If the graph is bipartite, the answer is $N-1$. Otherwise, let
for each vertex $1\le i\le N$, the same as "Sum of Distances" from the January contest. Also define
For convenience, define $s(a, b)$ to be the set of all nodes $i$ with $h(i) = (a, b)$.
A graph having $f_G$ equivalent to that of the given graph must have edges adjacent to vertex $i$ satisfying at least one of the following conditions for each $2\le i\le N$ (the condition is slightly different for vertex $1$ since $a_1=0$).
Edges that are not type A, B, or C edges cannot exist in the graph.
We can look at each layer (vertices with a fixed $a_i+b_i$) independently, as edges of type A are only relevant to the vertex in the higher layer, and edges of type B and C are between vertices in the same layer. For a given layer, let's satisfy the constraints in increasing order of $a_i$.
We'll describe two ways of doing this, of which the second approach is sufficient for full credit.
Suppose that we are currently deciding which edges to construct involving $s_{a,b}$. For simplicity, we'll only deal with the case that $|s_{a-1,b-1}|>0$ and $|s_{a+1,b-1}|>0$ (for the other cases, see the code for details). Our goal is to satisfy one of the first two conditions for each vertex $v$.
Approach 1: Define:
Then we'll need to add $\max(|s_{a,b}|-\min(j,k),0)$ additional edges of type A.
These observations are sufficient for an $\mathcal{O}(N^2)$ DP. Store $dp_{a,b}[j]$ for each $0\le j\le \max(|s_{a-1,b+1}|,|s_{a,b}|)$ and transition to $dp_{a+1,b-1}[k]$ for each $0\le k\le \max(|s_{a,b}|,|s_{a+1,b-1}|)$. See my $\texttt{solve_between}$ function below for details:
#include <bits/stdc++.h> using namespace std; using ll = long long; using db = long double; // or double, if TL is tight using str = string; // yay python! using pi = pair<int,int>; using pl = pair<ll,ll>; using pd = pair<db,db>; using vi = vector<int>; using vb = vector<bool>; using vl = vector<ll>; using vd = vector<db>; using vs = vector<str>; using vpi = vector<pi>; using vpl = vector<pl>; using vpd = vector<pd>; #define tcT template<class T #define tcTU tcT, class U // ^ lol this makes everything look weird but I'll try it tcT> using V = vector<T>; tcT, size_t SZ> using AR = array<T,SZ>; tcT> using PR = pair<T,T>; // pairs #define mp make_pair #define f first #define s second // vectors // oops size(x), rbegin(x), rend(x) need C++17 #define sz(x) int((x).size()) #define bg(x) begin(x) #define all(x) bg(x), end(x) #define rall(x) x.rbegin(), x.rend() #define sor(x) sort(all(x)) #define rsz resize #define ins insert #define ft front() #define bk back() #define pb push_back #define eb emplace_back #define pf push_front #define rtn return #define lb lower_bound #define ub upper_bound tcT> int lwb(V<T>& a, const T& b) { return int(lb(all(a),b)-bg(a)); } // loops #define FOR(i,a,b) for (int i = (a); i < (b); ++i) #define F0R(i,a) FOR(i,0,a) #define ROF(i,a,b) for (int i = (b)-1; i >= (a); --i) #define R0F(i,a) ROF(i,0,a) #define rep(a) F0R(_,a) #define each(a,x) for (auto& a: x) const int MOD = 1e9+7; // 998244353; const int MX = 2e5+5; const ll INF = 1e18; // not too close to LLONG_MAX const db PI = acos((db)-1); const int dx[4] = {1,0,-1,0}, dy[4] = {0,1,0,-1}; // for every grid problem!! mt19937 rng((uint32_t)chrono::steady_clock::now().time_since_epoch().count()); template<class T> using pqg = priority_queue<T,vector<T>,greater<T>>; // bitwise ops // also see https://gcc.gnu.org/onlinedocs/gcc/Other-Builtins.html constexpr int pct(int x) { return __builtin_popcount(x); } // # of bits set constexpr int bits(int x) { // assert(x >= 0); // make C++11 compatible until USACO updates ... return x == 0 ? 