Problem:
Score : 500 points

Problem Statement

In Takahashi Kingdom, which once existed, there are N cities, and some pairs of cities are connected bidirectionally by roads.
The following are known about the road network:

People traveled between cities only through roads. It was possible to reach any city from any other city, via intermediate cities if necessary.

Different roads may have had different lengths, but all the lengths were positive integers.

Snuke the archeologist found a table with N rows and N columns, A, in the ruin of Takahashi Kingdom.
He thought that it represented the shortest distances between the cities along the roads in the kingdom.

Determine whether there exists a road network such that for each u and v, the integer A_{u, v} at the u-th row and v-th column of A is equal to the length of the shortest path from City u to City v.
If such a network exist, find the shortest possible total length of the roads.

Constraints

1 \leq N \leq 300

If i ≠ j, 1 \leq A_{i, j} = A_{j, i} \leq 10^9.

A_{i, i} = 0

Inputs

Input is given from Standard Input in the following format:

N
A_{1, 1} A_{1, 2} ... A_{1, N}
A_{2, 1} A_{2, 2} ... A_{2, N}
...
A_{N, 1} A_{N, 2} ... A_{N, N}

Outputs

If there exists no network that satisfies the condition, print -1.
If it exists, print the shortest possible total length of the roads.

Sample Input 1

3
0 1 3
1 0 2
3 2 0

Sample Output 1

3

The network below satisfies the condition:

City 1 and City 2 is connected by a road of length 1.

City 2 and City 3 is connected by a road of length 2.

City 3 and City 1 is not connected by a road.

Sample Input 2

3
0 1 3
1 0 1
3 1 0

Sample Output 2

-1

As there is a path of length 1 from City 1 to City 2 and City 2 to City 3, there is a path of length 2 from City 1 to City 3.
However, according to the table, the shortest distance between City 1 and City 3 must be 3.

Thus, we conclude that there exists no network that satisfies the condition.

Sample Input 3

5
0 21 18 11 28
21 0 13 10 26
18 13 0 23 13
11 10 23 0 17
28 26 13 17 0

Sample Output 3

82

Sample Input 4

3
0 1000000000 1000000000
1000000000 0 1000000000
1000000000 1000000000 0

Sample Output 4

3000000000

Solution in C++14 (Clang 3.8.0) (judge verdict: Accepted)

//includes
#include<iostream>
#include<vector>
#include<set>
#include<tuple>
#include<algorithm>
#include<cmath>
#include<string>
#include<stack>
#include<map>
#include<queue>
#include<iomanip>
#include<numeric>
#include<assert.h>
#include<cstring>
#include<unordered_map>
#include<bitset>

//macros and consts
#define REP(i, n) for(int i = 0; i < (int)(n); i++)
#define all(a) (a).begin(),(a).end()
#define FOR(i,a,b) for(int i = (a); i < (b); i++)
#define FORE(i,a) for(auto &i : a)
#define pb push_back
#define mp make_pair
#define eb emplace_back
#define accm accumulate
#define Fi first
#define Se second
#define INF (1<<30)
#define mINF -(1<<30)
#define endl "\n"
#define cont continue
#define brk break

//shorter types
using namespace std;
using ll = long long;
using vi = std::vector<int>;
using vc = std::vector<char>;
using vll = std::vector<long long>;
using vs = std::vector<string>;
using Mi = map<int,int>;
using Mll = map<ll,ll>;
using UMi = unordered_map<int,int>;
using UMll = unordered_map<ll,ll>;
using Pi = pair<int,int>;
using Pll = pair<ll,ll>;
using vPi = vector<Pi>;
using vPll = vector<Pll>;
using vvi = vector<vector<int>>;
using vvll = vector<vector<ll>>;
using vvc = vector<vector<char>>;
using vvs = vector<vector<string>>;
using pqgi = priority_queue<int,vector<int>,greater<int>>;
using pqsi = priority_queue<int,vector<int>,less<int>>;
using pqgll = priority_queue<int,vector<int>,greater<int>>;
using pssll = priority_queue<int,vector<int>,less<int>>;
template<class T>
using vec = vector<T>;

