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 Go (1.6) (judge verdict: Accepted)

// ProblemURL : https://atcoder.jp/contests/abc074/tasks/arc083_b
// ---------------------------------------------
package main

import (
	"bufio"
	"fmt"
	"math"
	"os"
	"sort"
	"strconv"
	"strings"
)

func abs(x int) int {
	if x < 0 {
		return -x
	}
	return x
}
func diff(a, b int) int {
	if a > b {
		return a - b
	}
	return b - a
}
func larger(a, b int) int {
	if a > b {
		return a
	}
	return b
}
func smaller(a, b int) int {
	if a < b {
		return a
	}
	return b
}
func largest(a, b, c int) (lgst int) {
	if a > b {
		lgst = a
	} else {
		lgst = b
	}
	if c > lgst {
		lgst = c
	}
	return
}
func smallest(a, b, c int) (slst int) {
	if a < b {
		slst = a
	} else {
		slst = b
	}
	if c < slst {
		slst = c
	}
	return
}
func intsMax(a []int) (val int) {
	if len(a) == 0 {
		panic("func intsMax: argument slice length must not be zero")
	}
	val = a[0]
	for _, aa := range a {
		if aa > val {
			val = aa
		}
	}
	return
}
func intsMaxIdx(a []int) (idx, val int) {
	if len(a) == 0 {
		panic("func intsMaxIdx: argument slice length must not be zero")
	}
	val = a[0]
	for i, aa := range a {
		if aa > val {
			idx, val = i, aa
		}
	}
	return
}
func intsMin(a []int) (val int) {
	if len(a) == 0 {
		panic("func intsMin: argument slice length must not be zero")
	}
	val = a[0]
	for _, aa := range a {
		if aa < val {
			val = aa
		}
	}
	return
}
func intsMinIdx(a []int) (idx, val int) {
	if len(a) == 0 {
		panic("func intsMinIdx: argument slice length must not be zero")
	}
	val = a[0]
	for i, aa := range a {
		if aa < val {
			idx, val = i, aa
		}
	}
	return
}
func intsSum(a []int) int {
	res := 0
	for _, v := range a {
		res += v
	}
	return res
}
func intsAve(s []int) float64 {
	return float64(intsSum(s)) / float64(len(s))
}
func intsAdd(a *[]int, x int)    { *a = append(*a, x) }
func intsDelete(a *[]int, i int) { *a = append((*a)[:i], (*a)[i+1:]...) }
func intsJoin(a, b []int) []int  { return append(a, b...) }
func intsClear(a []int) []int    { return a[:0] }
func intsCopy(a []int) []int     { return append([]int(nil), a...) }
func intsCopySortAsc(a []int) []int {
	s := intsCopy(a)
	sort.Ints(s)
	return s
}
func intsCopySortDesc(a []int) []int {
	s := intsCopy(a)
	sort.Sort(sort.Reverse(sort.IntSlice(s)))
	return s
}
func intsReverse(a []int) {
	for i, j := 0, len(a)-1; i < j; i, j = i+1, j-1 {
		a[i], a[j] = a[j], a[i]
	}
}
func intsFill(a []int, val int) {
	for i := 0; i < len(a); i++ {
		a[i] = val
	}
}
func intsPeekBack(a []int) int {
	if len(a) == 0 {
		panic("func peekBack: zero length slice")
	}
	return a[len(a)-1]
}
func intsPeekFront(a []int) int {
	if len(a) == 0 {
		panic("func peekFront: zero length slice")
	}
	return a[0]
}
func intsPopBack(a *[]int) int {
	if len(*a) == 0 {
		panic("func popBack: zero length slice")
	}
	t := (*a)[len(*a)-1]
	*a = (*a)[:len(*a)-1]
	return t
}
func intsPopFront(a *[]int) int {
	if len(*a) == 0 {
		panic("func popFront: zero length slice")
	}
	h := (*a)[0]
	*a = (*a)[1:]
	return h
}
func intsPushBack(a *[]int, x int) { *a = append(*a, x) }
func intsPushFront(a *[]int, x int) {
	*a = append(*a, 0)