0 : 31-__builtin_clz(x); } // floor(log2(x)) constexpr int p2(int x) { return 1<<x; } constexpr int msk2(int x) { return p2(x)-1; } ll cdiv(ll a, ll b) { return a/b+((a^b)>0&&a%b); } // divide a by b rounded up ll fdiv(ll a, ll b) { return a/b-((a^b)<0&&a%b); } // divide a by b rounded down tcT> bool ckmin(T& a, const T& b) { return b < a ? a = b, 1 : 0; } // set a = min(a,b) tcT> bool ckmax(T& a, const T& b) { return a < b ? a = b, 1 : 0; } tcTU> T fstTrue(T lo, T hi, U f) { hi ++; assert(lo <= hi); // assuming f is increasing while (lo < hi) { // find first index such that f is true T mid = lo+(hi-lo)/2; f(mid) ? hi = mid : lo = mid+1; } return lo; } tcTU> T lstTrue(T lo, T hi, U f) { lo --; assert(lo <= hi); // assuming f is decreasing while (lo < hi) { // find first index such that f is true T mid = lo+(hi-lo+1)/2; f(mid) ? lo = mid : hi = mid-1; } return lo; } tcT> void remDup(vector<T>& v) { // sort and remove duplicates sort(all(v)); v.erase(unique(all(v)),end(v)); } tcTU> void erase(T& t, const U& u) { // don't erase auto it = t.find(u); assert(it != end(t)); t.erase(it); } // element that doesn't exist from (multi)set // INPUT #define tcTUU tcT, class ...U tcT> void re(complex<T>& c); tcTU> void re(pair<T,U>& p); tcT> void re(V<T>& v); tcT, size_t SZ> void re(AR<T,SZ>& a); tcT> void re(T& x) { cin >> x; } void re(double& d) { str t; re(t); d = stod(t); } void re(long double& d) { str t; re(t); d = stold(t); } tcTUU> void re(T& t, U&... u) { re(t); re(u...); } tcT> void re(complex<T>& c) { T a,b; re(a,b); c = {a,b}; } tcTU> void re(pair<T,U>& p) { re(p.f,p.s); } tcT> void re(V<T>& x) { each(a,x) re(a); } tcT, size_t SZ> void re(AR<T,SZ>& x) { each(a,x) re(a); } tcT> void rv(int n, V<T>& x) { x.rsz(n); re(x); } // TO_STRING #define ts to_string str ts(char c) { return str(1,c); } str ts(const char* s) { return (str)s; } str ts(str s) { return s; } str ts(bool b) { // #ifdef LOCAL // return b ? "true" : "false"; // #else return ts((int)b); // #endif } tcT> str ts(complex<T> c) { stringstream ss; ss << c; return ss.str(); } str ts(V<bool> v) { str res = "{"; F0R(i,sz(v)) res += char('0'+v[i]); res += "}"; return res; } template<size_t SZ> str ts(bitset<SZ> b) { str res = ""; F0R(i,SZ) res += char('0'+b[i]); return res; } tcTU> str ts(pair<T,U> p); tcT> str ts(T v) { // containers with begin(), end() #ifdef LOCAL bool fst = 1; str res = "{"; for (const auto& x: v) { if (!fst) res += ", "; fst = 0; res += ts(x); } res += "}"; return res; #else bool fst = 1; str res = ""; for (const auto& x: v) { if (!fst) res += " "; fst = 0; res += ts(x); } return res; #endif } tcTU> str ts(pair<T,U> p) { #ifdef LOCAL return "("+ts(p.f)+", "+ts(p.s)+")"; #else return ts(p.f)+" "+ts(p.s); #endif } // OUTPUT tcT> void pr(T x) { cout << ts(x); } tcTUU> void pr(const T& t, const U&... u) { pr(t); pr(u...); } void ps() { pr("\n"); } // print w/ spaces tcTUU> void ps(const T& t, const U&... u) { pr(t); if (sizeof...