Educational Codeforces Round 102 Editorial

#	User	Rating
1	tourist	3985
2	jiangly	3741
3	jqdai0815	3682
4	Benq	3529
5	orzdevinwang	3526
6	ksun48	3489
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8	Kevin114514	3442
9	ecnerwala	3392
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#	User	Contrib.
1	cry	169
2	maomao90	162
2	Um_nik	162
2	atcoder_official	162
5	djm03178	158
6	-is-this-fft-	157
7	adamant	155
8	awoo	154
8	Dominater069	154
10	nor	150

Tutorial

Solution (Neon)

#include <bits/stdc++.h>

using namespace std;

int main() {
  int tc;
  cin >> tc;
  while (tc--) {
    int n, d;
    cin >> n >> d;
    vector<int> a(n);
    for (int& x : a) cin >> x;
    sort(a.begin(), a.end());
    cout << (a.back() <= d || a[0] + a[1] <= d ? "YES" : "NO") << endl;
  }
}

1473B - String LCM

Idea: BledDest

Tutorial

Solution (Neon)

#include <bits/stdc++.h>

using namespace std;

int main() {
  auto mul = [](string s, int k) -> string {
    string res = "";
    while (k--) res += s;
    return res;
  };
  
  int tc;
  cin >> tc;
  while (tc--) {
    string s, t;
    cin >> s >> t;
    int n = s.length(), m = t.length();
    int g = __gcd(n, m);
    if (mul(s, m / g) == mul(t, n / g))
      cout << mul(s, m / g) << endl;
    else
      cout << "-1" << endl;
  }
}

1473C - No More Inversions

Idea: adedalic

Tutorial

Solution (adedalic)

fun main() {
    repeat(readLine()!!.toInt()) {
        val (n, k) = readLine()!!.split(' ').map { it.toInt() }

        for (i in 1 until (k - (n - k)))
            print("$i ")
        for (i in k downTo (k - (n - k)))
            print("$i ")
        println("")
    }
}

1473D - Program

Idea: BledDest

Tutorial

Solution (awoo)

#include <bits/stdc++.h>

#define forn(i, n) for (int i = 0; i < int(n); i++)

using namespace std;

int main() {
	cin.tie(0);
	ios_base::sync_with_stdio(0);
	int t;
	cin >> t;
	forn(_, t){
		int n, m;
		string s;
		cin >> n >> m;
		cin >> s;
		vector<int> sul(1, 0), sur(1, 0);
		for (int i = n - 1; i >= 0; --i){
			int d = s[i] == '+' ? 1 : -1;
			sul.push_back(min(0, sul.back() + d));
			sur.push_back(max(0, sur.back() + d));
		}
		reverse(sul.begin(), sul.end());
		reverse(sur.begin(), sur.end());
		vector<int> prl(1, 0), prr(1, 0), pr(1, 0);
		forn(i, n){
			int d = s[i] == '+' ? 1 : -1;
			pr.push_back(pr.back() + d);
			prl.push_back(min(prl.back(), pr.back()));
			prr.push_back(max(prr.back(), pr.back()));
		}
		forn(i, m){
			int l, r;
			cin >> l >> r;
			--l;
			int l1 = prl[l], r1 = prr[l];
			int l2 = sul[r] + pr[l], r2 = sur[r] + pr[l];
			printf("%d\n", max(r1, r2) - min(l1, l2) + 1);
		}
	}
	return 0;
}

1473E - Minimum Path

Idea: Neon

Tutorial

Solution (Neon)

