# ACM-ICPC 2017 模擬国内予選: F. マトリョーシカ

,

## solution

1. まず体積のことを無視すると、頂点path被覆になる
• これは二部matchingに帰着させて解ける
• 二部matchingは最大流で解ける
2. ここに重みを載せる
• 重み最大二部matching
• これは最小費用流: 重みを負で載せて、必ず$F$流せるように適当に辺を張る
• $\mathrm{source} \to$\mathrm{sink}$の費用$0$の辺を張るのが単純 • あるいは各マトリョーシカで$\mathrm{i} \to $\mathrm{i}$の費用$0$の辺を張るとsourceからの到達不能点が消えるため楽かも

## implementation

#include <algorithm>
#include <array>
#include <cstdio>
#include <queue>
#include <tuple>
#include <vector>
#define repeat(i, n) for (int i = 0; (i) < int(n); ++(i))
#define whole(f, x, ...) ([&](decltype((x)) whole) { return (f)(begin(whole), end(whole), ## __VA_ARGS__); })(x)
using namespace std;
template <class T> inline void setmin(T & a, T const & b) { a = min(a, b); }
template <class T> using reversed_priority_queue = priority_queue<T, vector<T>, greater<T> >;

template <class T>
struct edge_t { int to; T cap, cost; int rev; };
template <class T>
void add_edge(vector<vector<edge_t<T> > > & graph, int from, int to, T cap, T cost) {
graph[from].push_back((edge_t<T>) {   to, cap,  cost, int(graph[  to].size())     });
graph[  to].push_back((edge_t<T>) { from,  0, - cost, int(graph[from].size()) - 1 });
}
/**
* @brief minimum-cost flow with primal-dual method
* @note mainly O(V^2UC) for U is the sum of capacities and C is the sum of costs. and additional O(VE) if negative edges exist
*/
template <class T>
T min_cost_flow_destructive(int src, int dst, T flow, vector<vector<edge_t<T> > > & graph) {
T result = 0;
vector<T> potential(graph.size());
if (0 < flow) { // initialize potential when negative edges exist (slow). you can remove this if unnecessary
whole(fill, potential, numeric_limits<T>::max());
potential[src] = 0;
while (true) { // Bellman-Ford algorithm
bool updated = false;
repeat (e_from, graph.size()) for (auto & e : graph[e_from]) if (e.cap) {
if (potential[e_from] == numeric_limits<T>::max()) continue;
if (potential[e.to] > potential[e_from] + e.cost) {
potential[e.to] = potential[e_from] + e.cost; // min
updated = true;
}
}
if (not updated) break;
}
}
while (0 < flow) {
// update potential using dijkstra
vector<T> distance(graph.size(), numeric_limits<T>::max()); // minimum distance
vector<int> prev_v(graph.size()); // constitute a single-linked-list represents the flow-path
vector<int> prev_e(graph.size());
{ // initialize distance and prev_{v,e}
reversed_priority_queue<pair<T, int> > que; // distance * vertex
distance[src] = 0;
que.emplace(0, src);
while (not que.empty()) { // Dijkstra's algorithm
T d; int v; tie(d, v) = que.top(); que.pop();
if (potential[v] == numeric_limits<T>::max()) continue;
if (distance[v] < d) continue;
// look round the vertex
repeat (e_index, graph[v].size()) {
// consider updating
edge_t<T> e = graph[v][e_index];
int w = e.to;
if (potential[w] == numeric_limits<T>::max()) continue;
T d1 = distance[v] + e.cost + potential[v] - potential[w]; // updated distance
if (0 < e.cap and d1 < distance[e.to]) {
distance[w] = d1;
prev_v[w] = v;
prev_e[w] = e_index;
que.emplace(d1, w);
}
}
}
}
if (distance[dst] == numeric_limits<T>::max()) return -1; // no such flow
repeat (v, graph.size()) {
if (potential[v] == numeric_limits<T>::max()) continue;
potential[v] += distance[v];
}
// finish updating the potential
// let flow on the src->dst minimum path
T delta = flow; // capacity of the path
for (int v = dst; v != src; v = prev_v[v]) {
setmin(delta, graph[prev_v[v]][prev_e[v]].cap);
}
flow -= delta;
result += delta * potential[dst];
for (int v = dst; v != src; v = prev_v[v]) {
edge_t<T> & e = graph[prev_v[v]][prev_e[v]]; // reference
e.cap -= delta;
graph[v][e.rev].cap += delta;
}
}
return result;
}

int main() {
while (true) {
int n; scanf("%d", &n);
if (n == 0) break;
vector<array<int, 3> > size(n);
repeat (i, n) {
repeat (j, 3) {
scanf("%d", &size[i][j]);
}
whole(sort, size[i]);
}
auto volume = [&](int i) { return size[i][0] * size[i][1] * size[i][2]; };
int total_volume = 0;
vector<vector<edge_t<int> > > g(2 * n + 2);
const int src = 2 * n;
const int dst = 2 * n + 1;
repeat (i, n) {
add_edge(g, src, i, 1, 0);
repeat (j, n) {
if (    size[i][0] > size[j][0] and
size[i][1] > size[j][1] and
size[i][2] > size[j][2]) {
add_edge(g, i, n + j, 1, - volume(j));
}
}
add_edge(g, n + i, dst, 1, 0);
total_volume += volume(i);
}
add_edge(g, src, dst, n, 0);
int result = total_volume + min_cost_flow_destructive(src, dst, n, g);
printf("%d\n", result);
}
return 0;
}