(u)) pr(" "); ps(u...); } // DEBUG void DBG() { cerr << "]" << endl; } tcTUU> void DBG(const T& t, const U&... u) { cerr << ts(t); if (sizeof...(u)) cerr << ", "; DBG(u...); } #ifdef LOCAL // compile with -DLOCAL, chk -> fake assert #define dbg(...) cerr << "Line(" << __LINE__ << ") -> [" << #__VA_ARGS__ << "]: [", DBG(__VA_ARGS__) #define chk(...) if (!(__VA_ARGS__)) cerr << "Line(" << __LINE__ << ") -> function(" \ << __FUNCTION__ << ") -> CHK FAILED: (" << #__VA_ARGS__ << ")" << "\n", exit(0); #else #define dbg(...) 0 #define chk(...) 0 #endif void setPrec() { cout << fixed << setprecision(15); } void unsyncIO() { cin.tie(0)->sync_with_stdio(0); } // FILE I/O void setIn(str s) { freopen(s.c_str(),"r",stdin); } void setOut(str s) { freopen(s.c_str(),"w",stdout); } void setIO(str s = "") { unsyncIO(); setPrec(); // cin.exceptions(cin.failbit); // throws exception when do smth illegal // ex. try to read letter into int if (sz(s)) setIn(s+".in"), setOut(s+".out"); // for USACO } #define ints(...); int __VA_ARGS__; re(__VA_ARGS__); int N,M; vi adj[MX]; V<AR<int,2>> gen_dist() { V<AR<int,2>> dist(N,{MOD,MOD}); queue<pi> q; auto ad = [&](int a, int b) { if (dist[a][b%2] != MOD) return; dist[a][b%2] = b; q.push({a,b}); }; ad(0,0); while (sz(q)) { pi p = q.ft; q.pop(); each(t,adj[p.f]) ad(t,p.s+1); } return dist; } int ans = 0; set<pi> distinct; int div2(int x) { return (x+1)/2; } void solve_between(vi nums, vb exists, bool special) { vi dp{0}; int res = MOD; F0R(i,sz(nums)) { if (i == sz(nums)-1) { if (special) { F0R(j,sz(dp)) F0R(k,nums[i]+1) { int need_one = max(nums[i]-min(j,2*k),0); if (!exists[i] && need_one) continue; ckmin(res,dp[j]+need_one+k); } } else { assert(exists[i]); each(t,dp) ckmin(res,t+nums[i]); } } else { vi DP(max(nums[i],nums[i+1])+1,MOD); F0R(j,sz(dp)) F0R(k,max(nums[i],nums[i+1])+1) { int need_one = max(nums[i]-min(j,k),0); if (!exists[i] && need_one) continue; ckmin(DP[k],dp[j]+need_one+k); } swap(dp,DP); } } ans += res; } void solve_sum(int sum, vpi v) { dbg("SOLVE SUM",sum,v); assert(sz(v)); if (v[0].f == 0) { F0R(i,sz(v)-1) ans += max(v[i].s,v[i+1].s); ans += div2(v.bk.s); return; } for (int i = 0; i < sz(v); ++i) { vi nums{v[i].s}; vb exists{distinct.count({v[i].f-1,sum-v[i].f-1})}; while (i+1 < sz(v) && v[i+1].f == v[i].f+1) { ++i; nums.pb(v[i].s); exists.pb(distinct.count({v[i].f-1,sum-v[i].f-1})); } bool special = 0; if (2*v[i].f+1 == sum) special = 1; solve_between(nums,exists,special); } } void go() { auto a = gen_dist(); if (a[0][1] == MOD) { ps(N-1); return; } map<int,map<int,int>> cnt; each(t,a) { pi p{t[0],t[1]}; if (p.f > p.s) swap(p.f,p.s); distinct.ins(p); ++cnt[p.f+p.s][p.f]; } each(t,cnt) solve_sum(t.f,vpi(all(t.s))); ps(ans); } int main() { setIO(); int T; re(T); F0R(_,T) { re(N,M); F0R(i,N) adj[i].clear(); ans = 0; distinct.clear(); F0R(i,M) { int a,b; re(a,b); --a,--b; adj[a].pb(b), adj[b].pb(a); } go(); } } /* stuff you should look for * int overflow, array bounds * special cases (n=1?) * do smth instead of nothing and stay organized * WRITE STUFF DOWN * DON'T GET STUCK ON ONE APPROACH */
Approach 2: For full credit, we can assign edges greedily.
