]

    def _reduce_helper(self, start: int, end: int, node: int,
                       node_start: int, node_end: int) -> float:
        if start == node_start and end == node_end:
            return self._value[node]
        mid = (node_start + node_end) // 2
        if end <= mid:
            return self._reduce_helper(start, end, 2 * node, node_start, mid)
        else:
            if mid + 1 <= start:
                return self._reduce_helper(start, end, 2 * node + 1,
                                           mid + 1, node_end)
            else:
                return self._operation(
                    self._reduce_helper(start, mid, 2 * node, node_start, mid),
                    self._reduce_helper(
                        mid + 1, end, 2 * node + 1, mid + 1, node_end)
                )

    def reduce(self, start: int = 0, end: int = None) -> float:
        """
        Returns result of applying `self.operation`
        to a contiguous subsequence of the array.

            self.operation(arr[start], operation(arr[start+1]
            operation(... arr[end])))

        :param start: (int) beginning of the subsequence
        :param end: (int) end of the subsequences
        :return: (Any) result of reducing self.operation over the specified
            range of array elements.
        """
        if end is None:
            end = self._capacity
        if end < 0:
            end += self._capacity
        end -= 1
        return self._reduce_helper(start, end, 1, 0, self._capacity - 1)

    def __setitem__(self, idx: int, val: float) -> None:
        # index of the leaf
        idx += self._capacity
        self._value[idx] = val
        idx //= 2
        while idx >= 1:
            self._value[idx] = self._operation(
                self._value[2 * idx],
                self._value[2 * idx + 1]
            )
            idx //= 2

    def __getitem__(self, idx: int) -> float:
        assert 0 <= idx < self._capacity
        return self._value[self._capacity + idx]


class SumSegmentTree(SegmentTree):
    def __init__(self, capacity: int):
        super(SumSegmentTree, self).__init__(
            capacity=capacity,
            operation=operator.add,
            neutral_element=0.0
        )

    def sum(self, start: int = 0, end: int = None) -> float:
        """
        Returns arr[start] + ... + arr[end]

        :param start: (int) start position of the reduction (must be >= 0)
        :param end: (int) end position of the reduction (must be < len(arr),
            can be None for len(arr) - 1)
        :return: (Any) reduction of SumSegmentTree
        """
        return super(SumSegmentTree, self).reduce(start, end)

    def find_prefixsum_idx(self, prefixsum: float) -> int:
        """
        Find the highest index `i` in the array such that
            sum(arr[0] + arr[1] + ... + arr[i - i]) <= prefixsum

        if array values are probabilities, this function
        allows to sample indexes according to the discrete
        probability efficiently.

        :param prefixsum: (float) upperbound on the sum of array prefix
        :return: (int) highest index satisfying the prefixsum constraint
        """
        assert 0 <= prefixsum <= self.sum() + 1e-5
        idx = 1
        while idx < self._capacity:  # while non-leaf
            if self._value[2 * idx] > prefixsum:
                idx = 2 * idx
            else:
                prefixsum -= self._value[2 * idx]
                idx = 2 * idx + 1
        return idx - self._capacity


class MinSegmentTree(SegmentTree):
    def __init__(self, capacity: int):
        super(MinSegmentTree, self).__init__(
            capacity=capacity,
            operation=min,
            neutral_element=float('inf')
        )

    def min(self, start: int = 0, end: int = None) -> float:
        """
        Returns min(arr[start], ...,  arr[end])

        :param start: (int) start position of the reduction (must be >= 0)
        :param end: (int) end position of the reduction (must be < len(arr),
            can be None for len(arr) - 1)
        :return: (Any) reduction of MinSegmentTree
        """
        return super(MinSegmentTree, self).reduce(start, end)