//frequent functions and data
class Cpmath{
  public:
    template<typename T>
    static T gcd(T a, T b){
      if (a == 0) 
        return b; 
      return gcd(b % a, a); 
    }
    template<typename T>
    static T findGCD(vector<T> &arr, size_t n){
      T result = arr[0]; 
      for (size_t i = 1; i < n; i++) 
        result = gcd(arr[i], result); 
      return result; 
    }
    template<typename T>
    static bool is_prime(T n){
      if(n == 1){
        return false;
      }
      for(size_t i = 2 ; i <= pow(n,0.5) ; i++){
        if(n % i == 0){
          return false;
        }
      }
      return true;
    }
    static ll fact(ll n){
      if(n == 0){
        return 1LL;
      }
      else{
        return n * fact(n - 1);
      }
    }
    static ll permutation(int n, int r){
      assert(n >= r);
      ll ret = 1;
      for(int i = n ; i > n - r ; i--){
        ret *= i;
      }
      return ret;
    }
};

class NCR{
  private:
    static const int MAX = 210000;
    static const int MOD = 998244353;
    ll fac[MAX],finv[MAX],inv[MAX];
  public:
    void COMinit(){
      fac[0] = fac[1] = 1;
      finv[0] = finv[1] = 1;
      inv[1] = 1;
      for(int i = 2 ; i < MAX ; i++){
        fac[i] = fac[i-1] * i % MOD;
        inv[i] = MOD - inv[MOD % i] * (MOD / i) % MOD;
        finv[i] = finv[i-1] * inv[i] % MOD;
      }
    }
    ll COM(int n,int k){
      if(n < k) return 0;
      if(n < 0 || k < 0){
        return 0;
      }
      return fac[n] * (finv[k] * finv[n-k] % MOD) % MOD;
    }
};

struct BipartiteMatching {
    vector<vector<int>> E; int n, m; vector<int> match, dist;
    void init(int _n,int _m){n=_n,m=_m;E.resize(n+m+2);match.resize(n+m+2);dist.resize(n+m+2);}
    bool bfs() {
        queue<int> que;
        FOR(i, 1, n + 1) {
            if (!match[i]) dist[i] = 0, que.push(i);
            else dist[i] = INF;
        }
        dist[0] = INF;
        while (!que.empty()) {
            int u = que.front(); que.pop();
            if (u) for(auto &v : E[u]) if (dist[match[v]] == INF) {
                dist[match[v]] = dist[u] + 1;
                que.push(match[v]);
            }
        }
        return (dist[0] != INF);
    }
    bool dfs(int u) {
        if (u) {
            for(auto &v : E[u]) if (dist[match[v]] == dist[u] + 1) if (dfs(match[v])) {
                match[v] = u; match[u] = v;
                return true;
            }
            dist[u] = INF;
            return false;
        }
        return true;
    }
    void add(int a, int b) { b += n; E[a + 1].push_back(b + 1); E[b + 1].push_back(a + 1); }
    int whois(int x) { return match[x + 1] - 1; }
    int solve() {
        FOR(i, 0, n + m + 1) match[i] = 0;
        int res = 0;
        while (bfs()) FOR(i, 1, n + 1) if (!match[i] && dfs(i)) res++;
        return res;
    }
};

struct SegmentTree{
  private:
    int n;
    vector<int> node;
  public:
    SegmentTree(vector<int> v){
      int sz = v.size();
      n = 1;
      while(n < sz) n *= 2;
      node.resize(2 * n - 1 , INF);
      for(int i = 0 ; i < sz ; i++){
        node[i + n - 1] = v[i];
      }
      for(int i = n - 2 ; i >= 0 ; i--){
        node[i] = min(node[2*i+1],node[2*i+2]);
      }
    }
    void update(int x, int val){
      x += (n - 1);
      node[x] = val;
      while(x > 0){
        x = (x - 1) / 2;
        node[x] = min(node[2*x+1],node[2*x+2]);
      }
    }
    int getmin(int a, int b , int k = 0, int l = 0, int r = -1){
      if(r < 0){
        r = n;
      }
      if(r <= a || b <= l){
        return INF;
      }
      if(a <= l && r <= b){
        return node[k];
      }
      int vl = getmin(a,b,2*k+1,l,(l+r)/2);
      int vr = getmin(a,b,2*k+2,(l+r)/2,r);
      return min(vl,vr);
    }
};

template<class Abel> struct WUnionFind {
    vector<int> par;
    vector<int> rank;
    vector<Abel> diff_weight;