#include <bits/stdc++.h>

using namespace std;

const int N = 200 * 1000 + 13;

int n, m;
vector<pair<int, int>> g[N];
long long d[N][2][2];

int main() {
  scanf("%d%d", &n, &m);
  for (int i = 0; i < m; i++) {
    int v, u, w;
    scanf("%d%d%d", &v, &u, &w);
    --v; --u;
    g[v].emplace_back(u, w);
    g[u].emplace_back(v, w);
  }
  
  for (int i = 0; i < n; i++)
    for (int j = 0; j < 2; j++)
      for (int k = 0; k < 2; k++)
        d[i][j][k] = (long long)1e18;

  d[0][0][0] = 0;
  set<pair<long long, array<int, 3>>> q;
  q.insert({0, {0, 0, 0}});

  while (!q.empty()) {
    auto [v, mx, mn] = q.begin()->second;
    q.erase(q.begin());
    for (auto [u, w] : g[v]) {
      for (int i = 0; i <= 1 - mx; i++) {
        for (int j = 0; j <= 1 - mn; j++) {
          if (d[u][mx | i][mn | j] > d[v][mx][mn] + (1 - i + j) * w) {
            auto it = q.find({d[u][mx | i][mn | j], {u, mx | i, mn | j}});
            if (it != q.end())
              q.erase(it);
            d[u][mx | i][mn | j] = d[v][mx][mn] + (1 - i + j) * w;
            q.insert({d[u][mx | i][mn | j], {u, mx | i, mn | j}});
          }
        }
      }
    }
  }

  for (int i = 1; i < n; i++) {
    printf("%lld ", d[i][1][1]);
  }
  puts("");
}

1473F - Strange Set

Idea: BledDest

Tutorial

Solution (BledDest)

#include<bits/stdc++.h>

using namespace std;

template<typename T>
struct dinic {
    struct edge {
        int u, rev;
        T cap, flow;
    };
    
    int n, s, t;
    T flow;
    vector<int> lst;
    vector<int> d;
    vector<vector<edge>> g;
    
    T scaling_lim;
    bool scaling;
    
    dinic() {}
    
    dinic(int n, int s, int t, bool scaling = false) : n(n), s(s), t(t), scaling(scaling) {
        g.resize(n);
        d.resize(n);
        lst.resize(n);
        flow = 0;
    }

    void add_edge(int v, int u, T cap, bool directed = true) {
        g[v].push_back({u, g[u].size(), cap, 0});
        g[u].push_back({v, int(g[v].size()) - 1, directed ? 0 : cap, 0});
    }

    T dfs(int v, T flow) {
        if (v == t)
            return flow;
        if (flow == 0)
            return 0;
        T result = 0;
        for (; lst[v] < g[v].size(); ++lst[v]) {
            edge& e = g[v][lst[v]];
            if (d[e.u] != d[v] + 1)
                continue;
            T add = dfs(e.u, min(flow, e.cap - e.flow));
            if (add > 0) {
                result += add;
                flow -= add;
                e.flow += add;
                g[e.u][e.rev].flow -= add;
            }
            if (flow == 0)
                break;
        }
        return result;
    }

    bool bfs() {
        fill(d.begin(), d.end(), -1);
        queue<int> q({s});
        d[s] = 0;
        while (!q.empty() && d[t] == -1) {
            int v = q.front(); q.pop();
            for (auto& e : g[v]) {
                if (d[e.u] == -1 && e.cap - e.flow >= scaling_lim) {
                    q.push(e.u);
                    d[e.u] = d[v] + 1;
                }
            }
        }
        return d[t] != -1;
    }

    T calc() {
        T max_lim = numeric_limits<T>::max() / 2 + 1;
        for (scaling_lim = scaling ? max_lim : 1; scaling_lim > 0; scaling_lim >>= 1) {
            while (bfs()) {
                fill(lst.begin(), lst.end(), 0);
                T add;
                while((add = dfs(s, numeric_limits<T>::max())) > 0)
                flow += add;
            }
        }   
        return flow;
    }
    
    vector<bool> min_cut() {
        vector<bool> res(n);
        for(int i = 0; i < n; i++) 
            res[i] = (d[i] != -1);
        return res;
    }
};

