119 lines
2.7 KiB
Text
119 lines
2.7 KiB
Text
---
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id: treaps
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title: "Treaps"
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author: Benjamin Qi
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description: "?"
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prerequisites:
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- Platinum - Range Update Range Query
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frequency: 2
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---
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import { Problem } from "../models";
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export const metadata = {
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problems: {
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treeRot: [
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new Problem("POI", "Tree Rotations 2", "https://szkopul.edu.pl/problemset/problem/b0BM0al2crQBt6zovEtJfOc6/site/?key=statement", "Normal", false, [], ""),
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]
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}
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};
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## Merging
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## Splitting
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## Merging (Arbitrary Keys)
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<Problems problems={metadata.problems.treeRot} />
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<Spoiler title="Tree Rotations 2 Solution">
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Apparently [this paper](https://www.cs.cmu.edu/~scandal/papers/treaps-spaa98.pdf) proves that this runs in $O(n\log n)$.
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```cpp
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int n;
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ll ans, inv;
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typedef struct tnode* pt;
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struct tnode {
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int pri, val; pt c[2]; // essential
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int sz; // for range queries
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tnode (int _val) {
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pri = rng(); val = _val;
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sz = 1; c[0] = c[1] = NULL;
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}
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};
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int getsz(pt x) { return x?x->sz:0; }
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pt calc(pt x) {
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x->sz = 1+getsz(x->c[0])+getsz(x->c[1]);
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return x;
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}
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pair<pt,pt> split(pt t, int v) { // >= v goes to the right
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if (!t) return {t,t};
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if (t->val >= v) {
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auto p = split(t->c[0], v); t->c[0] = p.s;
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return {p.f,calc(t)};
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} else {
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auto p = split(t->c[1], v); t->c[1] = p.f;
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return {calc(t),p.s};
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}
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}
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pt merge(pt a, pt b) {
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if (!a || !b) return a?:b;
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if (a->pri > b->pri) {
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auto B = split(b,a->val);
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inv += (ll)(1+getsz(a->c[0]))*getsz(B.s);
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a->c[0] = merge(a->c[0],B.f); a->c[1] = merge(a->c[1],B.s);
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return calc(a);
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} else {
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auto A = split(a,b->val);
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inv += (ll)getsz(A.f)*(1+getsz(b->c[1]));
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b->c[0] = merge(A.f,b->c[0]); b->c[1] = merge(A.s,b->c[1]);
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return calc(b);
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}
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}
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pt go() {
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int x; re(x);
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if (x) return new tnode(x);
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pt L = go(); pt R = go();
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ll tot = (ll)L->sz*R->sz; inv = 0;
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pt cur = merge(L,R); assert(inv <= tot); // dbg("HUH",tot,inv);
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ans += min(inv,tot-inv);
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return cur;
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}
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int main() {
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setIO(); re(n); go();
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ps(ans);
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}
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```
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</Spoiler>
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<Optional>
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[This](https://codeforces.com/blog/entry/67980) CF post describes a more general version of the problem above.
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</Optional>
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## Tutorial
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- [Quora](https://threads-iiith.quora.com/Treaps-One-Tree-to-Rule-em-all-Part-1)
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- [PPT](https://docs.google.com/presentation/d/14xgtdDWnIBwmJRAuIdZ8FvLZcX9uRxnNoGOGAQRDIvc/edit?usp=sharing)
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- Samuel Hsiang Guide (see resources)
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## Problems
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- [Old Gold - Airplane Boarding](http://www.usaco.org/index.php?page=viewproblem2&cpid=402)
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- [Strings](https://csacademy.com/contest/archive/task/strings/) [](181)
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- [Points & Distances](https://www.hackerearth.com/problem/algorithm/septembereasy-points-and-distances-d30d0e6b/description/) [](185)
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IOI 2013 Game - https://oj.uz/submission/242393
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A2OJ: https://a2oj.com/category?ID=14
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? |