    WUnionFind(int n = 1, Abel SUM_UNITY = 0) {
        init(n, SUM_UNITY);
    }

    void init(int n = 1, Abel SUM_UNITY = 0) {
        par.resize(n); rank.resize(n); diff_weight.resize(n);
        for (int i = 0; i < n; ++i) par[i] = i, rank[i] = 0, diff_weight[i] = SUM_UNITY;
    }

    int root(int x) {
        if (par[x] == x) {
            return x;
        }
        else {
            int r = root(par[x]);
            diff_weight[x] += diff_weight[par[x]];
            return par[x] = r;
        }
    }

    Abel weight(int x) {
        root(x);
        return diff_weight[x];
    }

    bool issame(int x, int y) {
        return root(x) == root(y);
    }

    bool merge(int x, int y, Abel w) {
        w += weight(x); w -= weight(y);
        x = root(x); y = root(y);
        if (x == y) return false;
        if (rank[x] < rank[y]) swap(x, y), w = -w;
        if (rank[x] == rank[y]) ++rank[x];
        par[y] = x;
        diff_weight[y] = w;
        return true;
    }

    Abel diff(int x, int y) {
        return weight(y) - weight(x);
    }
};

struct UnionFind{
  vector<int> par;
  UnionFind(int n) : par(n,-1) {}
  void init(int n) {par.assign(n,-1);}
  int root(int x){
    if(par[x] < 0) return x;
    else return par[x] = root(par[x]);
  }
  bool issame(int x,int y){
    return root(x) == root(y);
  }
  bool merge(int x,int y){
    x = root(x);
    y = root(y);
    if (x == y) return false;
    if (par[x] > par[y]){
      swap(x,y);
    }
    par[x] += par[y];
    par[y] = x;
    return true;
  }
  int size(int x){
    return -par[root(x)];
  }
};

void YN(bool a){
  cout << (a?"YES":"NO") << endl;
}

void Yn(bool a){
  cout << (a?"Yes":"No") << endl;
}
void yn(bool a){
  cout << (a?"yes":"no") << endl;
}
template<class T>bool chmax(T &a, const T &b) { if (a<b) { a=b; return 1; } return 0; }
template<class T>bool chmin(T &a, const T &b) { if (b<a) { a=b; return 1; } return 0; }

int dx[4] = {1,0,-1,0};
int dy[4] = {0,1,0,-1};
template<typename T>
inline int digitsum(T num){
  int ret = 0;
  while(num){
    ret += num % 10;
    num /= 10;
  }
  return ret;
}
template<typename InputIt,typename T>
inline bool argexist(InputIt first,InputIt last,const T& x){
  if(std::find(first,last,x) != last){
    return true;
  }
  else{
    return false;
  }
}
template<typename InputIt,typename T>
inline int argfind(InputIt first, InputIt last, const T& x){
  auto it = find(first,last,x);
  return distance(first,it);
}
template<typename InputIt>
inline int argmax(InputIt first, InputIt last) {
    auto it = max_element(first, last);
    return distance(first, it);
}
template<typename InputIt>
inline int argmin(InputIt first, InputIt last) {
    auto it = min_element(first, last);
    return distance(first, it);
}
// you need sort before using this func
template<typename T>
inline void erasebv(vector<T>& c,T v){
  c.erase(remove(begin(c),end(c),v),end(c));
}
template<typename T>
inline void UNIQ(T& c) {
    c.erase(unique(begin(c), end(c)), end(c));
}
template<typename T>
inline T POP_BACK(vector<T>& que) {
    T x = que.back(); que.pop_back();
    return x;
}
template<typename T>
inline T POP_BACK(deque<T>& que) {
    T x = que.back(); que.pop_back();
    return x;
}
template<typename T>
inline T POP_FRONT(deque<T>& que) {
    T x = que.front(); que.pop_front();
    return x;
}
template<typename T, typename C>
inline T POP(stack<T,C>& stk) {
    T x = stk.top(); stk.pop();
    return x;
}
template<typename T, typename C>
inline T POP(queue<T,C>& que) {
    T x = que.front(); que.pop();
    return x;
}
template<typename T, typename Cont, typename Cmp>
inline T POP(priority_queue<T,Cont,Cmp>& que) {
    T x = que.top(); que.pop();
    return x;
}
template<typename T1, typename T2> ostream& operator<<(ostream& s, const pair<T1, T2>& p) {return s << "(" << p.first << ", " << p.second << ")";}
template<typename T> ostream& operator<<(ostream& s, const vector<T>& v) {
  int len = v.size();
  for (int i = 0; i < len; ++i) {
    s << v[i]; if (i < len - 1) s << "\t";
  }
  return s;
}
template<typename T> ostream& operator<<(ostream& s, const vector< vector<T> >& vv) {
  int len = vv.size();
  for (int i = 0; i < len; ++i) {
  s << vv[i] << endl;
  }
  return s;
}