	copy((*a)[1:], (*a)[0:])
	(*a)[0] = x
}
func intsAppendSentinelHead(a *[]int, sentinel int) { intsPushFront(a, sentinel) }
func intsAppendSentinelTail(a *[]int, sentinel int) { intsPushBack(a, sentinel) }
func intsAppendSentinel(a *[]int, sentinel int) {
	intsPushFront(a, sentinel)
	intsPushBack(a, sentinel)
}
func isEven(n int) bool    { return n&1 == 0 }
func isOdd(n int) bool     { return n&1 == 1 }
func floor(x float64) int  { return int(x) }
func ceil(x float64) int   { return int(math.Ceil(x)) }
func ceilDiv(x, y int) int { return (x + y - 1) / y }
func pow(x, y int) int {
	if y < 0 {
		panic("Exponent must be a natural number")
	}
	z := 1
	for ; y != 0; y >>= 1 {
		if y&1 == 1 {
			z *= x
		}
		x *= x
	}
	return z
}
func fact(x int) int {
	fact := 1
	for i := 1; i <= x; i++ {
		fact *= i
	}
	return fact
}
func sigmaK(n int) int  { return n * (n + 1) / 2 }
func sigmaKK(n int) int { return (n * (n + 1) / 2) * (2*n + 1) / 3 }
func euclideanDistance(a, b point) float64 {
	s := diff(a.x, b.x)
	s *= s
	t := diff(a.y, b.y)
	t *= t
	return math.Sqrt(float64(s) + float64(t))
}
func manhattanDistance(a, b point) int {
	s := diff(a.x, b.x)
	t := diff(a.y, b.y)
	return s + t
}
func swap(a, b *int) { *a, *b = *b, *a }
func chmax(a *int, b int) bool {
	if *a < b {
		*a = b
		return true
	}
	return false
}
func chmin(a *int, b int) bool {
	if *a > b {
		*a = b
		return true
	}
	return false
}
func cumSum(a []int) []int {
	b := append([]int{0}, a...)
	for i := 1; i < len(b); i++ {
		b[i] += b[i-1]
	}
	return b
}
func runesReverse(a []rune) {
	for i, j := 0, len(a)-1; i < j; i, j = i+1, j-1 {
		a[i], a[j] = a[j], a[i]
	}
	return
}
func bytesReverse(a []byte) {
	for i, j := 0, len(a)-1; i < j; i, j = i+1, j-1 {
		a[i], a[j] = a[j], a[i]
	}
	return
}
func strUpperCase(s string) string { return strings.ToUpper(s) }
func strLowerCase(s string) string { return strings.ToLower(s) }
func strToInt(s string) int {
	i, err := strconv.Atoi(s)
	if err != nil {
		panic(err)
	}
	return i
}
func intToStr(i int) string     { return strconv.Itoa(i) }
func CalcMod(a, mod int) int    { return (a%mod + mod) % mod }
func ModSum(a *int, b, mod int) { *a = (((*a) % mod) + mod + (b % mod)) % mod }
func ModMul(a *int, b, mod int) { *a = (((*a) % mod) * (b % mod)) % mod }
func ModPow(x, y, mod int) int {
	if y < 0 {
		panic("Exponent must be a natural number")
	}
	z := 1
	for ; y != 0; y >>= 1 {
		if y&1 == 1 {
			z = (z * x) % mod
		}
		x = (x * x) % mod
	}
	return z
}
func ModFact(x, mod int) int {
	fact := 1
	for i := 1; i <= x; i++ {
		fact = fact * i % mod
	}
	return fact
}
func ModInv(x, mod int) int { return ModPow(x, mod-2, mod) }
func initMatrixBool(parentSize, childSize int, initialValue bool) *[][]bool {
	res := make([][]bool, parentSize)
	for i := range res {
		res[i] = make([]bool, childSize)
		for j := range res[i] {
			res[i][j] = initialValue
		}
	}
	return &res
}
func initMatrixInt(parentSize, childSize int, initialValue int) *[][]int {
	res := make([][]int, parentSize)
	for i := range res {
		res[i] = make([]int, childSize)
		for j := range res[i] {
			res[i][j] = initialValue
		}
	}
	return &res
}

