SD Segment Tree Variation, A Shorter And More Efficient Segment Tree

Revision en11, by sword060, 2023-07-30 23:14:13

Segment Tree is a powerful data structure in programming, that is why it can still be optimized way more. In this blog I will explain one optimization that can make a basic segment tree slightly faster and easier to write. (idea and the code by me)

This does not work on range update range query segment trees.

Introduction:

Let's consider a point update range query segment tree, while querying we visit many of useless Nodes along the way in order to answer the query moving from the root downwards.

As you can see, there are nodes (marked in red) that are not needed during the recursion, and we only need to visit the important nodes (marked in green).

This is only true when querying in a point update segment tree or updating in a point query segment tree.

Main Idea:

We can solve a query range $$$[l, r]$$$ by noticing we can make it a smaller range $$$[l + X , r]$$$, where $$$X$$$ is any power of two but we need it to be maximum (in order to reduce the time complexity) and these two conditions should be true:

  • $$$l + X - 1 \le r$$$. (We cannot go out of the range)
  • $$$[l, l + X - 1]$$$ is a valid node in the segment tree.
The first condition:
The second condition

At the end, we can solve it now because $$$X$$$ is $$$2$$$ power the minimum between $$$log_2(r-l+1)$$$ and $$$log_2( M \& -M )$$$ because it satisfies the first and second conditions and is the maximum value possible.

C++ Code:

We can preprocess $$$log_2(K)$$$ for each $$$1 \le K \le N$$$ in an array.

Note that this only works when $$$N$$$ (the number of leaves) is a power of 2.

At each step we calculate the size of the movement $$$X$$$ which is equal to $$$2^K$$$

The following codes calculate sum in the range $$$L$$$ to $$$R$$$, assuming the segment tree is built after possibly several update queries.

Recursive:

long long query(int l, int r){
	if(l > r)return 0;
	int node = N + l - 1;
	int K = min(logs[node & -node], logs[r - l + 1]);
	return (query(l + (1 << K), r) + seg[node >> K]);
}

Iterative:

long long query(int l, int r){
	long long ret = 0;
	while(l<=r){
		int node = N + l - 1;
		int K = min(logs[node & -node], logs[r - l + 1]);
		ret = (ret + seg[node >> K]);
		l += (1 << K);
	}
	return ret;
}

This can also be applied to range update point query segment trees:

void update(int l, int r, int val){
	while(l<=r){
		int node = N + l - 1;
                int K = min(logs[node & -node], logs[r - l + 1]);
		seg[node >> K] += val;
		lazy[node >> K] += val;
		l += (1 << K);
	}
}

Benchmark:

Test-Cases Generator
SD-Segment-Tree Code
Iterative-Segment-Tree Code
Recursive-Segment-Tree Code
Size of the array and the number of queries Time of SD-Segment-Tree /S Time of Recursive-Segment-Tree /S Time of Iterative-Segment-Tree /S
$$$N,Q = 2^{16}$$$ 00.2847 00.3163 00.2292
$$$N,Q = 2^{17}$$$ 00.4311 00.5335 00.4414
$$$N,Q = 2^{18}$$$ 00.8322 00.9534 00.9729
$$$N,Q = 2^{19}$$$ 01.9915 02.1086 01.6837
$$$N,Q = 2^{20}$$$ 03.6747 04.4253 03.7347
$$$N,Q = 2^{21}$$$ 08.0204 08.6896 07.7844
$$$N,Q = 2^{22}$$$ 20.9266 27.0589 24.3542
$$$N,Q = 2^{23}$$$ 50.0656 61.9385 49.8065

Conclusion:

This variation has the same time complexity as the normal segment tree $$$O(log(N))$$$ per query, but might need more memory if you preprocess Logs array.

The constant factor is smaller because of the unnecessary nodes we don't visit but in practice the time it takes is not significant than the normal segment tree for smaller array sizes.

This can only be useful for squeezing in time limits or for becoming an easier way to implement segment trees because it is shorter.

UPD: Added Benchmark

History

 
 
 
 
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  Rev. Lang. By When Δ Comment
en11 English sword060 2023-07-30 23:14:13 30 Tiny change: 's shorter.' -> 's shorter.\n\n#### UPD: Added Benchmark'
en10 English sword060 2023-07-30 23:11:23 6088 (published)
en9 English sword060 2023-07-30 22:05:13 6257 Reverted to en7
en8 English sword060 2023-07-30 21:59:13 6257 (saved to drafts)
en7 English sword060 2023-07-27 12:22:05 8 Tiny change: 'ting in a range query seg' -> 'ting in a point query seg'
en6 English sword060 2023-07-27 12:18:47 125 Tiny change: ' to write.\n\nThis d' -> ' to write. (idea and the code by me)\n\nThis d'
en5 English sword060 2023-07-26 23:19:16 15
en4 English sword060 2023-07-26 21:56:37 38
en3 English sword060 2023-07-26 20:38:05 0 (published)
en2 English sword060 2023-07-26 20:36:32 1120 Tiny change: 're true:\n- $l + X' -> 're true:\n\n- $l + X'
en1 English sword060 2023-07-26 18:31:33 3213 Initial revision (saved to drafts)