int main() {
    int n;
    cin >> n;
    vector<int> a(n), b(n);
    for(int i = 0; i < n; i++)
        cin >> a[i];
    for(int i = 0; i < n; i++)
        cin >> b[i];
    int s = n;
    int t = n + 1;
    dinic<int> d(n + 2, s, t, true);
    vector<int> last(101, -1);
    for(int i = 0; i < n; i++){
        if(b[i] > 0)
            d.add_edge(s, i, b[i]);
        if(b[i] < 0)
            d.add_edge(i, t, -b[i]);
        for(int k = 1; k <= 100; k++)
            if(last[k] != -1 && a[i] % k == 0)
                d.add_edge(i, last[k], int(1e9));
        last[a[i]] = i;    
    }   
    int sum = 0;
    for(int i = 0; i < n; i++)
        sum += max(0, b[i]);
    cout << sum - d.calc() << endl;
}

1473G - Tiles

Idea: Neon, adedalic

Tutorial

Solution (Neon)

#include <bits/stdc++.h>

using namespace std;

#define forn(i, n) for (int i = 0; i < int(n); ++i)
#define fore(i, l, r) for (int i = int(l); i < int(r); ++i)
#define sz(a) int((a).size())

template<const int &MOD>
struct _m_int {
  int val;
 
  _m_int(int64_t v = 0) {
    if (v < 0) v = v % MOD + MOD;
    if (v >= MOD) v %= MOD;
    val = int(v);
  }
 
  _m_int(uint64_t v) {
    if (v >= MOD) v %= MOD;
    val = int(v);
  }
 
  _m_int(int v) : _m_int(int64_t(v)) {}
  _m_int(unsigned v) : _m_int(uint64_t(v)) {}
 
  static int inv_mod(int a, int m = MOD) {
    int g = m, r = a, x = 0, y = 1;
 
    while (r != 0) {
      int q = g / r;
      g %= r; swap(g, r);
      x -= q * y; swap(x, y);
    }
 
    return x < 0 ? x + m : x;
  }
 
  explicit operator int() const { return val; }
  explicit operator unsigned() const { return val; }
  explicit operator int64_t() const { return val; }
  explicit operator uint64_t() const { return val; }
  explicit operator double() const { return val; }
  explicit operator long double() const { return val; }
 
  _m_int& operator+=(const _m_int &other) {
    val -= MOD - other.val;
    if (val < 0) val += MOD;
    return *this;
  }
 
  _m_int& operator-=(const _m_int &other) {
    val -= other.val;
    if (val < 0) val += MOD;
    return *this;
  }
 
  static unsigned fast_mod(uint64_t x, unsigned m = MOD) {
#if !defined(_WIN32) || defined(_WIN64)
    return unsigned(x % m);
#endif
    // Optimized mod for Codeforces 32-bit machines.
    // x must be less than 2^32 * m for this to work, so that x / m fits in an unsigned 32-bit int.
    unsigned x_high = unsigned(x >> 32), x_low = unsigned(x);
    unsigned quot, rem;
    asm("divl %4\n"
      : "=a" (quot), "=d" (rem)
      : "d" (x_high), "a" (x_low), "r" (m));
    return rem;
  }
 
  _m_int& operator*=(const _m_int &other) {
    val = fast_mod(uint64_t(val) * other.val);
    return *this;
  }
 
  _m_int& operator/=(const _m_int &other) {
    return *this *= other.inv();
  }
 
  friend _m_int operator+(const _m_int &a, const _m_int &b) { return _m_int(a) += b; }
  friend _m_int operator-(const _m_int &a, const _m_int &b) { return _m_int(a) -= b; }
  friend _m_int operator*(const _m_int &a, const _m_int &b) { return _m_int(a) *= b; }
  friend _m_int operator/(const _m_int &a, const _m_int &b) { return _m_int(a) /= b; }
 
  _m_int& operator++() {
    val = val == MOD - 1 ? 0 : val + 1;
    return *this;
  }
 
  _m_int& operator--() {
    val = val == 0 ? MOD - 1 : val - 1;
    return *this;
  }
 