var (
	sc = bufio.NewScanner(os.Stdin)
	bw = bufio.NewWriterSize(os.Stdout, bwBufSize)
)

func init() {
	sc.Split(bufio.ScanWords)
	sc.Buffer(make([]byte, 1e7), 1e7)
}
func ru() (n int) {
	sc.Scan()
	if err := sc.Err(); err != nil {
		panic(err)
	}
	for _, v := range sc.Bytes() {
		n = n*10 + int(v-48)
	}
	return
}
func ri() (n int) {
	sc.Scan()
	if err := sc.Err(); err != nil {
		panic(err)
	}
	b := sc.Bytes()
	neg := false
	if b[0] == 45 {
		neg = true
		b = b[1:]
	}
	for _, v := range b {
		n = n*10 + int(v-48)
	}
	if neg {
		n = -n
	}
	return
}
func ri64() (n int64) {
	sc.Scan()
	if err := sc.Err(); err != nil {
		panic(err)
	}
	b := sc.Bytes()
	neg := false
	if b[0] == 45 {
		neg = true
		b = b[1:]
	}
	for _, v := range b {
		n = n*10 + int64(v-48)
	}
	if neg {
		n = -n
	}
	return
}
func rf64() float64 {
	sc.Scan()
	if err := sc.Err(); err != nil {
		panic(err)
	}
	f, err := strconv.ParseFloat(sc.Text(), 64)
	if err != nil {
		panic(err)
	}
	return f
}
func rs() string {
	sc.Scan()
	if err := sc.Err(); err != nil {
		panic(err)
	}
	return sc.Text()
}
func rb() []byte {
	sc.Scan()
	if err := sc.Err(); err != nil {
		panic(err)
	}
	return sc.Bytes()
}
func rr() []rune {
	sc.Scan()
	if err := sc.Err(); err != nil {
		panic(err)
	}
	return []rune(sc.Text())
}
func ris(n int) []int {
	s := make([]int, n)
	for i := range s {
		s[i] = ri()
	}
	return s
}
func risZeroBased(n int) []int {
	s := make([]int, n)
	for i := range s {
		s[i] = ri() - 1
	}
	return s
}
func ri64s(n int) []int64 {
	s := make([]int64, n)
	for i := range s {
		s[i] = ri64()
	}
	return s
}
func rss(n int) []string {
	s := make([]string, n)
	for i := range s {
		s[i] = rs()
	}
	return s
}
func pf(format string, a ...interface{}) {
	if _, err := fmt.Fprintf(bw, format, a...); err != nil {
		panic(err)
	}
}
func pln(a ...interface{}) {
	if _, err := fmt.Fprintln(bw, a...); err != nil {
		panic(err)
	}
}
func pints(a []int) {
	for _, v := range a {
		fmt.Fprintln(bw, v)
	}
}
func pintsol(a []int) {
	s := fmt.Sprint(a)
	if _, err := fmt.Fprintln(bw, s[1:len(s)-1]); err != nil {
		panic(err)
	}
}
func dbg(a ...interface{}) {
	if _, err := fmt.Fprintln(os.Stderr, a...); err != nil {
		panic(err)
	}
}
func dbgMatrix(a [][]int) {
	for i := 0; i < len(a); i++ {
		dbg(a[i])
	}
}
func YESorNO(b bool) string {
	if b {
		return "YES"
	} else {
		return "NO"
	}
}
func YesOrNo(b bool) string {
	if b {
		return "Yes"
	} else {
		return "No"
	}
}
func main() {
	solve()
	if err := bw.Flush(); err != nil {
		panic(err)
	}
}

const (
	bwBufSize = 1e6

	alphabetLower = "abcdefghijklmnopqrstuvwxyz"
	alphabetUpper = "ABCDEFGHIJKLMNOPQRSTUVWXYZ"

	// maxInt64  = 9223372036854775807  // 9e18
	// maxInt32  = 2147483647           // 2e9
	// maxUint64 = 18446744073709551615 // 1e19
	inf = 1 << 60
	mod = 1e9 + 7
)

type pair struct{ fi, se int }
type byFiSe []pair

func (p byFiSe) Len() int      { return len(p) }
func (p byFiSe) Swap(i, j int) { p[i], p[j] = p[j], p[i] }
func (p byFiSe) Less(i, j int) bool {
	if p[i].fi == p[j].fi {
		return p[i].se < p[j].se
	}
	return p[i].fi < p[j].fi
}

type point struct{ x, y int }

func solve() {
	n := ri()

	dist := make([][]int, n)
	edges := make([][]int, n)
	for i := 0; i < n; i++ {
		dist[i] = make([]int, n)
		edges[i] = make([]int, n)
		for j := 0; j < n; j++ {
			weight := ri()
			dist[i][j] = weight
			edges[i][j] = weight
		}
	}

	ok := true
	for k := 0; k < n; k++ {
		for i := 0; i < n; i++ {
			for j := 0; j < n; j++ {
				if k == i || k == j || i == j {
					continue
				}
				via := dist[i][k] + dist[k][j]
				if via < dist[i][j] {
					ok = false
				} else if via == dist[i][j] {
					edges[i][j] = inf
				}
			}
		}
	}

	if !ok {
		pln(-1)
		return
	}

	sum := 0
	for i := 0; i < n; i++ {
		for j := 0; j < n; j++ {
			if edges[i][j] == inf {
				continue
			}
			sum += edges[i][j]
		}
	}
	pln(sum / 2)
}