Hi!
Here are some implementations for solving RMQ (Tarjan’s algorithm) (Range Maximum / Minimum Query).
It’s very simple to implement and its time complexity is O((n + q)·a(n)), a() stands for Akerman inverse function used in DSU.
Problem: Given array a of n integers, and q queries, for each query print the maximum value in range [L, R].
Solution: We need an array of vectors, called assigned. assigned[r] contains queries that their R is r. When getting queries, push each query in assigned[R]. We need a dsu, first pari is i. We need a stack, named st.
For i from 0 to n, do:
While st is not empty and a[st.top] <= a[i]
Set i parent of st.top in dsu and pop this element from st.
Push i to st
For each query assigned to i
The answer to this query is a[root of L of this query in DSU].
Code here.
// God & me
// "Someone like you"?! Unbelievable ...
#include <bits/stdc++.h>
using namespace std;
const int maxn = 1e5 + 17;
int par[maxn], ans[maxn], n, a[maxn], l[maxn], q;
vector<int> qu[maxn];
int root(int v){
return par[v] == -1 ? v : par[v] = root(par[v]);
}
int main(){
ios::sync_with_stdio(0), cin.tie(0);
memset(par, -1, sizeof par);
cin >> n;
for(int i = 0; i < n; i++) cin >> a[i];
cin >> q;
for(int i = 0, r; i < q; i++){
cin >> l[i] >> r, r--, l[i]--;
qu[r].push_back(i);
}
stack<int> st;
for(int i = 0; i < n; i++){
while(st.size() && a[st.top()] <= a[i])
par[st.top()] = i, st.pop();
st.push(i);
for(auto qi : qu[i])
ans[qi] = a[root(l[qi])];
}
for(int i = 0; i < q; i++)
cout << ans[i] << ' ';
cout << '\n';
return 0;
}
Note that in the above code I used the path-compression technique for dsu only, size-comparing technique can be used too (but it has lower performance).
It’s obviously true because each time for any j ≤ i, a[root(j)] is the greatest value in range [j, i].
Performance test
This method (known as Arpas trick)
// God & me
// "Someone like you"?! Unbelievable ...
#include <bits/stdc++.h>
using namespace std;
const int maxn = 1 << 22;
int par[maxn], ans[maxn], n, a[maxn], l[maxn], q, head[maxn], prv[maxn], st[maxn], tail;
int root(int v){
return par[v] == -1 ? v : par[v] = root(par[v]);
}
int main(){
ios::sync_with_stdio(0), cin.tie(0);
memset(par, -1, sizeof par);
memset(head, -1, sizeof head);
cin >> n;
for(int i = 0; i < n; i++) cin >> a[i];
cin >> q;
for(int i = 0, r; i < q; i++){
cin >> l[i] >> r, r--, l[i]--;
prv[i] = head[r], head[r] = i;
}
for(int i = 0; i < n; i++){
while(tail-- && a[ st[tail] ] <= a[i])
par[ st[tail] ] = i;
st[++tail] = i;
for(int qi = head[i]; qi >= 0; qi = prv[qi])
ans[qi] = a[root(l[qi])];
}
for(int i = 0; i < q; i++)
cout << ans[i] << ' ';
cout << '\n';
return 0;
}
Vector + Binary search
// God & me
// "Someone like you"?! Unbelievable ...
#include <bits/stdc++.h>
using namespace std;
const int maxn = 1 << 22;
int n, q, a[maxn], ans[maxn], head[maxn], prv[maxn], l[maxn];
vector<int> v, qu[maxn];
int main(){
ios::sync_with_stdio(0), cin.tie(0);
memset(head, -1, sizeof head);
cin >> n;
for(int i = 0; i < n; i++) cin >> a[i];
cin >> q;
for(int i = 0, r; i < q; i++){
cin >> l[i] >> r;
prv[i] = head[r], head[r] = i;
}
for(int i = 0; i < n; i++){
while(v.size() && a[v.back()] <= a[i]) v.pop_back();
v.push_back(i);
for(int qi = head[i]; qi >= 0; qi = prv[qi])
ans[qi] = a[*lower_bound(v.begin(), v.end(), l[qi])];
}
for(int i = 0; i < q; i++)
cout << ans[i] << ' ';
cout << '\n';
return 0;
}
Sparse table
// God & me
// "Someone like you"?! Unbelievable ...