  _m_int operator++(int) { _m_int before = *this; ++*this; return before; }
  _m_int operator--(int) { _m_int before = *this; --*this; return before; }
 
  _m_int operator-() const {
    return val == 0 ? 0 : MOD - val;
  }
 
  friend bool operator==(const _m_int &a, const _m_int &b) { return a.val == b.val; }
  friend bool operator!=(const _m_int &a, const _m_int &b) { return a.val != b.val; }
  friend bool operator<(const _m_int &a, const _m_int &b) { return a.val < b.val; }
  friend bool operator>(const _m_int &a, const _m_int &b) { return a.val > b.val; }
  friend bool operator<=(const _m_int &a, const _m_int &b) { return a.val <= b.val; }
  friend bool operator>=(const _m_int &a, const _m_int &b) { return a.val >= b.val; }
 
  _m_int inv() const {
    return inv_mod(val);
  }
 
  _m_int pow(int64_t p) const {
    if (p < 0)
      return inv().pow(-p);
 
    _m_int a = *this, result = 1;
 
    while (p > 0) {
      if (p & 1)
        result *= a;
      a *= a;
      p >>= 1;
    }
 
    return result;
  }
  
  friend string to_string(_m_int<MOD> x) {
    return to_string(x.val);
  }
 
  friend ostream& operator<<(ostream &os, const _m_int &m) {
    return os << m.val;
  }
};

extern const int MOD = 998244353;
using Mint = _m_int<MOD>;

const int g = 3;
const int LOGN = 15;

vector<Mint> w[LOGN];
vector<int> rv[LOGN];

void prepare() {
  Mint wb = Mint(g).pow((MOD - 1) / (1 << LOGN));
  forn(st, LOGN - 1) {
    w[st].assign(1 << st, 1);
    Mint bw = wb.pow(1 << (LOGN - st - 1));
    Mint cw = 1;
    forn(k, 1 << st) {
      w[st][k] = cw;
      cw *= bw;
    }
  }
  forn(st, LOGN) {
    rv[st].assign(1 << st, 0);
    if (st == 0) {
      rv[st][0] = 0;
      continue;
    }
    int h = (1 << (st - 1));
    forn(k, 1 << st)
      rv[st][k] = (rv[st - 1][k & (h - 1)] << 1) | (k >= h);
  }
}

void ntt(vector<Mint> &a, bool inv) {
  int n = sz(a);
  int ln = __builtin_ctz(n);
  forn(i, n) {
    int ni = rv[ln][i];
    if (i < ni) swap(a[i], a[ni]);
  }
  forn(st, ln) {
    int len = 1 << st;
    for (int k = 0; k < n; k += (len << 1)) {
      fore(pos, k, k + len){
        Mint l = a[pos];
        Mint r = a[pos + len] * w[st][pos - k];
        a[pos] = l + r;
        a[pos + len] = l - r;
      }
    }
  }
  if (inv) {
    Mint rn = Mint(n).inv();
    forn(i, n) a[i] *= rn;
    reverse(a.begin() + 1, a.end());
  }
}

vector<Mint> mul(vector<Mint> a, vector<Mint> b) {
  int cnt = 1 << (32 - __builtin_clz(sz(a) + sz(b) - 1));
  a.resize(cnt);
  b.resize(cnt);
  ntt(a, false);
  ntt(b, false);
  vector<Mint> c(cnt);
  forn(i, cnt) c[i] = a[i] * b[i];
  ntt(c, true);
  return c;
}

int main() {
  prepare();
  
  vector<Mint> fact(1, 1), ifact(1, 1);
  auto C = [&](int n, int k) -> Mint {
    if (k < 0 || k > n) return 0;
    while (sz(fact) <= n) {
      fact.push_back(fact.back() * sz(fact));
      ifact.push_back(fact.back().inv());
    }
    return fact[n] * ifact[k] * ifact[n - k];
  };
  