Suppose that we have already added $\texttt{prev}$ edges from $s_{a,b}$ to $s_{a+1,b-1}$ in order to satisfy the conditions for vertices $w$. Let's assign each of these edges to different vertices $v$ if possible, meaning that we have at least $x=\min(\texttt{prev},|s_{a,b}|)$ vertices $v$ with edges to $(a-1,b+1)$.
Danny's code (which doesn't explicitly group vertices by $a+b$):
import java.io.BufferedReader; import java.io.IOException; import java.io.InputStreamReader; import java.util.*; public class MinimizingEdgesActuallyCorrect { public static void main(String[] args) throws IOException { BufferedReader in = new BufferedReader(new InputStreamReader(System.in)); StringTokenizer tokenizer = new StringTokenizer(in.readLine()); int n = Integer.parseInt(tokenizer.nextToken()); int m = Integer.parseInt(tokenizer.nextToken()); List<Integer>[] adj = new List[(2 * n) + 1]; for (int a = 1; a <= 2 * n; a++) { adj[a] = new ArrayList<>(); } for (int j = 1; j <= m; j++) { tokenizer = new StringTokenizer(in.readLine()); int a = Integer.parseInt(tokenizer.nextToken()); int b = Integer.parseInt(tokenizer.nextToken()); adj[a].add(b + n); adj[b + n].add(a); adj[a + n].add(b); adj[b].add(a + n); } int[] dist = new int[(2 * n) + 1]; Arrays.fill(dist, -1); dist[1] = 0; LinkedList<Integer> q = new LinkedList<>(); q.add(1); while (!q.isEmpty()) { int a = q.remove(); for (int b : adj[a]) { if (dist[b] == -1) { dist[b] = dist[a] + 1; q.add(b); } } } int answer = 0; if (dist[n + 1] == -1) { answer = n - 1; } else { TreeMap<Pair, Integer> freq = new TreeMap<>(); TreeMap<Pair, List<Integer>> buckets = new TreeMap<>(); for (int a = 1; a <= n; a++) { freq.merge(new Pair(Math.min(dist[a], dist[n + a]), Math.max(dist[a], dist[n + a])), 1, Integer::sum); buckets.computeIfAbsent(new Pair(Math.min(dist[a], dist[n + a]), Math.max(dist[a], dist[n + a])), __ -> new ArrayList<>()).add(a); } TreeMap<Pair, Integer> edgeAmt = new TreeMap<>(); for (Map.Entry<Pair, Integer> entry : freq.entrySet()) { Pair p = entry.getKey(); int f = entry.getValue(); int prev = edgeAmt.getOrDefault(new Pair(p.first - 1, p.second + 1), 0); if (p.second == p.first + 1) { if (p.first == 0) { answer += (f + 1) / 2; } else if (freq.containsKey(new Pair(p.first - 1, p.second - 1))) { answer += Math.max((f - prev) + ((prev + 1) / 2), (f + 1) / 2); } else { if (prev < f) { answer += f - prev; } answer += (f + 1) / 2; } } else { answer += f; if (p.first == 0) { edgeAmt.put(p, f); } else if (freq.containsKey(new Pair(p.first - 1, p.second - 1))) { edgeAmt.put(p, Math.min(f, prev)); } else { if (prev < f) { answer += f - prev; } edgeAmt.put(p, f); } } } } System.out.println(answer); } static class Pair implements Comparable<Pair> { final int first; final int second; Pair(int first, int second) { this.first = first; this.second = second; } @Override public int compareTo(Pair other) { if (first != other.first) { return first - other.first; } else { return second - other.second; } } } }
