$\newcommand{\O}{\mathrm{O}}$

おそらく誰も知らないようなアルゴリズム. supperbubble と呼ばれる構造を線形時間ですべて列挙するアルゴリズム.
元論文は "Linear-Time Superbubble Identification Algorithm for
Genome Assembly" [Brankovic et al. 2015]
DNA のゲノム解析などで使えるそうです.
構築: $\O (n)$, クエリ: $\O (1)$
template<typename T> class RMQ
{
private:
struct RMQNode
{
int id;
T val; //インデックス, 値
int par, left, right; //親, 左の子, 右の子
};
std::vector<RMQNode> tree;
std::vector<T> arr;
std::vector<int> euler_tour, depth, diff, visit_id;
int node_size, root, arr_len;
bool max_query;
std::vector<int> block_arr, diff_bit; //各ブロックで深さが最小になるようなインデックス 各ブロックのdiffの情報をビットに詰めたもの
std::vector<std::vector<int> > sparse_table; //i番目から長さ2^kの区間に含まれるdepthのうち最小のインデックス
std::vector<std::vector<std::vector<int> > > table_lookup;
int block_size, block_rem, block_cnt, log_block_cnt;
void make_cartesian_tree();
void make_euler_tour(int cur_node, int& id, int cur_depth);
void make_block_arr();
void make_sparse_table();
void make_diff_bit();
void make_table_lookup();
std::pair<int, T> PM_RMQ(int st, int ed);
public:
void build(std::vector<T>& arg1, bool max_query_=false); //trueのときrange_max_queryを表す
std::pair<int, T> query(int st, int ed); // (最小(大)値を達成するインデックス, 値)
};
template<typename T>
void RMQ<T>::build(std::vector<T>& arg1, bool max_query_){
node_size = (int)arg1.size();
assert(node_size >= 2);
arr.resize(node_size, 0);
max_query = max_query_;
if(max_query){
for(int i = 0; i < node_size; i++){
arr[i] = -arg1[i];
}
}else{
for(int i = 0; i < node_size; i++){
arr[i] = arg1[i];
}
}
make_cartesian_tree();
euler_tour.resize(2 * node_size - 1, -1), depth.resize(2 * node_size - 1, -1);
diff.resize(2 * node_size - 2, -1), visit_id.resize(node_size, -1);
int val = 0;
make_euler_tour(root, val, 0);
for(int i = 0;i < 2 * node_size - 2; i++){
diff[i] = depth[i+1] - depth[i];
}
//±RMQの実装
arr_len = 2 * node_size - 1;
make_block_arr();
make_sparse_table();
make_diff_bit();
make_table_lookup();
}
template<typename T>
void RMQ<T>::make_cartesian_tree()
{
tree.resize(node_size);
for(int i = 0; i < node_size; i++){
tree[i] = (RMQNode){i, arr[i], -1, -1, -1};
}
std::stack<std::pair<int, T> > st; //インデックス,値
st.push(std::make_pair(0,arr[0]));
root = 0;
for(int i = 1; i < node_size ; i++){
int buff;
while(1){
//iが根となる場合
if(st.empty()){
st.push(std::make_pair(i, arr[i]));
tree[i].left = buff;
tree[buff].par = i;
root = i;
break;
}
std::pair<int, T> top = st.top();
if(top.second <= arr[i]){
tree[i].par = top.first; //iの親を変更
int nd = tree[top.first].right; //iの親の右下のノード
//iの子となるものが存在する場合
if(nd != -1){
tree[i].left = nd; //iの親の右下のノードをiの左の子とする
tree[nd].par = i; //iの親の右下のノードの親をiとする
}
tree[top.first].right = i; //iの親の右下の子をiとする
st.push(std::make_pair(i,arr[i]));
break;
}
buff = top.first; //最後にpopされたものを暗記
st.pop();
}
}
}
template<typename T>
void RMQ<T>::make_euler_tour(int cur_node, int& id, int cur_depth)
{
visit_id[cur_node] = id;
euler_tour[id] = cur_node;
depth[id++] = cur_depth;
if(tree[cur_node].left >= 0){
make_euler_tour(tree[cur_node].left, id, cur_depth+1);
euler_tour[id] = cur_node;
depth[id++] = cur_depth;
}
if(tree[cur_node].right >= 0){
make_euler_tour(tree[cur_node].right, id, cur_depth+1);