  int n;
  cin >> n;
  vector<int> a(n), b(n);
  forn(i, n) cin >> a[i] >> b[i];
  
  vector<Mint> ans(1, 1);
  forn(i, n) {
    vector<Mint> Cs;
    for (int j = b[i] - sz(ans) + 1; j < sz(ans) + a[i]; ++j)
      Cs.push_back(C(a[i] + b[i], j));
    auto res = mul(ans, Cs);
    int cnt = sz(ans);
    ans.resize(cnt + a[i] - b[i]);
    forn(j, sz(ans)) ans[j] = res[cnt + j - 1];
  }
  
  cout << accumulate(ans.begin(), ans.end(), Mint(0)) << endl;
}

int torev = n-k+ 1; int si = k - torev, ei = k -1; for (int i = 0; i < torev / 2; i++) { int l = si + i, r = ei - i; int temp = a[l]; a[l] = a[r]; a[r] = temp; }

struct Node{ int l, r, curr; Node(){ l = r = curr = 0; } Node(char c){ if(c == '+') l = 0, r = 1, curr = 1; else l = -1, r = 0, curr = -1; } Node(int a, int b, int c){ l = a, r = b, curr = c; } }; inline Node join(Node a, Node b){ return Node(min(a.l, b.l + a.curr), max(a.r, b.r + a.curr), a.curr + b.curr); }

Comments (78)

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Lolboyyyy

4 years ago, # |

+67

The terminology of using "path" for problem E was a bit misleading, since path is normally defined as a walk in which all vertices are distinct, which may not be necessary in this case. Would have given greater clarity if "path" was defined within the problem.

Otherwise, great problem with an even better solution.

→ Reply

SimulatedAnnealing

4 years ago, # ^ |

+14

Completely agreed. I spent 30 minutes wondering how to eliminate repetitive edges during the contest (that would be a much harder problem, I believe). Since the term "path" is redefined, it deserves a more detailed distinction.

Into_Your_Arms

Can you please tell about time complexity for $$$E$$$. My doubt is regarding when graph will be like simple line then for $$$i^{th}$$$ vertex we will have $$$i_{C_2}$$$ many possible value of triplets for $$$i^{th}$$$ node so wouldn't it will go $$$O(n^2)$$$?

MadProgrammer

← Rev. 2 →

Let's count the number of vertices in the new graph. Since each vertex is defined by 3 independent values, the number of vertices will equal to the product of numbers of possible values. So, the first value is the number of vertex, so there are $$$n$$$ possible values. Every flag has 2 values, so there are only $$$4\cdot n$$$ vertices in the new graph. Without thinking much you can find an upper bound $$$16 \cdot m$$$ for number of edges. So, time complexity of Dijkstra algorithm on the new graph is still $$$O(m \log n)$$$

Ohh my bad. Thanks. We are not maintaining the edge removed/added instead we are using flag. Thank you!

tfg

+27

Interesting... In portuguese a "walk" is a "path" and a "path where no vertices are repeated" is a "simple path".

rgnerdplayer

There exist cases that passing one edge more than once would decrease the total weight?

Solution is 3 2 13

Oh! So the solution never pass the 4*n vertices more than once but may pass original vertices more than once. Thanks.

steadyEffort

fractal

fixed

Bojack_human

-15

If anyone is stuck at D :https://www.youtube.com/watch?v=mMFkvyeRFQY&t=1s

xyz34701

+32

I know this will sounds biased, but is n = 1000 necessary for G? I've read the official solution, and assuming that the constant factor of NTT is 1, then that solution would require somewhere around 9e7 operations in worst case (which is something around multiplying 2 polynomials of size 5e6). Of course, the official solution runs in a mere 654ms on the hardest test, but it kills pretty much any recursive NTT solution. (I swapped the solution's NTT with an recursive NTT, and it TLEd on custom invocation on the hardest test, and Codeforces gives 15000ms on custom invocation.) While using recursive NTT itself is folly, I've yet to seen a NTT problem on Educational rounds that forces the participant to use iterative NTT. Which leads to the original question: is this problem intended to introduce iterative NTT to more experienced participants, or was it just an overtly pushed limit to fight brute force?

mtw

I wouldn't call it overly pushed.

