From f31fa4ea7e26d8721be7e966ecf92fcbd87af65f Mon Sep 17 00:00:00 2001 From: AmirMohammad Hosseini Nasab <19665344+itsamirhn@users.noreply.github.com> Date: Tue, 16 Aug 2022 22:08:33 +0430 Subject: [PATCH] Fenwick Tree (#6319) * Enhance fenwick_tree.py * Change update to add in fenwick_tree.py * Some changes * Fix bug * Add O(N) initializer to FenwickTree * Add get method to Fenwick Tree * Change tree in Fenwick Tree * Add rank query to FenwickTree * Add get_array method to FenwickTree * Add some tests * Update data_structures/binary_tree/fenwick_tree.py Co-authored-by: Christian Clauss * Update data_structures/binary_tree/fenwick_tree.py Co-authored-by: Christian Clauss * Update data_structures/binary_tree/fenwick_tree.py Co-authored-by: Christian Clauss * change `List` to `list` Co-authored-by: Christian Clauss --- data_structures/binary_tree/fenwick_tree.py | 263 ++++++++++++++++++-- 1 file changed, 241 insertions(+), 22 deletions(-) diff --git a/data_structures/binary_tree/fenwick_tree.py b/data_structures/binary_tree/fenwick_tree.py index 54f0f07ac..96020d142 100644 --- a/data_structures/binary_tree/fenwick_tree.py +++ b/data_structures/binary_tree/fenwick_tree.py @@ -1,28 +1,247 @@ +from copy import deepcopy + + class FenwickTree: - def __init__(self, SIZE): # create fenwick tree with size SIZE - self.Size = SIZE - self.ft = [0 for i in range(0, SIZE)] + """ + Fenwick Tree - def update(self, i, val): # update data (adding) in index i in O(lg N) - while i < self.Size: - self.ft[i] += val - i += i & (-i) + More info: https://en.wikipedia.org/wiki/Fenwick_tree + """ - def query(self, i): # query cumulative data from index 0 to i in O(lg N) - ret = 0 - while i > 0: - ret += self.ft[i] - i -= i & (-i) - return ret + def __init__(self, arr: list[int] = None, size: int = None) -> None: + """ + Constructor for the Fenwick tree + + Parameters: + arr (list): list of elements to initialize the tree with (optional) + size (int): size of the Fenwick tree (if arr is None) + """ + + if arr is None and size is not None: + self.size = size + self.tree = [0] * size + elif arr is not None: + self.init(arr) + else: + raise ValueError("Either arr or size must be specified") + + def init(self, arr: list[int]) -> None: + """ + Initialize the Fenwick tree with arr in O(N) + + Parameters: + arr (list): list of elements to initialize the tree with + + Returns: + None + + >>> a = [1, 2, 3, 4, 5] + >>> f1 = FenwickTree(a) + >>> f2 = FenwickTree(size=len(a)) + >>> for index, value in enumerate(a): + ... f2.add(index, value) + >>> f1.tree == f2.tree + True + """ + self.size = len(arr) + self.tree = deepcopy(arr) + for i in range(1, self.size): + j = self.next(i) + if j < self.size: + self.tree[j] += self.tree[i] + + def get_array(self) -> list[int]: + """ + Get the Normal Array of the Fenwick tree in O(N) + + Returns: + list: Normal Array of the Fenwick tree + + >>> a = [i for i in range(128)] + >>> f = FenwickTree(a) + >>> f.get_array() == a + True + """ + arr = self.tree[:] + for i in range(self.size - 1, 0, -1): + j = self.next(i) + if j < self.size: + arr[j] -= arr[i] + return arr + + @staticmethod + def next(index: int) -> int: + return index + (index & (-index)) + + @staticmethod + def prev(index: int) -> int: + return index - (index & (-index)) + + def add(self, index: int, value: int) -> None: + """ + Add a value to index in O(lg N) + + Parameters: + index (int): index to