euler_tour[id] = cur_node;
depth[id++] = cur_depth;
}
}
template<typename T>
void RMQ<T>::make_block_arr()
{
block_size = ceil(log2(arr_len)/2);
block_cnt = (arr_len - 1) / block_size + 1;
block_rem = arr_len % block_size;
log_block_cnt = ceil(log2(block_cnt)) + 1;
block_arr.resize(block_cnt);
for(int i = 0; i < block_cnt; i++){
int mn = numeric_limits<int>::max();
int mn_id = -1;
for(int j = 0; j < block_size; j++){
int now_id = i * block_size + j;
if(now_id >= arr_len){
break;
}
if(depth[now_id] < mn){
mn = depth[now_id];
mn_id = now_id;
}
}
block_arr[i] = mn_id;
}
}
template<typename T>
void RMQ<T>::make_sparse_table()
{
sparse_table.resize(block_cnt, std::vector<int>(log_block_cnt, -1));
for(int i = 0; i < block_cnt; i++){
sparse_table[i][0] = block_arr[i];
}
for(int j = 0; j < log_block_cnt - 1; j++){
for(int i = 0; i < block_cnt; i++){
if(i + (1 << j) >= block_cnt){
sparse_table[i][j + 1] = sparse_table[i][j];
}else{
if(depth[sparse_table[i][j]] <= depth[sparse_table[i + (1 << j)][j]]){
sparse_table[i][j + 1] = sparse_table[i][j];
}else{
sparse_table[i][j + 1] = sparse_table[i + (1 << j)][j];
}
}
}
}
}
template<typename T>
void RMQ<T>::make_diff_bit()
{
diff_bit.resize(block_cnt, -1);
for(int i = 0; i < block_cnt; i++){
int num = 0;
for(int j = 0; j < block_size - 1; j++){
int now_id = i * block_size + j;
if(now_id >= arr_len){
break;
}
if(diff[now_id] > 0){
num += (1 << (block_size - j - 2));
}
}
diff_bit[i] = num;
}
}
template<typename T>
void RMQ<T>::make_table_lookup()
{
table_lookup.resize((1 << block_size), std::vector<std::vector<int> >(block_size + 1, std::vector<int>(block_size + 1, 0)));
//0は減少,1は増加
for(int i = 0; i < (1 << block_size); i++){
std::vector<int> vec(block_size - 1, -1);
for(int j = 0; j < block_size; j++){
if(i & (1 << (block_size - j - 2))){
vec[j] = 1;
}
}
for(int j = 0; j < block_size; j++){
int nw = 0, mn = 0, mn_id = j;
table_lookup[i][j][j+1] = j;
for(int k = j + 2; k <= block_size; k++){
nw += vec[k-2];
if(nw < mn){
mn = nw;
mn_id = k-1;
}
table_lookup[i][j][k] = mn_id;
}
}
}
}
//インデックスと値のpair
template<typename T>
std::pair<int, T> RMQ<T>::PM_RMQ(int st, int ed)
{
int st_block_id = (st == 0)?0:((st - 1) / block_size + 1); //ブロック区間の開始
int ed_block_id = ed / block_size - 1; //ブロック区間の終了
int st_id = st / block_size;
int ed_id = ed / block_size;
int st_rem = st % block_size;
int ed_rem = ed % block_size;
if(ed_block_id - st_block_id < 0){ //間にブロック区間がひとつもない場合
if(st_id == ed_id){ //同じブロック区間に存在する場合
int id_kari = block_size * st_id + table_lookup[diff_bit[st_id]][st_rem][ed_rem + 1];
return std::make_pair(euler_tour[id_kari], arr[euler_tour[id_kari]]);
}else{
int cand1 = block_size * st_id + table_lookup[diff_bit[st_id]][st_rem][block_size];
int cand2 = block_size * ed_id + table_lookup[diff_bit[ed_id]][0][ed_rem + 1];
if(depth[cand1] <= depth[cand2]){
return std::make_pair(euler_tour[cand1], arr[euler_tour[cand1]]);
}else{
return std::make_pair(euler_tour[cand2], arr[euler_tour[cand2]]);
}
}
}else{ //間にブロックっ区間が存在する場合
int num = floor(log2(ed_block_id - st_block_id + 1));
int cand1 = sparse_table[st_block_id][num];