Wait, so brute force actually pass lower limits? o_O

rananjay23

← Rev. 4 →

During the contest , what was your lane of thought while solving C ? How you approached it before reaching to the solution?

I saw that even SecondThread (in his screencast) and nskybytskyi (in comments) misread the problem C twice and same happened with me . Finally i wrote brute force and observed the pattern and passed somehow.

Though after seeing the solution it doesn't seems to be difficult , we just needed to observed that in palindrome number of inversions would remain same.

_uGOD

+11

I just figured out the number of inversions for the original array, messed around a bit and noticed the pattern, and tried my luck lol.

+49

I read the problem then guessed the solution based on "this is C so it should be easy". In fact it was the hardest problem to think about for me.

arbitrary_coder

hi! in problem C, no more inversion for the test case n= 6, k= 4, the expected output is -> 1 4 3 2 . but the answer 4 1 2 3 may also be correct. since the number of inversions are '3' ans also 4 1 2 3 is lexiographically greater than 1 4 3 2.

1 2 3 4 3 2

1 4 3 2 3 4

4 1 2 3 2 1

Manoj07

Hi, I want to ask you a thing, since you have watched screencast so, his code for C problem he created a "toReverse" variable whose value is n-k+1 whereas I declared it as 2k-n. Although I set different answers for k=2 and n=2, but not able to understand the logic behind that and so please explain this. It would be very helpful. I also included the same code here as well.

Thanks in Advance

Sorry , I watched his stream long back . I don't remember now . If you have any doubt in editorial , i can help you in that.

Adith082

Video Tutorial for problem A,B and D link of problem D : https://www.youtube.com/watch?v=botAQ-AZNOQ

link of problem B : https://www.youtube.com/watch?v=H3qBEDYwwX4

link of problem A : https://www.youtube.com/watch?v=YjC6CcxYxo8&t=263s

Hope you guys will enjoy and understand the intution behind the solutions !!!

_whoAreYou

← Rev. 3 →

+24

Here is an attempt to make an unofficial video editorial of Educational Round 102 by COPS IIT BHU (English) (Problem A-E).

Blog link : https://mirror.codeforces.com/blog/entry/86815

Do check it out. https://www.youtube.com/playlist?list=PLLt4yMoVgczWm1VzN3O4y29VElipDNJ1H

Ankush08

Can we solve the D problem using segment trees also ? If yes can someone explain their approach along with code link ?

iyua

https://mirror.codeforces.com/contest/1473/submission/104325953

PauloMiranda98

To solve using segtree, you just need to know how to join two segments. Each segment will store the smallest x, the largest x and the current x. Node and junction code are these:

Full code: https://mirror.codeforces.com/contest/1473/submission/104307027

ashu_3916

Yes, I did with segment tree , I used range sum segment tree to find sum for [l,r] , rest of the approach is same as editorial you can check here : https://mirror.codeforces.com/contest/1473/submission/104444800

ScarletS

Yep, 104324446 was what I did in-contest

Bhavyashah12050

can anyone explain the solution for problem E .

End_of_Dream

sorry for my poor English
Firstly,add "extra information" when you running dijkstra.Original dijkstra only cares about which point is visted,but now we add two new information on every point.Like copy every point 4 times.When dijkstra is over,we get the shortest path from 1 to others,and the conversions in editorial works based on the answer we get is the "shortest",and "shortest" means we must delete the maximum edge's cost,and multiple the minimum edge's cost twice.This exactly fits the problem.

Liviu2000

in what world G is a div 2 problem? (or under 2100)

nskybytskyi

Educational Contests are typically a bit harder than the regular Div. 2 Rounds (c) neal

+31

In the world where a div2 has problem G.

birchtree

Usually, F&G in Educational Contests focus on more complex algorithms(flow,fft...),but use it as a template.

gaurav5

Are there any problems similar to E ?

