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Commit 434a564

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Refactor segment tree implementation.
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‎README.md‎

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* [Binary Search Tree](https://github.com/trekhleb/javascript-algorithms/tree/master/src/data-structures/tree/binary-search-tree)
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* [AVL Tree](https://github.com/trekhleb/javascript-algorithms/tree/master/src/data-structures/tree/avl-tree)
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* [Red-Black Tree](https://github.com/trekhleb/javascript-algorithms/tree/master/src/data-structures/tree/red-black-tree)
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* [Segment Tree](https://github.com/trekhleb/javascript-algorithms/tree/master/src/data-structures/tree/segment-tree) - with min/max/sum range queries examples
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* [Graph](https://github.com/trekhleb/javascript-algorithms/tree/master/src/data-structures/graph) (both directed and undirected)
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* [Disjoint Set](https://github.com/trekhleb/javascript-algorithms/tree/master/src/data-structures/disjoint-set)
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‎src/data-structures/tree/segment-tree/README.md‎

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# Segment Tree
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A segment tree is a data structure designed to perform
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certain array operations efficiently - especially those
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involving range queries.
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A common application is the [Range Minimum Query](https://en.wikipedia.org/wiki/Range_minimum_query) (RMQ) problem,
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where we are given an array of numbers and need to
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support operations of updating values of the array and
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finding the minimum of a contiguous subarray.
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A segment tree implementation for the RMQ problem
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takes `O(n)` to initialize, and `O(log n)` per query or
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update. The "minimum" operation can be replaced by any
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array operation (such as sum).
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A segment tree is a binary tree with contiguous
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sub-arrays as nodes. The root of the tree represents the
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In computer science, a segment tree also known as a statistic tree
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is a tree data structure used for storing information about intervals,
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or segments. It allows querying which of the stored segments contain
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a given point. It is, in principle, a static structure; that is,
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it's a structure that cannot be modified once it's built. A similar
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data structure is the interval tree.
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A segment tree is a binary tree. The root of the tree represents the
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whole array. The two children of the root represent the
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first and second halves of the array. Similarly, the
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children of each node corresponds to the two halves of
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the array corresponding to the node. If the array has
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size `n`, we can prove that the segment tree has size at
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most `4n`. Each node stores the minimum of its
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corresponding sub-array.
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In the implementation, we do not explicitly store this
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tree structure, but represent it using a `4n` sized array.
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The left child of node i is `2i+1` and the right child
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is `2i+2`. This is a standard way to represent segment
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trees, and lends itself to an efficient implementation.
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the array corresponding to the node.
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We build the tree bottom up, with the value of each node
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being the minimum of its children's values. This will
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take time `O(n)`, with one operation for each node. Updates
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are also done bottom up, with values being recomputed
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starting from the leaf, and up to the root. The number
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being the "minimum" (or any other function) of its children's values. This will
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take `O(n log n)` time. The number
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of operations done is the height of the tree, which
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is `O(log n)`. To answer queries, each node splits the
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is `O(log n)`. To do range queries, each node splits the
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query into two parts, one sub-query for each child.
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If a query contains the whole subarray of a node, we
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can use the precomputed value at the node. Using this
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![Segment Tree](https://www.geeksforgeeks.org/wp-content/uploads/segment-tree1.png)
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## Application
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A segment tree is a data structure designed to perform
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certain array operations efficiently - especially those
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involving range queries.
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Applications of the segment tree are in the areas of computational geometry,
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and geographic information systems.
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Current implementation of Segment Tree implies that you may
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pass any binary (with two input params) function to it and
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thus you're able to do range query for variety of functions.
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In tests you may fins examples of doing `min`, `max` and `sam` range
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queries on SegmentTree.
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## References
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- [Wikipedia](https://en.wikipedia.org/wiki/Segment_tree)
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/**
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* Segment Tree implementation for Range Query data structure