int cand2 = sparse_table[ed_block_id - (1 << num) + 1][num];
int c1,c2;
if(depth[cand1] <= depth[cand2]){
c1 = cand1;
}else{
c1 = cand2;
}
int cand3 = block_size * st_id + table_lookup[diff_bit[st_id]][st_rem][block_size];
int cand4 = block_size * ed_id + table_lookup[diff_bit[ed_id]][0][ed_rem + 1];
if(depth[cand3] <= depth[cand4]){
c2 = cand3;
}else{
c2 = cand4;
}
if(depth[c1] <= depth[c2]){
return std::make_pair(euler_tour[c1], arr[euler_tour[c1]]);
}else{
return std::make_pair(euler_tour[c2], arr[euler_tour[c2]]);
}
}
}
template<typename T>
std::pair<int, T> RMQ<T>::query(int st, int ed)
{
assert(st < ed);
std::pair<int, T> res = PM_RMQ(std::min(visit_id[st], visit_id[ed-1]), std::max(visit_id[st], visit_id[ed-1]));
if(max_query) res.second = -res.second;
return res;
}
void graph_input(int edge_size, std::vector<std::vector<int> >& graph, std::vector<std::vector<int> >& rgraph)
{
for(int i = 0; i < edge_size; i++){
int arg1, arg2;
scanf("%d%d", &arg1, &arg2);
graph[arg1].push_back(arg2);
rgraph[arg2].push_back(arg1);
}
}
void recursive_topological_sort(int u, int& index, std::vector<int>& visit, std::vector<int>& ord,
std::vector<int>& ordD, std::vector<std::vector<int> >& graph)
{
if(visit[u] == 1){
return;
}
visit[u] = 0;
for(int nx : graph[u]){
recursive_topological_sort(nx, index, visit, ord, ordD, graph);
}
visit[u] = 1;
ord[index] = u;
ordD[u] = index--;
}
void topological_sort(int node_size, std::vector<int>& ord, std::vector<int>& ordD, std::vector<std::vector<int> >& graph)
{
//与えられたグラフがconnected かつ DAGであることを仮定
std::vector<int> visit(node_size, -1);
int index = node_size - 1;
for(int i = 0; i < node_size; i++){
if(visit[i] == -1){
recursive_topological_sort(i, index, visit, ord, ordD, graph);
}
}
}
void prepare_superbubble(int node_size, std::vector<bool>& ent, std::vector<bool>& exi,std::vector<int>& ordD, std::vector<int>& outparent,
std::vector<int>& outchild, std::vector<std::vector<int> >& graph, std::vector<std::vector<int> >& rgraph)
{
for(int i = 0; i < node_size; i++){
bool ok = false;
for(int j = 0; j < (int)rgraph[i].size(); j++){
if((int)graph[rgraph[i][j]].size() == 1){
ok = true;
break;
}
}
exi[i] = ok;
ok = false;
for(int j = 0; j < (int)graph[i].size(); j++){
if((int)rgraph[graph[i][j]].size() == 1){
ok = true;
break;
}
}
ent[i] = ok;
}
for(int i = 0; i < node_size; i++){
int mn_ordD = node_size;
for(int j = 0; j < (int)rgraph[i].size(); j++){
mn_ordD = std::min(mn_ordD, ordD[rgraph[i][j]]);
}
outparent[ordD[i]] = mn_ordD;
int mx_ordD = -1;
for(int j = 0; j < (int)graph[i].size(); j++){
mx_ordD = std::max(mx_ordD, ordD[graph[i][j]]);
}
outchild[ordD[i]] = mx_ordD;
}
}
int validate_superbubble(int st_node, int ed_node, std::vector<bool>& ent, std::vector<int>& ord, std::vector<int>& ordD,
std::vector<int>& prev_ent, RMQ<int>& parent_rmq, RMQ<int>& child_rmq)
{
int st_id = ordD[st_node];
int ed_id = ordD[ed_node];
int mx_child = child_rmq.query(st_id, ed_id).second;
int mn_parent = parent_rmq.query(st_id + 1, ed_id + 1).second;
if(mx_child != ed_id){
return -1;
}
if(mn_parent == st_id){
return st_node;
}else if(ent[ord[mn_parent]]){
return ord[mn_parent];