Deepesson

https://acm.timus.ru/problem.aspx?space=1&num=1527 You can use a similar algorithm here.

JOICOMP

Problem C, maybe I'm misreading forever and I can't figure out where it is, can someone help me?

For example n = 5, k = 3, sequence a will be "12321", and inversions of "12321" equals to 3.

The answer p is "321" then sequence b will be "32123", and inversions of "32123" equals to 4 ??

It exceeds the total number of inversions in a??

nipul1

12321 has 4 inversions. (2, 1)(First appearance of 2), (3, 2), (3, 1), (2, 1) // second 2

Oh!!! I understand that I couldn't find second (2, 1) for many hours...

Very thanks!

+23

finally...... rating:1599 :(

What kind of solutions does the ML in F intend to block?

O(N^2) edges I guess.

Thanks. I was thinking that the ML was to block something other than flow because that edge-decreasing observation seemed rather obvious.

jiggly_puff

Can someone please explain D , went thorugh the editorial multiple times but still ain't clear ? Thanks !

the video tutorial mentioned in the comments might help you

epsilon_573

You can plot a graph from the given string. As the graph is continuous , the answer will be $$$max + abs(min) + 1$$$. Now notice what happens when you remove a segment. Graph on the right side of the segment gets shifted up or down by $$$abs(\text{sum of segment l,r})$$$ , i.e. if you know the max and min on right side of the segment, you can calculate the new max and min. Solve this in O(1) using prefix, suffix arrays.

manoj_269

i have tried to explain it here

Shameimaru_Aya

+10

May I ask why $$$ans_{i, j} = \sum\limits_{k=1}^{m} \binom{a_i+b_i}{b_i-k+j} ans_{i-1,k}$$$ is established in problem G?

Nero

+13

Consider an infinity grid.

Let $$$ans_{i-1,k}$$$ be at $$$(x_1, y_1) = (k-1+OFFSET_{i-1}, 1-k+OFFSET_{i-1})$$$.

Let $$$ans_{i,j}$$$ be at $$$(x_2, y_2) = (j-1+b_i+OFFSET_{i-1}, 1-j+a_i+OFFSET_{i-1})$$$.

Where $$$OFFSET_{i-1}$$$ is some accumulated offset from $$$1$$$ to $$$i-1$$$.

The number of ways walking from $$$(x_1, y_1)$$$ to $$$(x_2, y_2)$$$ using only right and down is $$$\binom{x2-x1+y2-y1}{x2-x1} = \binom{a_i+b_i}{b_i-k+j}$$$.

Sum from $$$k=1$$$ to $$$k=m$$$ we get the formula.

thanks!

OInfly

sorry...I can't understand the meaning of the coordinate. Could you help me? Thanks a lot

A diagonal line here(say x+y=CONSTANT) corresponds to a column in the original problem.

For G:

1e3 * 1e5 * 17 with constant factor C = 1.7e9 * C.

For NTT, obviously that C > 1.

And it's quite easy to reach the worst case.

How dare you set a 1e3 there.

It's not even close to 1e5, it was something that slowly increases from 1 to 5001 in the worst case. Still, I think this is a pretty tight limit for G though.

I see. My mistook. Thanks for pointing it out.

it's $$$\sum_{i=1}^{1000}{(10i-3)log_2{(10i-3)}}$$$.

Too tight +1.

ftiasch

Finally squeezed within ~300ms. See submission 104694945.

QwQ So fast!

bitcoin

Hi @BledDest,

For the problem F and the input:

2
1 2
100 -200

I guess the network should look like

Where is min-cut here? And where is the flow?

