Baltic OI 2015 - Editor
Author: Benjamin Qi
Time Complexity: .
Let lev[x]=max(0,-a[x])
.
Let active_y[x]=true
if operation x
is active after y
operations, and
false
otherwise. Obviously active_y[y]=true
.
Let activepath_y[x]=true
if operation x
could possibly be undone after
another operation, and false
otherwise. In other words, active_y[x]=true
and
no z>x
exists with active_y[z]=true
and lev[z]<=lev[x]
. If this is the
case, we'll say that "x
lies on the active path of y
." Obviously,
activepath_y[y]=true
.
Claim: If activepath_y[x]
then active_y[1..x]=active_x[1..x]
.
Proof: We use induction. Suppose that this is true for y=1..q
and we we
want to prove it for y=q+1
. If lev[q+1]=0
then q+1
is the only vertex on
the active path for q+1
, so this obviously holds. Otherwise, assume that
operation q+1
undoes operation z
on the active path for q
. By the
inductive hypothesis, active_q[1..z]=active_z[1..z]
, which implies that
active_{q+1}[1..z-1]=active_{z-1}[1..z-1]
.
All operations on the active path for q+1
aside from q+1
itself also lie on
the active path for z-1
. Also, for any t
on the active path for z-1
,
active_t[1..t]=active_{z-1}[1..t]=active_{q+1}[1..t]
. This completes the
proof.
So the solution is to maintain the active path for every i
. If there exists an
operation with level 0 on the active path for i
, then its state is the answer
for i
; otherwise, the answer for i
is 0.
#include <bits/stdc++.h>using namespace std;using ll = long long;using ld = long double;using db = double;using str = string; // yay python!using pi = pair<int, int>;using pl = pair<ll, ll>;
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