//graphs
namespace GraphLib{
  using Weight = int;
  struct Edge{
    int src,dst;
    Weight weight;
    Edge(int src,int dst,Weight weight):
      src(src),dst(dst),weight(weight){}
  };
  bool operator < (const Edge &e, const Edge &f) {
    return e.weight != f.weight ? e.weight > f.weight :
      e.src != f.src ? e.src < f.src : e.dst < f.dst;
  }
  using Edges = vector<Edge>;
  using Graph = vector<Edges>;
  using Array = vector<Weight>;
  using Matrix = vector<Array>;
  
  void shortestPath(const Graph &g, int s,
      vector<Weight> &dist, vector<int> &prev) {
    int n = g.size();
    dist.assign(n, INF); dist[s] = 0;
    prev.assign(n, -1);
    priority_queue<Edge> Q; // "e < f" <=> "e.weight > f.weight"
    for (Q.push(Edge(-2, s, 0)); !Q.empty(); ) {
      Edge e = Q.top(); Q.pop();
      if (prev[e.dst] != -1) continue;
      prev[e.dst] = e.src;
      FORE(f,g[e.dst]) {
        if (dist[f.dst] > e.weight+f.weight) {
          dist[f.dst] = e.weight+f.weight;
          Q.push(Edge(f.src, f.dst, e.weight+f.weight));
        }
      }
    }
  }
  vector<int> DijkstraBuildPath(const vector<int> &prev, int t) {
    vector<int> path;
    for (int u = t; u >= 0; u = prev[u])
      path.push_back(u);
    reverse(path.begin(), path.end());
    return path;
  }
  void visit(const Graph & g, int v, int u,
      Edges& brdg, vector< vector<int> >& tecomp,
      stack<int>& roots, stack<int>& S, vector<bool>& inS,
      vector<int>& num, int& time) {
    num[v] = ++time;
    S.push(v); inS[v] = true;
    roots.push(v);
    FORE(e, g[v]) {
      int w = e.dst;
      if (num[w] == 0)
        visit(g, w, v, brdg, tecomp, roots, S, inS, num, time);
      else if (u != w && inS[w])
        while (num[roots.top()] > num[w]) roots.pop();
    }
    if (v == roots.top()) {
      brdg.push_back(Edge(u, v, 1)); // any weight will do
      tecomp.push_back(vector<int>());
      while (1) {
        int w = S.top(); S.pop(); inS[w] = false;
        tecomp.back().push_back(w);
        if (v == w) break;
      }
      roots.pop();
    }
  }
  void bridge(const Graph& g, Edges& brdg, vector< vector<int> >& tecomp) {
    const int n = g.size();
    vector<int> num(n);
    vector<bool> inS(n);
    stack<int> roots, S;
    int time = 0;
    REP(u, n) if (num[u] == 0) {
      visit(g, u, n, brdg, tecomp, roots, S, inS, num, time);
      brdg.pop_back();
    }
  }
}


//here begins your code
int N;
int d[310][310];
int a[310][310];
signed main(){
  std::ios::sync_with_stdio(false);
  std::cin.tie(0);
  cin >> N;
  REP(i,N){
    REP(j,N){
      cin >> a[i][j];
      d[i][j] = a[i][j];
    }
  }
  
  REP(k,N){
    REP(i,N){
      REP(j,N){
        if(i == j || j == k || k == i || d[i][k] == INF || d[k][j] == INF){
          continue;
        }
        if(d[i][k] + d[k][j] == a[i][j]){
          d[i][j] = INF;
        }
        else if(d[i][k] + d[k][j] < a[i][j]){
          cout << -1 << endl;
          return 0;
        }
      }
    }
  }
  ll sum = 0;
  REP(i,N){
    REP(j,N){
      if(d[i][j] != INF){
        sum+=d[i][j];
      }
    }
  }
  cout << sum / 2 << endl;
  return 0;

}