My code (which constructs a solution):
#include <bits/stdc++.h> using namespace std; using ll = long long; using db = long double; // or double, if TL is tight using str = string; // yay python! using pi = pair<int,int>; using pl = pair<ll,ll>; using pd = pair<db,db>; using vi = vector<int>; using vb = vector<bool>; using vl = vector<ll>; using vd = vector<db>; using vs = vector<str>; using vpi = vector<pi>; using vpl = vector<pl>; using vpd = vector<pd>; #define tcT template<class T #define tcTU tcT, class U // ^ lol this makes everything look weird but I'll try it tcT> using V = vector<T>; tcT, size_t SZ> using AR = array<T,SZ>; tcT> using PR = pair<T,T>; // pairs #define mp make_pair #define f first #define s second // vectors // oops size(x), rbegin(x), rend(x) need C++17 #define sz(x) int((x).size()) #define bg(x) begin(x) #define all(x) bg(x), end(x) #define rall(x) x.rbegin(), x.rend() #define sor(x) sort(all(x)) #define rsz resize #define ins insert #define ft front() #define bk back() #define pb push_back #define eb emplace_back #define pf push_front #define rtn return #define lb lower_bound #define ub upper_bound tcT> int lwb(V<T>& a, const T& b) { return int(lb(all(a),b)-bg(a)); } // loops #define FOR(i,a,b) for (int i = (a); i < (b); ++i) #define F0R(i,a) FOR(i,0,a) #define ROF(i,a,b) for (int i = (b)-1; i >= (a); --i) #define R0F(i,a) ROF(i,0,a) #define rep(a) F0R(_,a) #define each(a,x) for (auto& a: x) const int MOD = 1e9+7; // 998244353; const int MX = 2e5+5; const ll INF = 1e18; // not too close to LLONG_MAX const db PI = acos((db)-1); const int dx[4] = {1,0,-1,0}, dy[4] = {0,1,0,-1}; // for every grid problem!! mt19937 rng((uint32_t)chrono::steady_clock::now().time_since_epoch().count()); template<class T> using pqg = priority_queue<T,vector<T>,greater<T>>; // bitwise ops // also see https://gcc.gnu.org/onlinedocs/gcc/Other-Builtins.html constexpr int pct(int x) { return __builtin_popcount(x); } // # of bits set constexpr int bits(int x) { // assert(x >= 0); // make C++11 compatible until USACO updates ... return x == 0 ? 0 : 31-__builtin_clz(x); } // floor(log2(x)) constexpr int p2(int x) { return 1<<x; } constexpr int msk2(int x) { return p2(x)-1; } ll cdiv(ll a, ll b) { return a/b+((a^b)>0&&a%b); } // divide a by b rounded up ll fdiv(ll a, ll b) { return a/b-((a^b)<0&&a%b); } // divide a by b rounded down tcT> bool ckmin(T& a, const T& b) { return b < a ? a = b, 1 : 0; } // set a = min(a,b) tcT> bool ckmax(T& a, const T& b) { return a < b ? a = b, 1 : 0; } tcTU> T fstTrue(T lo, T hi, U f) { hi ++; assert(lo <= hi); // assuming f is increasing while (lo < hi) { // find first index such that f is true T mid = lo+(hi-lo)/2; f(mid) ? hi = mid : lo = mid+1; } return lo; } tcTU> T lstTrue(T lo, T hi, U f) { lo --; assert(lo <= hi); // assuming f is decreasing while (lo < hi) { // find first index such that