Problem:
Score : 500 points

Problem Statement

There is a rooted tree (see Notes) with N vertices numbered 1 to N.
Each of the vertices, except the root, has a directed edge coming from its parent.
Note that the root may not be Vertex 1.

Takahashi has added M new directed edges to this graph.
Each of these M edges, u \rightarrow v, extends from some vertex u to its descendant v.

You are given the directed graph with N vertices and N-1+M edges after Takahashi added edges.
More specifically, you are given N-1+M pairs of integers, (A_1, B_1), ..., (A_{N-1+M}, B_{N-1+M}), which represent that the i-th edge extends from Vertex A_i to Vertex B_i.

Restore the original rooted tree.

Notes

For "tree" and other related terms in graph theory, see the article in Wikipedia, for example.

Constraints

3 \leq N

1 \leq M

N + M \leq 10^5

1 \leq A_i, B_i \leq N

A_i \neq B_i

If i \neq j, (A_i, B_i) \neq (A_j, B_j).

The graph in input can be obtained by adding M edges satisfying the condition in the problem statement to a rooted tree with N vertices.

Input

Input is given from Standard Input in the following format:

N M
A_1 B_1
:
A_{N-1+M} B_{N-1+M}

Output

Print N lines.
In the i-th line, print 0 if Vertex i is the root of the original tree, and otherwise print the integer representing the parent of Vertex i in the original tree.

Note that it can be shown that the original tree is uniquely determined.

Sample Input 1

3 1
1 2
1 3
2 3

Sample Output 1

0
1
2

The graph in this input is shown below:

It can be seen that this graph is obtained by adding the edge 1 \rightarrow 3 to the rooted tree 1 \rightarrow 2 \rightarrow 3.

Sample Input 2

6 3
2 1
2 3
4 1
4 2
6 1
2 6
4 6
6 5

Sample Output 2

6
4
2
0
6
2

Solution in C# (Mono 4.6.2.0) (judge verdict: Time Limit Exceeded)

using System;
using System.Collections.Generic;
using System.IO;
using System.Linq;

class Program
{
    static void Main(string[] args)
    {
        var N = sc.ReadInt();
        var M = sc.ReadInt();
        var AB = sc.ReadTupArray(N + M - 1, -1);
        var g = new List<int>[N];
        for (int i = 0; i < N; i++)
        {
            g[i] = new List<int>();
        }
        var bel = new BellmanFord(N);
        var dij = new Dijkstra(N, true);
        for (int i = 0; i < N + M - 1; i++)
        {
            var a = AB[i].Item1;
            var b = AB[i].Item2;
            dij.Add(a, b, -1);
            bel.Add(a, b, -1);
            g[a].Add(b);
        }
        var root = Enumerable.Range(0, N).Except(AB.Select(ab => ab.Item2)).First();
        bel.CalcDistance(root);
        var dist = bel.Distances;
        var edges = new List<Tuple<int, int>>();
        var q = new Queue<Tuple<int, int>>();
        q.Enqueue(Tuple.Create(root, 0));
        while (q.Count > 0)
        {
            var t = q.Dequeue();
            var now = t.Item1;
            var d = t.Item2;
            if (d == -dist[now])
            {
                foreach (var next in g[now])
                {
                    if (-dist[next] == d + 1)
                    {
                        edges.Add(Tuple.Create(now, next));
                        q.Enqueue(Tuple.Create(next, d + 1));
                    }
                }
            }
        }
        edges.Add(Tuple.Create(-1, root));
        foreach (var e in edges.OrderBy(e => e.Item2))
        {
            Console.WriteLine(e.Item1 + 1);
        }
    }

    class BellmanFord
    {
        public List<Edge> Edges { get; set; }
        public int V { get; set; }
        public long[] Distances { get; set; }
        public bool[] Negative { get; set; }
        public BellmanFord(int v)
        {
            this.V = v;
            this.Edges = new List<Edge>();
        }

        public void Add(int from, int to, int cost)
        {
            this.Edges.Add(new Edge(from, to, cost));
        }

        public void CalcDistance(int start)
        {
            this.Distances = new long[V];
            for (int i = 0; i < V; i++)
            {
                this.Distances[i] = long.MaxValue;
            }
            this.Distances[start] = 0;

            for (int i = 0; i < this.V; i++)
            {
                foreach (var e in this.Edges)
                {
                    if (this.Distances[e.From] == long.MaxValue) continue;
                    var cost = this.Distances[e.From] + e.Cost;
                    if (cost < this.Distances[e.To])
                    {
                        this.Distances[e.To] = cost;
                    }
                }
            }

