Implemented binary indexed tree.

Binary Indexed Tree (Fenwick Tree)/Binary Indexed Tree.playground/Contents.swift

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+let array = [1.0, 2.0, 3.0, 4.0, 5.0, 6.0]
+
+let tree = BinaryIndexedTree<Double>(array: array)
+
+// calculates the sum within [0-5] inclusive. Should be 21
+tree.query(from: 0, to: 5)
+
+// updates value at index 4 from 5.0 to 0.5
+tree.update(at: 4, with: 0.5)
+
+// calculates the sum within [0-5] inclusive. Should be 16.5
+tree.query(from: 0, to: 5)
+

Binary Indexed Tree (Fenwick Tree)/Binary Indexed Tree.playground/Sources/BinaryIndexedTree.swift

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+/**
+    Data structure that can efficiently update elements and calculate prefix sums in a table of numbers
+ */
+public class BinaryIndexedTree<T: Numeric> {
+
+    private var values: [T]
+    private var tree: [T]
+
+    public init(array: [T]) {
+        self.values = Array(repeating: 0, count: array.count)
+
+        // first element (ie index 0) is root
+        self.tree = Array(repeating: 0, count: array.count + 1)
+
+        for (index, value) in array.enumerated() {
+            update(at: index, with: value)
+        }
+    }
+
+    // Updates value at given index with a new value
+    // Complexity: O(ln(n))
+    public func update(at index: Int, with value: T) {
+        updateInternally(at: index, with: value - values[index])
+    }
+
+    // Calculates sum of the given interval.
+    // Inteval given by [from - to] inclusive
+    // Complexity: O(ln(n))
+    public func query(from lowerIndex: Int, to upperIndex: Int) -> T {
+        return queryInternally(to: upperIndex) - queryInternally(to: lowerIndex - 1)
+    }
+
+    private func updateInternally(at index: Int, with diff: T) {
+        self.values[index] += diff
+
+        var treeIndex = index + 1
+        while isValid(index: treeIndex) {
+            tree[treeIndex] += diff
+            treeIndex = sibling(for: treeIndex)
+        }
+    }
+
+    private func queryInternally(to upperIndex: Int) -> T {
+        var sum: T = 0
+
+        var treeIndex = upperIndex + 1
+
+        while isValid(index: treeIndex) {
+            sum += tree[treeIndex]
+            treeIndex = parent(for: treeIndex)
+        }
+
+        return sum
+    }
+
+    private func isValid(index: Int) -> Bool {
+        return index > 0 && index < tree.count
+    }
+
+    private func parent(for index: Int) -> Int {
+        return index - index & (~index + 1);
+    }
+
+    private func sibling(for index: Int) -> Int {
+        return index + index & (~index + 1);
+    }
+
+}

Binary Indexed Tree (Fenwick Tree)/Binary Indexed Tree.playground/contents.xcplayground

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+<?xml version="1.0" encoding="UTF-8" standalone="yes"?>
+<playground version='5.0' target-platform='macos' buildActiveScheme='true'>
+    <timeline fileName='timeline.xctimeline'/>
+</playground>

Binary Indexed Tree (Fenwick Tree)/BinaryIndexedTree.swift

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+/**
+    Data structure that can efficiently update elements and calculate prefix sums in a table of numbers
+ */
+public class BinaryIndexedTree<T: Numeric> {
+
+    private var values: [T]
+    private var tree: [T]
+
+    public init(array: [T]) {
+        self.values = Array(repeating: 0, count: array.count)
+
+        // first element (ie index 0) is root
+        self.tree = Array(repeating: 0, count: array.count + 1)
+
+        for (index, value) in array.enumerated() {
+            update(at: index, with: value)
+        }
+    }
+
+    // Updates value at given index with a new value
+    // Complexity: O(ln(n))
+    public func update(at index: Int, with value: T) {
+        updateInternally(at: index, with: value - values[index])
+    }
+
+    // Calculates sum of the given interval.
+    // Inteval given by [from - to] inclusive
+    // Complexity: O(ln(n))
+    public func query(from lowerIndex: Int, to upperIndex: Int) -> T {
+        return queryInternally(to: upperIndex) - queryInternally(to: lowerIndex - 1)
+    }
+
+    private func updateInternally(at index: Int, with diff: T) {
+        self.values[index] += diff
+
+        var treeIndex = index + 1
+        while isValid(index: treeIndex) {
+            tree[treeIndex] += diff
+            treeIndex = sibling(for: treeIndex)
+        }
+    }
+
+    private func queryInternally(to upperIndex: Int) -> T {
+        var sum: T = 0
+
+        var treeIndex = upperIndex + 1
+
+        while isValid(index: treeIndex) {
+            sum += tree[treeIndex]
+            treeIndex = parent(for: treeIndex)
+        }
+
+        return sum
+    }
+
+    private func isValid(index: Int) -> Bool {
+        return index > 0 && index < tree.count
+    }
+
+    private func parent(for index: Int) -> Int {
+        return index - index & (~index + 1);
+    }
+
+    private func sibling(for index: Int) -> Int {
+        return index + index & (~index + 1);
+    }
+
+}

