Problem:
MathJax.Hub.Config({ tex2jax: { inlineMath: [["$","$"], ["\\(","\\)"]], processEscapes: true }});

Single Source Shortest Path

For a given weighted graph $G = (V, E)$, find the shortest path from a source to each vertex. For each vertex $u$, print the total weight of edges on the shortest path from vertex $0$ to $u$.

Input

In the first line, an integer $n$ denoting the number of vertices in $G$ is given. In the following $n$ lines, adjacency lists for each vertex $u$ are respectively given in the following format:

$u$ $k$ $v_1$ $c_1$ $v_2$ $c_2$ ... $v_k$ $c_k$

Vertices in $G$ are named with IDs $0, 1, ..., n-1$. $u$ is ID of the target vertex and $k$ denotes its degree. $v_i (i = 1, 2, ... k)$ denote IDs of vertices adjacent to $u$ and $c_i$ denotes the weight of a directed edge connecting $u$ and $v_i$ (from $u$ to $v_i$).

Output

For each vertex, print its ID and the distance separated by a space character in a line respectively. Print in order of vertex IDs.

Constraints

$1 \leq n \leq 100$

$0 \leq c_i \leq 100,000$

$|E| \leq 10,000$

All vertices are reachable from vertex $0$

Sample Input 1

5
0 3 2 3 3 1 1 2
1 2 0 2 3 4
2 3 0 3 3 1 4 1
3 4 2 1 0 1 1 4 4 3
4 2 2 1 3 3

Sample Output 1

0 0
1 2
2 2
3 1
4 3

Reference

Introduction to Algorithms, Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. The MIT Press.

Solution in JavaScript (judge verdict: Accepted)

//config = { input: 'tmp', newline: '\r\n' }; // win
config = { input: '/dev/stdin', newline: '\n' }; // linux

line = require('fs').readFileSync(config.input, 'ascii')
  .trim()
  .split(config.newline);
n = Number(line.shift());
adj = {};
function add(obj, u, v, c) {
  if (!obj.hasOwnProperty(u)) obj[u] = {};
  obj[u][v] = c;
}
for (i in line) {
  ary = line[i].split(' ').map(Number);
  u = ary[0];
  for (j = 2; j < ary.length; j += 2)
    add(adj, u, ary[j], ary[j+1]);
}

d = new Array(n);
pi = new Array(n);

function initializeSingleSource(s) {
  var i;
  for (i = 0; i < n; i++) {
    d[i] = Number.POSITIVE_INFINITY;
    pi[i] = null;
  }
  d[s] = 0;
}

function relax(u, v, w) {
  if (d[v] > d[u] + w) {
    d[v] = d[u] + w;
    pi[v] = u;
  }
}

function bellmanFord(adj, s) {
  initializeSingleSource(s);
  for (i = 1; i < n; i++) {
    for (u in adj) {
      for (v in adj[u]) {
        relax(u, v, adj[u][v]);
      }
    }
  }
  for (u in adj) {
    for (v in adj[v]) {
      if (d[v] > d[u] + adj[u][v])
        return false;
    }
  }
  return true;
}

bellmanFord(adj, 0);
for (i in d) console.log("%s %d", i, d[i]);