1442C - Graph Transpositions - CodeForces Solution

// In the name of Allah
// Khodaya khodet komak kon
// Hope is last to die
// Let's go
// Hello today I decided to solve at least 5 great problems per day for 5 months
// 2023/11/24

#include <bits/stdc++.h>
using namespace std;

typedef long long ll;
typedef long double ld;
typedef pair<ll, ll> pll;

#define pb push_back
#define S second
#define F first
#define mp make_pair
#define smax(x, y) (x) = max((x), (y))
#define smin(x, y) (x) = min((x), (y))
#define all(x) (x).begin(), (x).end()
#define set_dec(x) cout << fixed << setprecision(x);
#define kill(x) cout << x << endl, exit(0)
#define sz(x) ((int)(x).size())
#define file_io freopen("input.txt", "r", stdin); freopen("output.txt", "w", stdout);

const ll maxn = 4e5 + 100, lg = 21, mod = 998244353;
const ll inf = 1e15 + 7;
ll n, m, dist2[maxn * (lg + 1)];
pll dist1[maxn];
vector<ll> adj[maxn];
vector<pair<ll, pll>> from[maxn];
vector<pll> fr[maxn * (lg + 1)];

ll tavan(ll a, ll b){
    if(!b) return 1;
    ll x = tavan(a, b / 2);
    x = (x * x) % mod;
    if(b % 2) x = (x * a) % mod;
    return x;
}

void dij1(){
    memset(dist1, 63, sizeof dist1);
    dist1[0] = mp(0, 0);
    set<pair<pll, ll>> st;
    st.insert(mp(mp(0, 0), 0));
    while(sz(st)){
        ll v = (*st.begin()).S;
        st.erase(st.begin());
        for(auto e: from[v]){
            pll p = dist1[v];
            p.F += e.S.F;
            p.S += e.S.S;
            if(p < dist1[e.F]){
                st.erase(mp(dist1[e.F], e.F));
                dist1[e.F] = p;
                st.insert(mp(dist1[e.F], e.F));
            }
        }
    }
    // cout << dist1[n - 1].F << ' ' << dist1[n - 1].S << '\n';
    // cout << dist1[2 * n - 1].F << ' ' << dist1[2 * n - 1].S << '\n';
}

void dij2(){
    memset(dist2, 63, sizeof dist2);
    set<pll> st;
    st.insert(mp(0, 0));
    while(sz(st)){
        ll v = (*st.begin()).S, d = (*st.begin()).F;
        st.erase(st.begin());
        if(dist2[v] <= inf) continue;
        dist2[v] = d;
        for(auto [u, w]: fr[v]){
            if(dist2[u] > d + w){
                st.insert(mp(d + w, u));
            }
        }
    }
    // for(int i = 0; i <= lg; i++) cout << dist2[i * n + n - 1] << '\n';
}

int32_t main()
{
    ios_base::sync_with_stdio(0); cin.tie(0); cout.tie(0);

    cin >> n >> m;
    for(int i = 0; i < m; i++){
        ll u, v; cin >> u >> v;
        adj[--u].pb(--v);
    }
    for(int i = 0; i < n; i++){
        from[i].pb(mp(i + n, mp(1, 0)));
        from[i + n].pb(mp(i, mp(1, 0)));
        for(auto u: adj[i]){
            from[i].pb(mp(u, mp(0, 1)));
            from[u + n].pb(mp(i + n, mp(0, 1)));
        }
    }
    dij1();
    ll t1 = dist1[n - 1].F, r1 = dist1[n - 1].S;
    ll t2 = dist1[2 * n - 1].F, r2 = dist1[2 * n - 1].S;
    if(min(t1, t2) >= 30){
        if(t1 == t2){
            cout << (tavan(2, t1) - 1 + min(r1, r2)) % mod;
        }
        else if(t1 < t2){
            cout << (tavan(2, t1) - 1 + r1) % mod;
        }
        else cout << (tavan(2, t2) - 1 + r2) % mod;
        return 0;
    }
    for(int b = 0; b <= lg; b++){
        for(int i = 0; i < n; i++){
            for(auto u: adj[i]){
                if(b % 2 == 0) fr[b * n + i].pb(mp(b * n + u, 1));
                else fr[b * n + u].pb(mp(b * n + i, 1));
            }
        }
        if(b != lg){
            for(int i = 0; i < n; i++){
                fr[b * n + i].pb(mp((b + 1) * n + i, (1ll << b)));
            }
        }
    }
    dij2();
    ll ans = inf;
    for(int i = 0; i <= lg; i++) smin(ans, dist2[i * n + n - 1]);
    smin(ans, (1ll << dist1[n - 1].F) + dist1[n - 1].S);
    smin(ans, (1ll << dist1[2 * n - 1].F) + dist1[2 * n - 1].S);
    cout << (ans % mod);

    return 0;
}