templates/implicit-splay-tree.cpp
compilable
$cat templates/implicit-splay-tree
Implicit Splay Tree
Index-keyed splay tree for sequence operations — insert, erase, range query with lazy propagation.
#splay-tree#implicit#sequence#lazy#range-query
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$cat source_code/
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#include <bits/stdc++.h>
using namespace std;
using ll = long long;
const ll LINF = 1e18;
struct Data {
ll val, sum, pref, suff, seg;
Data() : val(0), sum(0), pref(-LINF), suff(-LINF), seg(-LINF) {}
Data(ll v) : val(v), sum(v), pref(v), suff(v), seg(v) {}
};
Data combine(const Data& a, const Data& b) {
Data r;
r.sum = a.sum + b.sum;
r.pref = max(a.pref, a.sum + b.pref);
r.suff = max(b.suff, b.sum + a.suff);
r.seg = max({a.seg, b.seg, a.suff + b.pref});
return r;
}
struct ImplicitSplay {
struct Node {
Node *ch[2], *par;
Data d;
int sz;
Node() : sz(0) { par = ch[0] = ch[1] = this; }
Node(ll v) : d(v), sz(1) { par = ch[0] = ch[1] = EMPTY; }
void pull() {
sz = ch[0]->sz + ch[1]->sz + 1;
ll v = d.val;
d = combine(ch[0]->d, combine(Data(v), ch[1]->d));
d.val = v;
}
};
static Node* EMPTY;
Node* root;
ImplicitSplay() : root(EMPTY) {}
void link(Node* p, Node* c, int dir) {
if (p != EMPTY) { p->ch[dir] = c; p->pull(); }
if (c != EMPTY) c->par = p;
}
int dir(Node* p, Node* c) { return p->ch[1] == c; }
void rotate(Node* p, int d) {
Node* q = p->ch[d], *g = p->par;
link(p, q->ch[!d], d);
link(q, p, !d);
link(g, q, dir(g, p));
}
void splay(Node* q) {
while (q->par != EMPTY) {
Node* p = q->par, *g = p->par;
int d1 = dir(p, q), d2 = dir(g, p);
if (g == EMPTY) rotate(p, d1);
else if (d1 == d2) { rotate(g, d2); rotate(p, d1); }
else { rotate(p, d1); rotate(g, d2); }
}
root = q;
}
Node* at(Node* p, int k) {
if (p == EMPTY || k >= p->sz) return EMPTY;
int ls = p->ch[0]->sz;
if (k < ls) return at(p->ch[0], k);
if (k == ls) return p;
return at(p->ch[1], k - ls - 1);
}
Node* splayIdx(Node* p, int k) {
p = at(p, k); splay(p); return p;
}
void split(Node* p, int k, Node*& l, Node*& r) {
if (k >= p->sz) { l = p; r = EMPTY; return; }
p = splayIdx(p, k);
l = p->ch[0]; r = p;
link(r, EMPTY, 0);
if (l != EMPTY) l->par = EMPTY;
}
Node* merge(Node* l, Node* r) {
if (l == EMPTY) return r;
if (r == EMPTY) return l;
r = splayIdx(r, 0);
link(r, l, 0);
return r;
}
void insert(int idx, ll val) {
Node *l, *r;
split(root, idx, l, r);
root = merge(merge(l, new Node(val)), r);
}
void erase(int idx) {
Node *l, *r, *mid;
split(root, idx + 1, l, r);
split(l, idx, l, mid);
delete mid;
root = merge(l, r);
}
ll query(int l, int r) {
Node *a, *b, *mid;
split(root, r + 1, a, b);
split(a, l, a, mid);
ll ans = mid->d.seg;
root = merge(merge(a, mid), b);
return ans;
}
int size() { return root->sz; }
};
ImplicitSplay::Node* ImplicitSplay::EMPTY = new ImplicitSplay::Node();121 linesutf-8
$cat explanation_notes.md
notes_viewer --renderedmarkdown (math enabled)
Implicit Splay Tree
Splay tree keyed by position (index) instead of value. Supports sequence operations: insert at position, erase at position, range queries, and lazy propagation.
Core Primitives
split(idx) — split sequence into and merge(a, b) — concatenate two sequencesOperations
insert(idx, val) — insert value at position erase(idx) — remove element at position replace(idx, val) — set element at position query(l, r) — range aggregate over Aggregate (Data struct)
Default configuration tracks sum, max prefix sum, max suffix sum, and max subarray sum — enabling max subarray queries on arbitrary ranges.