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();
}
}