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* Tracks a array of numbers. 0 indexed
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* operation is a binary function (eg sum, min) - needs to be associative
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* identity is the identity of the operation
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* i.e, operation(x, identity) = x (eg 0 for sum, Infinity for min)
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* Supports methods
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* update(index, val) - set value of index
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* query(l, r) - finds operation(values in range [l, r]) (both inclusive)
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*
11-
* As is customary, we store the tree implicitly with i being the parent of 2i, 2i+1.
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*/
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import isPowerOfTwo from '../../../algorithms/math/is-power-of-two/isPowerOfTwo';
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export default class SegmentTree {
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/**
16-
* array initialises the numbers
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* @param {number[]} array
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* @param {number[]} inputArray
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* @param {function} operation - binary function (i.e. sum, min)
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* @param {number} operationFallback - operation fallback value (i.e. 0 for sum, Infinity for min)
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*/
19-
constructor(array, operation, identity) {
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this.n = array.length;
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this.array = array;
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this.tree = new Array(4 * this.n);
23-
9+
constructor(inputArray, operation, operationFallback) {
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this.inputArray = inputArray;
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this.operation = operation;
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this.identity = identity;
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27-
// use Range Min Query by default
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if (this.operation === undefined) {
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this.operation = Math.min;
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this.identity = Infinity;
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}
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this.operationFallback = operationFallback;
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14+
// Init array representation of segment tree.
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this.segmentTree = this.initSegmentTree(this.inputArray);
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34-
this.build();
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this.buildSegmentTree();
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}
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/**
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* Stub for recursive call
21+
* @param {number[]} inputArray
22+
* @return {number[]}
3923
*/
40-
build() {
41-
this.buildRec(1,0,this.n-1);
42-
}
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initSegmentTree(inputArray) {
25+
letsegmentTreeArrayLength;
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constinputArrayLength=inputArray.length;
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44-
/**
45-
* Left child index
46-
* @param {number} root
47-
*/
48-
left(root) {
49-
return 2 * root;
28+
if (isPowerOfTwo(inputArrayLength)) {
29+
// If original array length is a power of two.
30+
segmentTreeArrayLength = (2 * inputArrayLength) - 1;
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} else {
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// If original array length is not a power of two then we need to find
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// next number that is a power of two and use it to calculate
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// tree array size. This is happens because we need to fill empty children
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// in perfect binary tree with nulls.And those nulls need extra space.
36+
const currentPower = Math.floor(Math.log2(inputArrayLength));
37+
const nextPower = currentPower + 1;
38+
const nextPowerOfTwoNumber = 2 ** nextPower;
39+
segmentTreeArrayLength = (2 * nextPowerOfTwoNumber) - 1;
40+
}
41+
42+
return new Array(segmentTreeArrayLength).fill(null);
5043
}
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5245
/**
53-
* Right child index
54-
* @param {number} root
46+
* Build segment tree.
5547
*/
56-
right(root) {
57-
return (2 * root) + 1;
48+
buildSegmentTree() {
49+
const leftIndex = 0;
50+
const rightIndex = this.inputArray.length - 1;
51+
const position = 0;
52+
this.buildTreeRecursively(leftIndex, rightIndex, position);
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}
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6055
/**
61-
* root is the index in the tree, [l,r] (inclusive) is the current array segment being built
62-
* @param {number} root
63-
* @param {number} l
64-
* @param {number} r
56+
* Build segment tree recursively.
57+
*
58+
* @param {number} leftInputIndex
59+
* @param {number} rightInputIndex
60+
* @param {number} position
6561
*/
66-
buildRec(root, l, r) {
67-
if (l === r) {
68-
this.tree[root] = this.array[l];
69-
} else {
70-
const mid = Math.floor((l + r) / 2);
71-
// build left and right nodes
72-
this.buildRec(this.left(root), l, mid);
73-
this.buildRec(this.right(root), mid + 1, r);
74-
this.tree[root] = this.operation(this.tree[this.left(root)], this.tree[this.right(root)]);
62+
buildTreeRecursively(leftInputIndex, rightInputIndex, position) {
63+
// If low input index and high input index are equal that would mean
64+
// the we have finished splitting and we are already came to the leaf
65+
// of the segment tree. We need to copy this leaf value from input
66+
// array to segment tree.
67+
if (leftInputIndex === rightInputIndex) {
68+
this.segmentTree[position] = this.inputArray[leftInputIndex];
69+
return;
7570
}
71+
72+
// Split input array on two halves and process them recursively.
73+
const middleIndex = Math.floor((leftInputIndex + rightInputIndex) / 2);
74+
// Process left half of the input array.
75+
this.buildTreeRecursively(leftInputIndex, middleIndex, this.getLeftChildIndex(position));
76+
// Process right half of the input array.
77+
this.buildTreeRecursively(middleIndex + 1, rightInputIndex, this.getRightChildIndex(position));
78+
79+
// Once every tree leaf is not empty we're able to build tree bottom up using
80+
// provided operation function.
81+
this.segmentTree[position] = this.operation(
82+
this.segmentTree[this.getLeftChildIndex(position)],
83+
this.segmentTree[this.getRightChildIndex(position)],
84+
);
7685
}
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7887
/**
79-
* Stub for recursive call
80-
* @param {number} lindex
81-
* @param {number} rindex
88+
* Do range query on segment tree in context of this.operation function.
89+
*
90+
* @param {number} queryLeftIndex
91+
* @param {number} queryRightIndex
92+
* @return {number}
8293
*/
83-
query(lindex, rindex) {
84-
return this.queryRec(1, lindex, rindex, 0, this.n - 1);
94+
rangeQuery(queryLeftIndex, queryRightIndex) {
95+
const leftIndex = 0;
96+
const rightIndex = this.inputArray.length - 1;
97+
const position = 0;
98+
99+
return this.rangeQueryRecursive(
100+
queryLeftIndex,
101+
queryRightIndex,
102+
leftIndex,
103+
rightIndex,
104+
position,
105+
);
85106
}
86107