#include <bits/stdc++.h>
using namespace std;
const int maxn = 1 << 22, lg = 22;
int par[maxn], ans[maxn], n, q, lg2[maxn];
int spt[lg][maxn];
int main(){
ios::sync_with_stdio(0), cin.tie(0);
memset(par, -1, sizeof par);
cin >> n;
for(int i = 0; i < n; i++)
cin >> spt[0][i];
for(int i = 2; i < n; i++) lg2[i] = lg2[i >> 1] + 1;
for(int k = 1; k < lg; k++)
for(int i = 0; i + (1 << k) <= n; i++)
spt[k][i] = max(spt[k - 1][i], spt[k - 1][i + 1 << (k - 1)]);
cin >> q;
for(int i = 0, l, r; i < q; i++){
cin >> l >> r, r++;
cout << max(spt[ lg2[r - l] ][l], spt[ lg2[r - l] ][ r - (1 << lg2[r - l]) ]) << ' ';
}
cout << '\n';
return 0;
}
O(n) method
// God & me
// "Someone like you"?! Unbelievable ...
#include <bits/stdc++.h>
using namespace std;
const int maxn = 1 << 22;
template<typename Val, typename Compare = std::less<Val>, int BlockSize = 10>
class DirectRMQ {
public:
typedef int Index; //今のところ大きくともintを仮定している(queryとか)
typedef char InBlockIndex;
typedef InBlockIndex(*BlockTypeRef)[BlockSize];
DirectRMQ(Compare comp_ = Compare()) :
blockTypes(0), innerBlockTable(0), sparseTable(0) {
comp = comp_;
calcBallotNumbers();
buildInnerBlockTable();
}
~DirectRMQ() {
delete[] innerBlockTable;
delete[] blockTypes; delete[] sparseTable;
}
void build(const Val *a, Index n) {
blocks = (n + BlockSize - 1) / BlockSize;
stHeight = 0; while(1 << stHeight < blocks) ++ stHeight;
delete[] blockTypes; delete[] sparseTable;
blockTypes = new BlockTypeRef[blocks];
calcBlockTypes(a, n);
buildInnerBlockTable(a, n);
sparseTable = new Index[blocks * stHeight];
buildSparseTable(a);
}
//[l,r]の閉区間
Index query(const Val *a, Index l, Index r) const {
Index x = l / BlockSize, y = r / BlockSize, z = y - x;
if(z == 0) return x * BlockSize + blockTypes[x][l % BlockSize][r % BlockSize];
if(z == 1) return assumeleft_minIndex(a,
x * BlockSize + blockTypes[x][l % BlockSize][BlockSize - 1],
y * BlockSize + blockTypes[y][0][r % BlockSize]);
z -= 2;
Index k = 0, s;
s = ((z & 0xffff0000) != 0) << 4; z >>= s; k |= s;
s = ((z & 0x0000ff00) != 0) << 3; z >>= s; k |= s;
s = ((z & 0x000000f0) != 0) << 2; z >>= s; k |= s;
s = ((z & 0x0000000c) != 0) << 1; z >>= s; k |= s;
s = ((z & 0x00000002) != 0) << 0; z >>= s; k |= s;
return assumeleft_minIndex(a
, assumeleft_minIndex(a,
x * BlockSize + blockTypes[x][l % BlockSize][BlockSize - 1],
sparseTable[x + 1 + blocks * k])
, assumeleft_minIndex(a,
sparseTable[y + blocks * k - (1 << k)],
y * BlockSize + blockTypes[y][0][r % BlockSize])
);
}
Val queryVal(const Val *a, Index l, Index r) const {
Index x = l / BlockSize, y = r / BlockSize, z = y - x;
if(z == 0) return a[x * BlockSize + blockTypes[x][l % BlockSize][r % BlockSize]];
Val edge = minVal(
a[x * BlockSize + blockTypes[x][l % BlockSize][BlockSize - 1]],
a[y * BlockSize + blockTypes[y][0][r % BlockSize]]);
if(z == 1) return edge;
z -= 2;
Index k = 0, s;
s = ((z & 0xffff0000) != 0) << 4; z >>= s; k |= s;
s = ((z & 0x0000ff00) != 0) << 3; z >>= s; k |= s;
s = ((z & 0x000000f0) != 0) << 2; z >>= s; k |= s;
s = ((z & 0x0000000c) != 0) << 1; z >>= s; k |= s;
s = ((z & 0x00000002) != 0) << 0; z >>= s; k |= s;
return minVal(edge, minVal(
a[sparseTable[x + 1 + blocks * k]],
a[sparseTable[y + blocks * k - (1 << k)]]));
}
private:
Compare comp;
int ballotNumbers[BlockSize + 1][BlockSize + 1];
InBlockIndex(*innerBlockTable)[BlockSize][BlockSize];
Index blocks;
int stHeight;
BlockTypeRef *blockTypes;
Index *sparseTable;
inline Index minIndex(const Val *a, Index x, Index y) const {
return comp(a[x], a[y]) || (a[x] == a[y] && x < y) ? x : y;
}