            this.Negative = new bool[V];
            for (int i = 0; i < this.V; i++)
            {
                foreach (var e in this.Edges)
                {
                    if (this.Distances[e.From] == long.MaxValue) continue;
                    var cost = this.Distances[e.From] + e.Cost;
                    if (cost < this.Distances[e.To])
                    {
                        this.Distances[e.To] = cost;
                        this.Negative[e.To] = true;
                    }
                }
            }
        }
        public struct Edge
        {
            public int From { get; set; }
            public int To { get; set; }
            public int Cost { get; set; }
            public Edge(int from, int to, int cost)
            {
                this.From = from;
                this.To = to;
                this.Cost = cost;
            }
        }
    }
    class Dijkstra
    {
        public int V { get; set; }
        public int E { get; set; }
        public List<Edge>[] G { get; set; }
        public bool Directed { get; set; }
        public Dijkstra(int n, bool directed = false)
        {
            this.V = n;
            this.Directed = directed;
            this.G = new List<Edge>[V];
            for (int i = 0; i < V; i++)
            {
                this.G[i] = new List<Edge>();
            }
        }
        public void Add(int a, int b, long cost = 1)
        {
            this.G[a].Add(new Edge(b, cost));
            if (!this.Directed) this.G[b].Add(new Edge(a, cost));
            this.E++;
        }
        public long GetMinCost(int start, int goal)
        {
            var cost = new long[V];
            for (int i = 0; i < V; i++) cost[i] = long.MaxValue;
            cost[start] = 0;

            var capacity = Math.Max(1, E);
            var q = new PriorityQueue<P>(capacity, false);
            q.Enqueue(new P(start, 0));

            while (q.Count > 0)
            {
                var p = q.Dequeue();
                if (p.Cost != cost[p.A]) continue;
                foreach (var e in this.G[p.A])
                {
                    if (cost[e.To] > p.Cost + e.Cost)
                    {
                        cost[e.To] = p.Cost + e.Cost;
                        q.Enqueue(new P(e.To, cost[e.To]));
                    }
                }
            }
            return cost[goal] == long.MaxValue ? -1 : cost[goal];
        }
        public long[] Calc(int start)
        {
            var cost = new long[V];
            for (int i = 0; i < V; i++) cost[i] = long.MaxValue;
            cost[start] = 0;

            var capacity = Math.Max(1, E);
            var q = new PriorityQueue<P>(capacity, false);
            q.Enqueue(new P(start, 0));

            while (q.Count > 0)
            {
                var p = q.Dequeue();
                if (p.Cost != cost[p.A]) continue;
                foreach (var e in this.G[p.A])
                {
                    if (cost[e.To] > p.Cost + e.Cost)
                    {
                        cost[e.To] = p.Cost + e.Cost;
                        q.Enqueue(new P(e.To, cost[e.To]));
                    }
                }
            }
            return cost;
        }
        public struct Edge
        {
            public int To;
            public long Cost;
            public Edge(int to, long cost)
            {
                this.To = to;
                this.Cost = cost;
            }
        }
        public struct P : IComparable<P>
        {
            public int A;
            public long Cost;
            public P(int a, long cost)
            {
                this.A = a;
                this.Cost = cost;
            }
            public int CompareTo(P other)
            {
                return this.Cost.CompareTo(other.Cost);
            }
        }
    }
    class PriorityQueue<T>
    {
        public readonly T[] Array;
        public readonly bool Greater;
        public int Count { get; private set; } = 0;
        private IComparer<T> Cmp;
        public PriorityQueue(int capacity, bool greater = true, IComparer<T> cmp = null)
        {
            this.Cmp = cmp;
            if (cmp == null)
            {
                this.Cmp = Comparer<T>.Default;
            }
            this.Array = new T[capacity];
            this.Greater = greater;
        }
        public void Enqueue(T item)
        {
            this.Array[this.Count] = item;
            this.Count += 1;
            this.UpHeap();
        }
        public T Dequeue()
        {
            this.Swap(0, this.Count - 1);
            this.Count -= 1;
            this.DownHeap();

            return this.Array[this.Count];
        }
        public T Peek()
        {
            return this.Array[0];
        }
        private void UpHeap()
        {
            var n = this.Count - 1;
            while (n != 0)
            {
                var parent = (n - 1) / 2;

                if (this.Compare(this.Array[n], this.Array[parent]) > 0)
                {
                    this.Swap(n, parent);
                    n = parent;
                }
                else
                {
                    break;
                }
            }
        }
        private void DownHeap()
        {
            var parent = 0;
            while (true)
            {
                var child = 2 * parent + 1;
                if (child > this.Count - 1) break;