add value to + value (int): value to add to index + + Returns: + None + + >>> f = FenwickTree([1, 2, 3, 4, 5]) + >>> f.add(0, 1) + >>> f.add(1, 2) + >>> f.add(2, 3) + >>> f.add(3, 4) + >>> f.add(4, 5) + >>> f.get_array() + [2, 4, 6, 8, 10] + """ + if index == 0: + self.tree[0] += value + return + while index < self.size: + self.tree[index] += value + index = self.next(index) + + def update(self, index: int, value: int) -> None: + """ + Set the value of index in O(lg N) + + Parameters: + index (int): index to set value to + value (int): value to set in index + + Returns: + None + + >>> f = FenwickTree([5, 4, 3, 2, 1]) + >>> f.update(0, 1) + >>> f.update(1, 2) + >>> f.update(2, 3) + >>> f.update(3, 4) + >>> f.update(4, 5) + >>> f.get_array() + [1, 2, 3, 4, 5] + """ + self.add(index, value - self.get(index)) + + def prefix(self, right: int) -> int: + """ + Prefix sum of all elements in [0, right) in O(lg N) + + Parameters: + right (int): right bound of the query (exclusive) + + Returns: + int: sum of all elements in [0, right) + + >>> a = [i for i in range(128)] + >>> f = FenwickTree(a) + >>> res = True + >>> for i in range(len(a)): + ... res = res and f.prefix(i) == sum(a[:i]) + >>> res + True + """ + if right == 0: + return 0 + result = self.tree[0] + right -= 1 # make right inclusive + while right > 0: + result += self.tree[right] + right = self.prev(right) + return result + + def query(self, left: int, right: int) -> int: + """ + Query the sum of all elements in [left, right) in O(lg N) + + Parameters: + left (int): left bound of the query (inclusive) + right (int): right bound of the query (exclusive) + + Returns: + int: sum of all elements in [left, right) + + >>> a = [i for i in range(128)] + >>> f = FenwickTree(a) + >>> res = True + >>> for i in range(len(a)): + ... for j in range(i + 1, len(a)): + ... res = res and f.query(i, j) == sum(a[i:j]) + >>> res + True + """ + return self.prefix(right) - self.prefix(left) + + def get(self, index: int) -> int: + """ + Get value at index in O(lg N) + + Parameters: + index (int): index to get the value + + Returns: + int: Value of element at index + + >>> a = [i for i in range(128)] + >>> f = FenwickTree(a) + >>> res = True + >>> for i in range(len(a)): + ... res = res and f.get(i) == a[i] + >>> res + True + """ + return self.query(index, index + 1) + + def rank_query(self, value: int) -> int: + """ + Find the largest index with prefix(i) <= value in O(lg N) + NOTE: Requires that all values are non-negative! + + Parameters: + value (int): value to find the largest index of + + Returns: + -1: if value is smaller than all elements in prefix sum + int: largest index with prefix(i) <= value + + >>> f = FenwickTree([1, 2, 0, 3, 0, 5]) + >>> f.rank_query(0) + -1 + >>> f.rank_query(2) + 0 + >>> f.rank_query(1) + 0 + >>> f.rank_query(3) + 2 + >>> f.rank_query(5) + 2 + >>> f.rank_query(6) + 4 + >>> f.rank_query(11) + 5 + """ + value -= self.tree[0] + if value < 0: + return -1 + + j = 1 # Largest power of 2 <= size + while j * 2 < self.size: + j *= 2 + + i = 0 + + while j > 0: + if i + j < self.size and self.tree[i + j] <= value: + value -= self.tree[i + j] + i += j + j //= 2 + return i if __name__ == "__main__": - f = FenwickTree(100) - f.update(1, 20) - f.update(4, 4) - print(f.query(1)) - print(f.query(3)) - print(f.query(4)) - f.update(2, -5) - print(f.query(1)) - print(f.query(3)) + import doctest + + doctest.testmod()