Binary Indexed Tree (Fenwick Tree)/README.markdown

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+# Binary Indexed (Fenwick) tree
+
+Fenwick Tree is a data structure that can efficiently update elements and calculate prefix sums in a table of numbers.
+
+## Motivation
+
+Let's begin with a small example. Imagine we have an array of numbers:
+
+```
+[1, 2, 3, 4, 5, 6]
+```
+
+We would like to support:
+
+1. query the sum on some interval
+
+2. update the value of a specified element
+
+There are a lot of ways to implement these operations:
+
+- The most straightforward way is to keep array "as is". Complexity of sum operation would be **O(n)** as we need to calculate sum each time we query it, but the complexity of update will be **O(1)**.
+
+- Next approach comes from dynamic programming. We can precalculate prefix sums inside of some buffer. For the given example precalculated prefix sum is `[1, 3, 6, 10, 15, 21]`. Then we can observe that cumulative sum on interval `(lower-upper)` is `precalculateSum[upper] - precalculateSum[lower] + values[lower]`. Complexity of sum query is **O(1)**, but supporting update operation requires to update all precalculated sums ahead of given index, so, it is **O(n)** for update operation.
+
+- Fenwick tree allows to support these operations in efficient way. Both complexities are **O(ln(n))**.
+
+## Basic
+
+Fenwick Tree supports two basic operations:
+
+1. **update(index, newValue)**: Updates value inside of an array at the given index
+
+2. **query(lowerBound, upperBound)**: Calculates desired cumulative sum on a give range. 
+
+
+## Basic idea
+
+Binary indexed tree can be represented as an array of size **n + 1**, where **n** is a length of initial array. The data structure is called tree as there is nice representation the data structure as tree.
+
+Basic idea of binary indexed tree is based on a fact that any number can be represented as the sum of powers of 2. For example:
+
+```
+7 = 2^2 + 2^1 + 2^0
+32 = 2^5
+13 = 2^3 + 2^2 + 2^0
+```
+
+Each node of the tree stores the sum on the interval from **node_index - 2^b** exclusive to **node_index** inclusive. The **b** stands for the position of last significant bit of the **node_index**. The __0th__ node is a dummy node and keep 0.
+
+```
+array <- [ 1, 2, 3, 4, 5, 6] (array indexed starting from 1)
+
+stored range <-  [ 0 - 0 (dummy, no associated array values), 0 - 1, 0 - 2, 2 - 3, 0 - 4, 4 - 5, 4 - 6]
+
+tree <-          [ 0 (dummy), 1, 3, 3, 10, 5, 11]
+
+```
+
+## Sum
+
+To calculate the sum on the interval from **1** up to **val**:
+
+1. While the given index bigger than `0`:
+
+1.1 Find the end of the current interval by extracting last significant byte. To extract last byte you can use the following rule `val - val & (~val + 1)`. This number is the starting point of interval associated with node, when the original one is the end. **3** node keeps the representation of the interval.
+
+1.2 Accumulate the current node value, which is stored inside of a tree. The desired value is tree[index]
+
+1.3 Update the number to the start of it's interval.
+
+1.4 Repeat
+
+For example to calculate the sum from **1** to **3** for the array [**1, 2, 3**, 4, 5, 6] we should do:
+
+0. Tree for thus array looks like [ 0 (dummy), 1, 3, 3, 10, 5, 11].
+
+1. Index is **3**
+
+1.1 `3 = 3 - 3 & (~3 + 1) = 3 - 3 & (0 + 1) = 3 - 3 & 1 = 3 - 1 = 2`, so the interval of the node tree[3] (tree is 0 indexed) is [2-3]
+
+1.2 Sum is 3
+
+1.3 Update the index to **2**
+
+2. Index is **2**
+
+2.1 `2 = 2 - 2 & (~2 + 1) = 2 - 2 & (1 + 1) = 2 - 2 & 2 = 0`. The interval is [0-2], node is tree[2]
+
+2.2  Sum is 3 (from step 1) + 3 (from current step, tree[2])
+
+2.3 Update the index to **0**
+
+3. Index is **0**. Returned sum is **6**.
+
+## Update
+
+Update works like the sum, except it looking for a intervals ahead.
+
+1. While the given index smaller than `n + 1`:
+
+1.1 Find the start of the new interval by extracting last significant byte. To do it you can use the following rule `val + val & (~val + 1)`.
+
+1.2 Accumulate the current node value
+
+1.3 Update the number to the start of following interval.
+
+
+## See also
+
+See the playground for more examples of how to use this data structure.
+
+[Fenwick Tree at Wikipedia](https://en.wikipedia.org/wiki/Fenwick_tree)
+
+*Written for Swift Algorithm Club by [Alexander Dadukin](https://github.com/st235)