inline Index assumeleft_minIndex(const Val *a, Index x, Index y) const {
return comp(a[y], a[x]) ? y : x;
}
inline Val minVal(Val x, Val y) const {
return comp(y, x) ? y : x;
}
void buildSparseTable(const Val *a) {
Index *b = sparseTable;
if(stHeight) for(Index i = 0; i < blocks; i ++)
b[i] = i * BlockSize + blockTypes[i][0][BlockSize - 1];
for(Index t = 1; t * 2 < blocks; t *= 2) {
std::memcpy(b + blocks, b, blocks * sizeof(Index));
b += blocks;
for(Index i = 0; i < blocks - t; ++ i)
b[i] = assumeleft_minIndex(a, b[i], b[i + t]);
}
}
void buildInnerBlockTable(const Val *a, Index n) {
for(Index i = 0; i < blocks; i ++) {
BlockTypeRef table = blockTypes[i];
if(table[0][0] != -1) continue;
const Val *p = getBlock(a, n, i);
for(InBlockIndex left = 0; left < BlockSize; left ++) {
Val minV = p[left];
InBlockIndex minI = left;
for(InBlockIndex right = left; right < BlockSize; right ++) {
if(comp(p[right], minV)) {
minV = p[right];
minI = right;
}
table[left][right] = minI;
}
}
}
}
//端っこのブロック用に関数内staticなテンポラリ配列を返す
const Val *getBlock(const Val *a, Index n, Index i) {
Index offset = i * BlockSize;
if(offset + BlockSize <= n)
return a + offset;
else {
static Val tmp_a[BlockSize];
std::copy(a + offset, a + n, tmp_a);
Val maxVal = Val();
for(Index j = i; j < n; j ++) //iでなくoffsetでは？(動作には問題ないし計算量もほとんど変わらないけれど…)(バグるのが嫌なので(今まで動いていたので)直すのは後にする)
if(comp(maxVal, a[j])) maxVal = a[j];
std::fill(tmp_a + (n - offset), tmp_a + BlockSize, maxVal);
return tmp_a;
}
}
void calcBlockTypes(const Val *a, Index n) {
Val tmp_rp[BlockSize + 1];
for(Index i = 0; i < blocks; i ++)
blockTypes[i] = calcBlockType(getBlock(a, n, i), tmp_rp);
}
BlockTypeRef calcBlockType(const Val *a, Val *rp) {
int q = BlockSize, N = 0;
for(int i = 0; i < BlockSize; i ++) {
while(q + i - BlockSize > 0 && comp(a[i], rp[q + i - BlockSize])) {
N += ballotNumbers[BlockSize - i - 1][q];
q --;
}
rp[q + i + 1 - BlockSize] = a[i];
}
return innerBlockTable[N];
}
void calcBallotNumbers() {
for(int p = 0; p <= BlockSize; p ++) {
for(int q = 0; q <= BlockSize; q ++) {
if(p == 0 && q == 0)
ballotNumbers[p][q] = 1;
else if(p <= q)
ballotNumbers[p][q] =
(q ? ballotNumbers[p][q - 1] : 0) +
(p ? ballotNumbers[p - 1][q] : 0);
else
ballotNumbers[p][q] = 0;
}
}
}
void buildInnerBlockTable() {
int numberOfTrees = ballotNumbers[BlockSize][BlockSize];
innerBlockTable = new InBlockIndex[numberOfTrees][BlockSize][BlockSize];
for(int i = 0; i < numberOfTrees; i ++)
innerBlockTable[i][0][0] = -1;
}
};
int n, q, a[maxn];
int main(){
ios::sync_with_stdio(0), cin.tie(0);
cin >> n;
for(int i = 0; i < n; i++)
cin >> a[i];
DirectRMQ<int> rmq;
rmq.build(a, n);
cin >> q;
for(int i = 0, l, r; i < q; i++){
cin >> l >> r;
cout << rmq.queryVal(a, l, r) << ' ';
}
cout << '\n';
return 0;
}
generator
// God & me
// "Someone like you"?! Unbelievable ...
#include <bits/stdc++.h>
using namespace std;
const int n = 1 << 22;
int a[n];
int main(){
ios::sync_with_stdio(0), cin.tie(0);
srand(time(0));
cout << n << '\n';
iota(a, a + n, 0);
random_shuffle(a, a + n); // used for generating random test
for(int i = 0; i < n; i++)
cout << a[i] << " \n"[ i == n - 1 ];
cout << n << '\n';
for(int i = 0; i < n; i++){
int l = rand() % n, r = rand() % n;
if(l > r) swap(l, r);
cout << l << ' ' << r << '\n';
}
return 0;
}
Here is the result:
| Method\Time(milliseconds) | Strictly increasing array | Strictly decreasing array | Random |
| This method (known as Arpa's trick) | 2943 | 2890 | 2946 |
| Sparse table | 3612 | 3595 | 3807 |
| Vector + Binary search | 3101 | 6130 | 3153 |
| O(n) method | 3788 | 3920 | 3610 |
