#[allow(unused_imports)] use std::cmp::*; #[allow(unused_imports)] use std::collections::*; use std::io::{Write, BufWriter}; // https://qiita.com/tanakh/items/0ba42c7ca36cd29d0ac8 macro_rules! input { ($($r:tt)*) => { let stdin = std::io::stdin(); let mut bytes = std::io::Read::bytes(std::io::BufReader::new(stdin.lock())); let mut next = move || -> String{ bytes.by_ref().map(|r|r.unwrap() as char) .skip_while(|c|c.is_whitespace()) .take_while(|c|!c.is_whitespace()) .collect() }; input_inner!{next, $($r)*} }; } macro_rules! input_inner { ($next:expr) => {}; ($next:expr,) => {}; ($next:expr, $var:ident : $t:tt $($r:tt)*) => { let $var = read_value!($next, $t); input_inner!{$next $($r)*} }; } macro_rules! read_value { ($next:expr, ( $($t:tt),* )) => { ($(read_value!($next, $t)),*) }; ($next:expr, [ $t:tt ; $len:expr ]) => { (0..$len).map(|_| read_value!($next, $t)).collect::>() }; ($next:expr, chars) => { read_value!($next, String).chars().collect::>() }; ($next:expr, usize1) => (read_value!($next, usize) - 1); ($next:expr, [ $t:tt ]) => {{ let len = read_value!($next, usize); read_value!($next, [$t; len]) }}; ($next:expr, $t:ty) => ($next().parse::<$t>().expect("Parse error")); } // Dinic's algorithm for maximum flow problem. // This implementation uses O(n) stack space. // Verified by: // - yukicoder No.177 (http://yukicoder.me/submissions/148371) // - ABC239-G (https://atcoder.jp/contests/abc239/submissions/29497217) #[derive(Clone)] struct Edge { to: usize, cap: T, rev: usize, // rev is the position of the reverse edge in graph[to] } struct Cut { is_t: Vec, } #[allow(unused)] impl Cut { pub fn is_cut(&self, s: usize, t: usize) -> bool { !self.is_t[s] && self.is_t[t] } pub fn t(&self) -> Vec { (0..self.is_t.len()).filter(|&v| self.is_t[v]).collect() } pub fn s(&self) -> Vec { (0..self.is_t.len()).filter(|&v| !self.is_t[v]).collect() } } struct Dinic { graph: Vec>>, iter: Vec, zero: T, } impl Dinic where T: Clone, T: Copy, T: Ord, T: std::ops::Add, T: std::ops::Sub, T: std::ops::AddAssign, T: std::ops::SubAssign, { fn bfs(&self, s: usize, t: usize, level: &mut [Option]) { let n = level.len(); for i in 0..n { level[i] = None; } let mut que = std::collections::VecDeque::new(); level[s] = Some(0); que.push_back(s); while let Some(v) = que.pop_front() { for e in self.graph[v].iter() { if e.cap > self.zero && level[e.to] == None { level[e.to] = Some(level[v].unwrap() + 1); if e.to == t { return; } que.push_back(e.to); } } } } // search an augment path with dfs. // if f == None, f is treated as infinity. fn dfs(&mut self, v: usize, s: usize, f: Option, level: &mut [Option]) -> T { if v == s { return f.unwrap(); } let mut res = self.zero; while self.iter[v] < self.graph[v].len() { let i = self.iter[v]; let e = self.graph[v][i].clone(); let cap = self.graph[e.to][e.rev].cap; if cap > self.zero && level[e.to].is_some() && level[v] > level[e.to] { let newf = std::cmp::min(f.unwrap_or(cap + res) - res, cap); let d = self.dfs(e.to, s, Some(newf), level); if d > self.zero { self.graph[v][i].cap += d; self.graph[e.to][e.rev].cap -= d; res += d; if Some(res) == f { return res; } } } self.iter[v] += 1; } res } pub fn new(n: usize, zero: T) -> Self { Dinic { graph: vec![Vec::new(); n], iter: vec![0; n], zero: zero, } } pub fn add_edge(&mut self, from: usize, to: usize, cap: T) { if from == to { return; } let added_from = Edge { to: to, cap: cap, rev: self.graph[to].len() }; let added_to = Edge { to: from, cap: self.zero, rev: self.graph[from].len() }; self.graph[from].push(added_from); self.graph[to].push(added_to); } pub fn max_flow(&mut self, s: usize, t: usize) -> (T, Cut) { let mut flow = self.zero; let n = self.graph.len(); let mut level = vec![None; n]; loop { self.bfs(s, t, &mut level); if level[t] == None { let is_t: Vec = (0..n).map(|v| level[v].is_none()) .collect(); return (flow, Cut { is_t: is_t }); } self.iter.clear(); self.iter.resize(n, 0); let f = self.dfs(t, s, None, &mut level); flow += f; } } } fn main() { // In order to avoid potential stack overflow, spawn a new thread. let stack_size = 104_857_600; // 100 MB let thd = std::thread::Builder::new().stack_size(stack_size); thd.spawn(|| solve()).unwrap().join().unwrap(); } // https://yukicoder.me/problems/no/1479 (3) // 値が単一である問題だけ解ければ、元の問題はその答えの和である。 // これはこのような問題である: H 行 W 列のグリッドの何点かに印がついている。印がついた点を // 行と列を何個か選ぶことでカバーしたい。カバーに必要な行と列の個数の最小値は? // これは最小辺カバーと呼ばれる線形計画問題であり、二部グラフの場合は最大マッチングに帰着することができる。 // 計算量は O(E sqrt(V)) である。 // 全体の計算量はグラフの構築が O(HW(H+W)) で、最大マッチングの計算が O(HW sqrt(H + W)) である。 // 合計 O(HW(H+W)) である。 fn solve() { let out = std::io::stdout(); let mut out = BufWriter::new(out.lock()); macro_rules! puts {($($format:tt)*) => (let _ = write!(out,$($format)*););} #[allow(unused)] macro_rules! putvec { ($v:expr) => { for i in 0..$v.len() { puts!("{}{}", $v[i], if i + 1 == $v.len() {"\n"} else {" "}); } } } input! { h: usize, w: usize, a: [[usize; w]; h], } const W: usize = 500_000; let mut occ = vec![vec![]; W]; for i in 0..h { for j in 0..w { if a[i][j] != 0 { occ[a[i][j] - 1].push((i, j)); } } } let mut ans = 0; for i in 0..W { if occ[i].is_empty() { continue; } let mut din = Dinic::new(2 + h + w, 0i64); for i in 0..h { din.add_edge(0, 2 + i, 1); } for i in 0..w { din.add_edge(2 + h + i, 1, 1); } for &(x, y) in &occ[i] { din.add_edge(2 + x, 2 + h + y, 1); } let (ma, _) = din.max_flow(0, 1); ans += ma; } println!("{}", ans); }