f is true T mid = lo+(hi-lo+1)/2; f(mid) ? lo = mid : hi = mid-1; } return lo; } tcT> void remDup(vector<T>& v) { // sort and remove duplicates sort(all(v)); v.erase(unique(all(v)),end(v)); } tcTU> void erase(T& t, const U& u) { // don't erase auto it = t.find(u); assert(it != end(t)); t.erase(it); } // element that doesn't exist from (multi)set // INPUT #define tcTUU tcT, class ...U tcT> void re(complex<T>& c); tcTU> void re(pair<T,U>& p); tcT> void re(V<T>& v); tcT, size_t SZ> void re(AR<T,SZ>& a); tcT> void re(T& x) { cin >> x; } void re(double& d) { str t; re(t); d = stod(t); } void re(long double& d) { str t; re(t); d = stold(t); } tcTUU> void re(T& t, U&... u) { re(t); re(u...); } tcT> void re(complex<T>& c) { T a,b; re(a,b); c = {a,b}; } tcTU> void re(pair<T,U>& p) { re(p.f,p.s); } tcT> void re(V<T>& x) { each(a,x) re(a); } tcT, size_t SZ> void re(AR<T,SZ>& x) { each(a,x) re(a); } tcT> void rv(int n, V<T>& x) { x.rsz(n); re(x); } // TO_STRING #define ts to_string str ts(char c) { return str(1,c); } str ts(const char* s) { return (str)s; } str ts(str s) { return s; } str ts(bool b) { // #ifdef LOCAL // return b ? "true" : "false"; // #else return ts((int)b); // #endif } tcT> str ts(complex<T> c) { stringstream ss; ss << c; return ss.str(); } str ts(V<bool> v) { str res = "{"; F0R(i,sz(v)) res += char('0'+v[i]); res += "}"; return res; } template<size_t SZ> str ts(bitset<SZ> b) { str res = ""; F0R(i,SZ) res += char('0'+b[i]); return res; } tcTU> str ts(pair<T,U> p); tcT> str ts(T v) { // containers with begin(), end() #ifdef LOCAL bool fst = 1; str res = "{"; for (const auto& x: v) { if (!fst) res += ", "; fst = 0; res += ts(x); } res += "}"; return res; #else bool fst = 1; str res = ""; for (const auto& x: v) { if (!fst) res += " "; fst = 0; res += ts(x); } return res; #endif } tcTU> str ts(pair<T,U> p) { #ifdef LOCAL return "("+ts(p.f)+", "+ts(p.s)+")"; #else return ts(p.f)+" "+ts(p.s); #endif } // OUTPUT tcT> void pr(T x) { cout << ts(x); } tcTUU> void pr(const T& t, const U&... u) { pr(t); pr(u...); } void ps() { pr("\n"); } // print w/ spaces tcTUU> void ps(const T& t, const U&... u) { pr(t); if (sizeof...(u)) pr(" "); ps(u...); } // DEBUG void DBG() { cerr << "]" << endl; } tcTUU> void DBG(const T& t, const U&... u) { cerr << ts(t); if (sizeof...(u)) cerr << ", "; DBG(u...); } #ifdef LOCAL // compile with -DLOCAL, chk -> fake assert #define dbg(...) cerr << "Line(" << __LINE__ << ") -> [" << #__VA_ARGS__ << "]: [", DBG(__VA_ARGS__) #define chk(...) if (!(__VA_ARGS__)) cerr << "Line(" << __LINE__ << ") -> function(" \ << __FUNCTION__ << ") -> CHK FAILED: (" << #__VA_ARGS__ << ")" << "\n", exit(0); #else #define dbg(...) 0 #define chk(...) 