                if (child < this.Count - 1 && this.Compare(this.Array[child], this.Array[child + 1]) < 0)
                {
                    child += 1;
                }
                if (this.Compare(this.Array[parent], this.Array[child]) < 0)
                {
                    this.Swap(parent, child);
                    parent = child;
                }
                else
                {
                    break;
                }
            }
        }
        private int Compare(T a, T b)
        {
            var c = this.Cmp.Compare(a, b);
            if (!this.Greater)
            {
                c = -c;
            }
            return c;
        }
        private void Swap(int a, int b)
        {
            var tmp = this.Array[a];
            this.Array[a] = this.Array[b];
            this.Array[b] = tmp;
        }
    }
    static Scanner sc = new Scanner();
}

class Scanner
{
    string[] S = new string[0];
    int Index = 0;
    char[] Separators = new char[] { ' ' };

    public string Next()
    {
        if (this.Index < this.S.Length) return this.S[this.Index++];
        var line = "";
        while (line == "") line = Console.ReadLine();
        this.S = line.Split(this.Separators, StringSplitOptions.RemoveEmptyEntries);
        if (this.S.Length == 0) return this.Next();
        this.Index = 0;
        return this.S[this.Index++];
    }
    public string ReadStr()
    {
        return this.Next();
    }
    public char ReadChar()
    {
        return this.Next()[0];
    }
    public int ReadInt()
    {
        return int.Parse(this.Next());
    }
    public uint ReadUInt()
    {
        return uint.Parse(this.Next());
    }
    public long ReadLong()
    {
        return long.Parse(this.Next());
    }
    public double ReadDouble()
    {
        return double.Parse(this.Next());
    }
    public Tuple<int, int> ReadTup(int add = 0)
    {
        return Tuple.Create(this.ReadInt() + add, this.ReadInt() + add);
    }
    public Tuple<long, long> ReadTupLong(int add = 0)
    {
        return Tuple.Create(this.ReadLong() + add, this.ReadLong() + add);
    }
    public Tuple<int, int, int> ReadTup3(int add = 0)
    {
        return Tuple.Create(this.ReadInt() + add, this.ReadInt() + add, this.ReadInt() + add);
    }
    public Tuple<int, int, int, int> ReadTup4(int add = 0)
    {
        return Tuple.Create(this.ReadInt() + add, this.ReadInt() + add, this.ReadInt() + add, this.ReadInt() + add);
    }
    public int[] ReadIntArray(int n)
    {
        var array = new int[n];
        for (int i = 0; i < array.Length; i++)
        {
            array[i] = this.ReadInt();
        }
        return array;
    }
    public long[] ReadLongArray(int n)
    {
        var array = new long[n];
        for (int i = 0; i < array.Length; i++)
        {
            array[i] = this.ReadLong();
        }
        return array;
    }
    public double[] ReadDoubleArray(int n)
    {
        var array = new double[n];
        for (int i = 0; i < array.Length; i++)
        {
            array[i] = this.ReadDouble();
        }
        return array;
    }
    public char[] ReadCharArray(int n)
    {
        var array = new char[n];
        for (int i = 0; i < array.Length; i++)
        {
            array[i] = this.ReadChar();
        }
        return array;
    }
    public string[] ReadStrArray(int n)
    {
        var array = new string[n];
        for (int i = 0; i < array.Length; i++)
        {
            array[i] = this.ReadStr();
        }
        return array;
    }
    public Tuple<long, long>[] ReadTupLongArray(int n, int add = 0)
    {
        var array = new Tuple<long, long>[n];
        for (int i = 0; i < n; i++)
        {
            array[i] = this.ReadTupLong(add);
        }
        return array;
    }
    public Tuple<int, int>[] ReadTupArray(int n, int add = 0)
    {
        var array = new Tuple<int, int>[n];
        for (int i = 0; i < n; i++)
        {
            array[i] = this.ReadTup(add);
        }
        return array;
    }
    public Tuple<int, int, int>[] ReadTup3Array(int n, int add = 0)
    {
        var array = new Tuple<int, int, int>[n];
        for (int i = 0; i < n; i++)
        {
            array[i] = this.ReadTup3(add);
        }
        return array;
    }
    public Tuple<int, int, int, int>[] ReadTup4Array(int n, int add = 0)
    {
        var array = new Tuple<int, int, int, int>[n];
        for (int i = 0; i < n; i++)
        {
            array[i] = this.ReadTup4(add);
        }
        return array;
    }
}