The min cut is 0 (take the set {s, 1}) and the answer is 100-0.

sharath101

Hey can someone provide me a case where my submission for E fails? Almost all tests other than samples are too big to simulate by hand and I'm not able to think of a case :(

_w_

I tried this sample case on your submission

Actual Output: 3 5 6

Expected Output: 3 3 2

oo im so dumb, thanks a lot!

cote

Can anyone give me a visualization of new graph in problem E? Maybe using the first sample test please. I could not imagine how that graph may look like.

Thanks in advance.

There is no new graph . Here dp[u][0][0],dp[u][0][1],dp[u][1][0],dp[u][1][1] are not effected by each other and thus can be seen as independent nodes with all edges attached to them similar to u.

Few definitions (similar to editorial) :

dp[u][0][0] :Minimum Path length from vertex 1 to u such that each edge in path is counted once .

dp[u][0][1] :Minimum Path length from vertex 1 to u such that one edge in path is counted one more time.

dp[u][1][0] :Minimum Path length from vertex 1 to u such that one edge in path is subtracted.

dp[u][1][1] :Minimum Path length from vertex 1 to u such that one edge in path is counted one more time and one edge in path is subtracted (Note that both type of edges can be same and in that case it will be equivalent to dp[u][0][0]).

Answer for node u will be dp[u][1][1] because it fits definition in the problem and the edge subtracted will be always maximum and edge added one more time will be of minimum length (see proof in editorial).

Let's consider graph with n=3,m=2 .Edge 1----2 with weight 3 , 2----3 with weight 6 . Initially dp[u][x][y] = Infinity for all nodes and all values of x and y .

clearly dp[1][0][0] = 0 . Now transitions :

dp[2][0][0] = 3 + dp[1][0][0] = 3 . dp[2][0][1] = 3 + 3 + dp[1][0][0] = 6 . dp[2][1][0] = dp[1][0][0] = 0 . dp[2][1][1] = 3 + 3 — 3 + dp[0][0] = 3 . As expected answer for node 2 is 3.

dp[3][0][0] = 6 + dp[2][0][0] = 9 . dp[3][0][1] = min(dp[2][0][0]+6+6,dp[2][0][1]+6) = min(15,12) = 12 . dp[3][1][0] = min(dp[2][1][0]+6,dp[2][0][0]) = 3 . dp[3][1][1] = min(dp[2][1][1]+6,dp[2][0][1]) = min(9,6) = 6 . As expected answer for node 3 is 6.

It can coded exactly as Dijkstra where we start from dp[1][0][0]. See the code in editorial it's very short and easy to understand.

Thanks for your detailed example. Such a creative use of Dijkstra.

True. It can be best described by demoralizer his comment in his stream

Dijkstra is like a mixture of DP and Greedy. DP type technique with Greedy order of evaluation. That's why a lot of DP things can be used in Dijkstra, as long as the greediness is maintained.

I wonder if there are other problems similar to this one. I just want to see different kinds of dp combined with Dijkstra.

demoralizer

https://cses.fi/problemset/task/1196

Try this problem from cses.

1000101

Also similar stuffs are discussed in the cses book Pg No.-153.

is there any video editorial for F ??

JEEADVANCED

can someone explain where i am wrong in Probem E

i am getting WA on test 5

although i did similar as mentioned in editorial.

SUBMISSION

yeye

Problem F is Closure problem.

Can someone explain this to me? (taken from editorial problem E) We can notice that on the shortest path, the maximum weight edge was subtracted and the minimum weight edge was added. Let's assume that this is not the case, and an edge of non-maximum weight was subtracted from the path, then we can reduce the length of the path by choosing an edge of maximum weight. But this is not possible, because we considered the shortest path. Similarly, it is proved that the added edge was of minimal weight.

How exactly do we reduce path length by choosing the edge of maximum weight ???

MmusoM

3 years ago, # |

What is this, could you explain better?

nonrice

13 months ago, # |

Never mind

AvaraKedavra

9 months ago, # |

Don't know why you are taking max and min with 0 in solution of D, instead of taking max and min with d. Taking d would have been a better choice as it is a more general way of computing suffix max and suffix min.

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