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Commit434a564

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Refactor segment tree implementation.
1 parent5784a4a commit434a564

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-251
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4 files changed

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‎README.md

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Original file line numberDiff line numberDiff line change
@@ -32,6 +32,7 @@ the data.
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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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@@ -1,41 +1,23 @@
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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,
8-
where we are given an array of numbers and need to
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support operations of updating values of the array and
10-
finding the minimum of a contiguous subarray.
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A segment tree implementation for the RMQ problem
12-
takes`O(n)` to initialize, and`O(log n)` per query or
13-
update. The "minimum" operation can be replaced by any
14-
array operation (such as sum).
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16-
A segment tree is a binary tree with contiguous
17-
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
4+
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
6+
a given point. It is, in principle, a static structure; that is,
7+
it's a structure that cannot be modified once it's built. A similar
8+
data structure is the interval tree.
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10+
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
23-
most`4n`. Each node stores the minimum of its
24-
corresponding sub-array.
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26-
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
29-
is`2i+2`. This is a standard way to represent segment
30-
trees, and lends itself to an efficient implementation.
14+
the array corresponding to the node.
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We build the tree bottom up, with the value of each node
33-
being the minimum of its children's values. This will
34-
take time`O(n)`, with one operation for each node. Updates
35-
are also done bottom up, with values being recomputed
36-
starting from the leaf, and up to the root. The number
17+
being the "minimum" (or any other function) of its children's values. This will
18+
take`O(n log n)` time. The number
3719
of operations done is the height of the tree, which
38-
is`O(log n)`. Toanswer queries, each node splits the
20+
is`O(log n)`. Todo 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
@@ -44,6 +26,21 @@ operations are done.
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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
32+
certain array operations efficiently - especially those
33+
involving range queries.
34+
35+
Applications of the segment tree are in the areas of computational geometry,
36+
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
42+
queries on SegmentTree.
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4744
##References
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4946
-[Wikipedia](https://en.wikipedia.org/wiki/Segment_tree)
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/**
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* Segment Tree implementation for Range Query data structure
3-
* Tracks a array of numbers. 0 indexed
4-
* operation is a binary function (eg sum, min) - needs to be associative
5-
* identity is the identity of the operation
6-
* i.e, operation(x, identity) = x (eg 0 for sum, Infinity for min)
7-
* Supports methods
8-
* update(index, val) - set value of index
9-
* query(l, r) - finds operation(values in range [l, r]) (both inclusive)
10-
*
11-
* As is customary, we store the tree implicitly with i being the parent of 2i, 2i+1.
12-
*/
1+
importisPowerOfTwofrom'../../../algorithms/math/is-power-of-two/isPowerOfTwo';
132

143
exportdefaultclassSegmentTree{
154
/**
16-
* array initialises the numbers
17-
*@param {number[]} array
5+
*@param {number[]} inputArray
6+
*@param {function} operation - binary function (i.e. sum, min)
7+
*@param {number} operationFallback - operation fallback value (i.e. 0 for sum, Infinity for min)
188
*/
19-
constructor(array,operation,identity){
20-
this.n=array.length;
21-
this.array=array;
22-
this.tree=newArray(4*this.n);
23-
9+
constructor(inputArray,operation,operationFallback){
10+
this.inputArray=inputArray;
2411
this.operation=operation;
25-
this.identity=identity;
26-
27-
// use Range Min Query by default
28-
if(this.operation===undefined){
29-
this.operation=Math.min;
30-
this.identity=Infinity;
31-
}
12+
this.operationFallback=operationFallback;
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14+
// Init array representation of segment tree.
15+
this.segmentTree=this.initSegmentTree(this.inputArray);
3316

34-
this.build();
17+
this.buildSegmentTree();
3518
}
3619

3720
/**
38-
* Stub for recursive call
21+
*@param {number[]} inputArray
22+
*@return {number[]}
3923
*/
40-
build(){
41-
this.buildRec(1,0,this.n-1);
42-
}
24+
initSegmentTree(inputArray){
25+
letsegmentTreeArrayLength;
26+
constinputArrayLength=inputArray.length;
4327

44-
/**
45-
* Left child index
46-
*@param {number} root
47-
*/
48-
left(root){
49-
return2*root;
28+
if(isPowerOfTwo(inputArrayLength)){
29+
// If original array length is a power of two.
30+
segmentTreeArrayLength=(2*inputArrayLength)-1;
31+
}else{
32+
// If original array length is not a power of two then we need to find
33+
// next number that is a power of two and use it to calculate
34+
// tree array size. This is happens because we need to fill empty children
35+
// in perfect binary tree with nulls.And those nulls need extra space.
36+
constcurrentPower=Math.floor(Math.log2(inputArrayLength));
37+
constnextPower=currentPower+1;
38+
constnextPowerOfTwoNumber=2**nextPower;
39+
segmentTreeArrayLength=(2*nextPowerOfTwoNumber)-1;
40+
}
41+
42+
returnnewArray(segmentTreeArrayLength).fill(null);
5043
}
5144

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+
constleftIndex=0;
50+
constrightIndex=this.inputArray.length-1;
51+
constposition=0;
52+
this.buildTreeRecursively(leftIndex,rightIndex,position);
5853
}
5954

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-
constmid=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+
constmiddleIndex=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
}
7786

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-
returnthis.queryRec(1,lindex,rindex,0,this.n-1);
94+
rangeQuery(queryLeftIndex,queryRightIndex){
95+
constleftIndex=0;
96+
constrightIndex=this.inputArray.length-1;
97+
constposition=0;
98+
99+
returnthis.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-
returnthis.identity;
118+
rangeQueryRecursive(queryLeftIndex,queryRightIndex,leftIndex,rightIndex,position){
119+
if(queryLeftIndex<=leftIndex&&queryRightIndex>=rightIndex){
120+
// Total overlap.
121+
returnthis.segmentTree[position];
102122
}
103-
if(l===lindex&&r===rindex){
104-
// query region matches current region - use tree value
105-
returnthis.tree[root];
123+
124+
if(queryLeftIndex>rightIndex||queryRightIndex<leftIndex){
125+
// No overlap.
126+
returnthis.operationFallback;
106127
}
107-
constmid=Math.floor((l+r)/2);
108-
// get left and right results and combine
109-
constleftResult=this.queryRec(this.left(root),lindex,Math.min(rindex,mid),l,mid);
110-
constrightResult=this.queryRec(
111-
this.right(root),Math.max(mid+1,lindex),rindex,
112-
mid+1,r,
128+
129+
// Partial overlap.
130+
constmiddleIndex=Math.floor((leftIndex+rightIndex)/2);
131+
132+
constleftOperationResult=this.rangeQueryRecursive(
133+
queryLeftIndex,
134+
queryRightIndex,
135+
leftIndex,
136+
middleIndex,
137+
this.getLeftChildIndex(position),
113138
);
114-
returnthis.operation(leftResult,rightResult);
139+
140+
constrightOperationResult=this.rangeQueryRecursive(
141+
queryLeftIndex,
142+
queryRightIndex,
143+
middleIndex+1,
144+
rightIndex,
145+
this.getRightChildIndex(position),
146+
);
147+
148+
returnthis.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-
constmid=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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