87108
/**
88-
* [lindex, rindex] is the query region
89-
* [l,r] is the current region being processed
90-
* Guaranteed that [lindex,rindex] contained in [l,r]
91-
* @param {number} root
92-
* @param {number} lindex
93-
* @param {number} rindex
94-
* @param {number} l
95-
* @param {number} r
109+
* Do range query on segment tree recursively in context of this.operation function.
110+
*
111+
* @param {number} queryLeftIndex - left index of the query
112+
* @param {number} queryRightIndex - right index of the query
113+
* @param {number} leftIndex - left index of input array segment
114+
* @param {number} rightIndex - right index of input array segment
115+
* @param {number} position - root position in binary tree
116+
* @return {number}
96117
*/
97-
queryRec(root, lindex, rindex, l, r) {
98-
// console.log(root, lindex, rindex, l, r);
99-
if (lindex > rindex) {
100-
// happens when mid+1 > r - no segment
101-
return this.identity;
118+
rangeQueryRecursive(queryLeftIndex, queryRightIndex, leftIndex, rightIndex, position) {
119+
if (queryLeftIndex <= leftIndex && queryRightIndex >= rightIndex) {
120+
// Total overlap.
121+
return this.segmentTree[position];
102122
}
103-
if (l === lindex && r === rindex) {
104-
// query region matches current region - use tree value
105-
return this.tree[root];
123+
124+
if (queryLeftIndex > rightIndex || queryRightIndex < leftIndex) {
125+
// No overlap.
126+
return this.operationFallback;
106127
}
107-
const mid = Math.floor((l + r) / 2);
108-
// get left and right results and combine
109-
const leftResult = this.queryRec(this.left(root), lindex, Math.min(rindex, mid), l, mid);
110-
const rightResult = this.queryRec(
111-
this.right(root), Math.max(mid + 1, lindex), rindex,
112-
mid + 1, r,
128+
129+
// Partial overlap.
130+
const middleIndex = Math.floor((leftIndex + rightIndex) / 2);
131+
132+
const leftOperationResult = this.rangeQueryRecursive(
133+
queryLeftIndex,
134+
queryRightIndex,
135+
leftIndex,
136+
middleIndex,
137+
this.getLeftChildIndex(position),
113138
);
114-
return this.operation(leftResult, rightResult);
139+
140+
const rightOperationResult = this.rangeQueryRecursive(
141+
queryLeftIndex,
142+
queryRightIndex,
143+
middleIndex + 1,
144+
rightIndex,
145+
this.getRightChildIndex(position),
146+
);
147+
148+
return this.operation(leftOperationResult, rightOperationResult);
115149
}
116150

117151
/**
118-
* Set array[index] to value
119-
* @param {number} index
120-
* @param {number} value
152+
* Left child index.
153+
* @param {number} parentIndex
154+
* @return {number}
121155
*/
122-
update(index, value) {
123-
this.array[index] = value;
124-
this.updateRec(1, index, value, 0, this.n - 1);
156+
getLeftChildIndex(parentIndex) {
157+
return (2 * parentIndex) + 1;
125158
}
126159

127160
/**
128-
* @param {number} root
129-
* @param {number} index
130-
* @param {number} value
131-
* @param {number} l
132-
* @param {number} r
161+
* Right child index.
162+
* @param {number} parentIndex
163+
* @return {number}
133164
*/
134-
updateRec(root, index, value, l, r) {
135-
if (l === r) {
136-
// we are at tree node containing array[index]
137-
this.tree[root] = value;
138-
} else {
139-
const mid = Math.floor((l + r) / 2);
140-
// update whichever child index is in, update this.tree[root]
141-
if (index <= mid) {
142-
this.updateRec(this.left(root), index, value, l, mid);
143-
} else {
144-
this.updateRec(this.right(root), index, value, mid + 1, r);
145-
}
146-
this.tree[root] = this.operation(this.tree[this.left(root)], this.tree[this.right(root)]);
147-
}
165+
getRightChildIndex(parentIndex) {
166+
return (2 * parentIndex) + 2;
148167
}
149168
}

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