0 #endif void setPrec() { cout << fixed << setprecision(15); } void unsyncIO() { cin.tie(0)->sync_with_stdio(0); } // FILE I/O void setIn(str s) { freopen(s.c_str(),"r",stdin); } void setOut(str s) { freopen(s.c_str(),"w",stdout); } void setIO(str s = "") { unsyncIO(); setPrec(); // cin.exceptions(cin.failbit); // throws exception when do smth illegal // ex. try to read letter into int if (sz(s)) setIn(s+".in"), setOut(s+".out"); // for USACO } #define ints(...); int __VA_ARGS__; re(__VA_ARGS__); int N,M; vi adj[MX]; V<AR<int,2>> gen_dist() { V<AR<int,2>> dist(N,{MOD,MOD}); queue<pi> q; auto ad = [&](int a, int b) { if (dist[a][b%2] != MOD) return; dist[a][b%2] = b; q.push({a,b}); }; ad(0,0); while (sz(q)) { pi p = q.ft; q.pop(); each(t,adj[p.f]) ad(t,p.s+1); } return dist; } vpi ans_ed; map<pi,int> distinct; vpi group; int div2(int x) { return (x+1)/2; } void satisfy(vi a, vi b) { F0R(i,max(sz(a),sz(b))) ans_ed.pb({a[min(i,sz(a)-1)],b[min(i,sz(b)-1)]}); } void satisfy_self(vi a) { for (int i = 0; i < sz(a); i += 2) { if (i == sz(a)-1) ans_ed.pb({a[i],a[i]}); else ans_ed.pb({a[i],a[i+1]}); } } void satisfy_lower_left(vi v) { each(t,v) { pi p = group[t]; --p.f,--p.s; assert(distinct.count(p)); ans_ed.pb({t,distinct[p]}); } } void satisfy_upper_left(vi v) { each(t,v) { pi p = group[t]; --p.f,++p.s; assert(distinct.count(p)); ans_ed.pb({t,distinct[p]}); } } void solve_between(V<vi> nums, vb exists, bool special) { vi bef; F0R(i,sz(nums)) { vi yes = nums[i], no; while (sz(yes) > sz(bef)) { no.pb(yes.bk); yes.pop_back(); } satisfy(bef,yes); if (i == sz(nums)-1) { if (special) { // ans += nums[i]-sz(bef); if (exists[i]) { satisfy_lower_left(no); // for each in yes: loop // for each in no: go to // ans += div2(bef); satisfy_self(yes); } else { satisfy_upper_left(no); satisfy_self(nums[i]); } } else { assert(exists[i]); satisfy_lower_left(nums[i]); } } else if (exists[i]) { bef = yes; satisfy_lower_left(no); } else { satisfy_upper_left(no); bef = nums[i]; } } } void solve_sum(int sum, V<pair<int,vi>> v) { dbg("SOLVE SUM",sum,v); assert(sz(v)); if (v[0].f == 0) { F0R(i,sz(v)-1) satisfy(v[i].s,v[i+1].s); satisfy_self(v.bk.s); return; } for (int i = 0; i < sz(v); ++i) { V<vi> nums{v[i].s}; vb exists{distinct.count({v[i].f-1,sum-v[i].f-1})}; while (i+1 < sz(v) && v[i+1].f == v[i].f+1) { ++i; nums.pb(v[i].s); exists.pb(distinct.count({v[i].f-1,sum-v[i].f-1})); } bool special = 0; if (2*v[i].f+1 == sum) special = 1; solve_between(nums,exists,special); } } void solve() { auto a = gen_dist(); // dbg(a); if (a[0][1] == MOD) { ps(N-1); return; } map<int,map<int,vi>> cnt; F0R(i,N) { AR<int,2> t = a[i]; pi p{t[0],t[1]}; if (p.f > p.s) swap(p.f,p.s); group.pb(p); distinct[p] = i; cnt[p.f+p.s][p.f].pb(i); } each(t,cnt) solve_sum(t.f,V<pair<int,vi>>(all(t.s))); F0R(i,N) adj[i].clear(); each(t,ans_ed) adj[t.f].pb(t.s), adj[t.s].pb(t.f); assert(a == gen_dist()); ps(sz(ans_ed)); assert(sz(ans_ed) <= M); } int main() { setIO(); int TC; re(TC); F0R(_,TC) { re(N,M); distinct.clear(); F0R(i,N) adj[i].clear(); ans_ed.clear(); group.clear(); F0R(i,M) { int a,b; re(a,b); --a,--b; adj[a].pb(b), adj[b].pb(a); } solve(); } }