}else{
return prev_ent[ord[mn_parent]];
}
}
void report_superbubble(std::list<std::pair<int, int> >& cand, int st_node, int ed_node, std::vector<bool>& ent, std::vector<bool>& exi,
std::vector<int>& ord, std::vector<int>& ordD, std::vector<int>& prev_ent, std::vector<int>& next_ent, std::vector<int>& alt_ent,
RMQ<int>& parent_rmq, RMQ<int>& child_rmq, std::vector<std::pair<int, int> >& superbubble_set)
//ex_nodeで終わるsuperbubbleを存在すれば報告する
{
if(st_node < 0 || ed_node < 0 || ordD[st_node] >= ordD[ed_node]){
cand.pop_back();
return;
}
int prevent_node = prev_ent[ed_node];
int valid = -1;
while(ordD[prevent_node] >= ordD[st_node]){
valid = validate_superbubble(prevent_node, ed_node, ent, ord, ordD, prev_ent, parent_rmq, child_rmq);
if(valid == prevent_node || valid == alt_ent[prevent_node] || valid == -1){
break;
}
alt_ent[prevent_node] = valid;
prevent_node = valid;
}
cand.pop_back();
if(valid == prevent_node){
superbubble_set.push_back(std::make_pair(prevent_node, ed_node));
while((cand.back()).first != prevent_node){
if((cand.back()).second == 1){
//nestされたsuperbubbleについて調べる
report_superbubble(cand, next_ent[prevent_node], (cand.back()).first, ent, exi, ord, ordD, prev_ent, next_ent, alt_ent,
parent_rmq, child_rmq, superbubble_set);
}else{
cand.pop_back();
}
}
}
}
void superbubble(int node_size, std::vector<std::vector<int> >& graph, std::vector<std::vector<int> >& rgraph,
std::vector<std::pair<int, int> >& superbubble_set)
{
std::vector<int> ord(node_size, -1);
std::vector<int> ordD(node_size, -1);
topological_sort(node_size, ord, ordD, graph);
std::vector<bool> ent(node_size, false), exi(node_size, false);
std::vector<int> outparent(node_size, -1), outchild(node_size, -1);
std::vector<int> alt_ent(node_size, -1), prev_ent(node_size, -1), next_node(node_size, -1);
prepare_superbubble(node_size, ent, exi, ordD, outparent, outchild, graph, rgraph);
std::list<std::pair<int, int> > cand; //(頂点の値,entかexiのフラグ)
std::vector<int> next_ent(node_size, -1);
RMQ<int> parent_rmq, child_rmq;
parent_rmq.build(outparent, 0), child_rmq.build(outchild, 1);
int prev_entrance = -1;
int next_entrance = -1;
for(int ver : ord){
alt_ent[ver] = -1;
prev_ent[ver] = prev_entrance;
if(exi[ver]){
cand.push_back(std::make_pair(ver, 1));
}
if(ent[ver]){
cand.push_back(std::make_pair(ver, 0));
prev_entrance = ver;
if(next_entrance >= 0){
next_ent[next_entrance] = ver;
}
next_entrance = ver;
}
}
while(!cand.empty()){
if((cand.back()).second == 0){
cand.pop_back();
}else{
report_superbubble(cand, (cand.front()).first, (cand.back()).first, ent, exi, ord, ordD, prev_ent, next_ent, alt_ent,
parent_rmq, child_rmq, superbubble_set);
}
}
}
int main()
{
int node_size, edge_size;
scanf("%d%d", &node_size, &edge_size);
std::vector<std::vector<int> > graph(node_size);
std::vector<std::vector<int> > rgraph(node_size);
graph_input(edge_size, graph, rgraph);
std::vector<std::pair<int, int> > superbubble_set;
superbubble(node_size, graph, rgraph, superbubble_set);
for(int i = 0; i < (int)superbubble_set.size(); i++){
printf("%d %d\n",superbubble_set[i].first, superbubble_set[i].second);
}
}
verify していません(verify 問題を知らない)