Problem:
Score : 400 points

Problem Statement

You are given a tree with N vertices.

Here, a tree is a kind of graph, and more specifically, a connected undirected graph with N-1 edges, where N is the number of its vertices.

The i-th edge (1≤i≤N-1) connects Vertices a_i and b_i, and has a length of c_i.

You are also given Q queries and an integer K. In the j-th query (1≤j≤Q):

find the length of the shortest path from Vertex x_j and Vertex y_j via Vertex K.

Constraints

3≤N≤10^5

1≤a_i,b_i≤N (1≤i≤N-1)

1≤c_i≤10^9 (1≤i≤N-1)

The given graph is a tree.

1≤Q≤10^5

1≤K≤N

1≤x_j,y_j≤N (1≤j≤Q)

x_j≠y_j (1≤j≤Q)

x_j≠K,y_j≠K (1≤j≤Q)

Input

Input is given from Standard Input in the following format:

N
a_1 b_1 c_1
:
a_{N-1} b_{N-1} c_{N-1}
Q K
x_1 y_1
:
x_{Q} y_{Q}

Output

Print the responses to the queries in Q lines.

In the j-th line j(1≤j≤Q), print the response to the j-th query.

Sample Input 1

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

Sample Output 1

3
2
4

The shortest paths for the three queries are as follows:

Query 1: Vertex 2 → Vertex 1 → Vertex 2 → Vertex 4 : Length 1+1+1=3

Query 2: Vertex 2 → Vertex 1 → Vertex 3 : Length 1+1=2

Query 3: Vertex 4 → Vertex 2 → Vertex 1 → Vertex 3 → Vertex 5 : Length 1+1+1+1=4

Sample Input 2

7
1 2 1
1 3 3
1 4 5
1 5 7
1 6 9
1 7 11
3 2
1 3
4 5
6 7

Sample Output 2

5
14
22

The path for each query must pass Vertex K=2.

Sample Input 3

10
1 2 1000000000
2 3 1000000000
3 4 1000000000
4 5 1000000000
5 6 1000000000
6 7 1000000000
7 8 1000000000
8 9 1000000000
9 10 1000000000
1 1
9 10

Sample Output 3

17000000000

Solution in Java8 (OpenJDK 1.8.0) (judge verdict: Accepted)

import java.io.InputStream;
import java.io.IOException;
import java.io.PrintStream;
import java.util.*;
import java.util.function.Function;
import java.util.function.Predicate;
import java.util.stream.Collectors;
import java.util.stream.IntStream;

public class Main {

  public static void main(String[] args) {
    solve(System.in, System.out);
  }

  static void solve(InputStream is, PrintStream os) {
    Scanner sc = new Scanner(is);

    /* read */

    int n = sc.nextInt();

    List<Set<Graph.Edge<Integer>>> lists = new ArrayList<>(n);
    for (int i = 0; i < n; i++) {
      lists.add(new HashSet<>());
    }

    for (int i = 0; i < n - 1; i++) {
      int a = sc.nextInt() - 1;
      int b = sc.nextInt() - 1;
      int c = sc.nextInt();
      lists.get(a).add(new Graph.Edge<>(a, b, c));
      lists.get(b).add(new Graph.Edge<>(b, a, c));
    }

    Graph<Integer> graph = new AdjacencyListGridPointGraph(n, lists);

    PathQuery<Integer> query = new MemoDfsPathQuery(graph) {

      private final Graph.VertexPath<Integer>[] memo = new Graph.VertexPath[n];

      @Override
      Graph.VertexPath<Integer> fetch(Integer begin, Integer end) {
        if (end != null && memo[end] != null) {
          return memo[end];
        }
        return null;
      }

      @Override
      void memo(Graph.VertexPath<Integer> path) {
        memo[path.getEnd()] = path;
      }
    };

    int q = sc.nextInt();
    int k = sc.nextInt() - 1;

    query.path(k, null);

    for (int i = 0; i < q; i++) {
      int x = sc.nextInt() - 1;
      int y = sc.nextInt() - 1;
      os.println(query.path(k, x).getWeight() + query.path(k, y).getWeight());
    }
  }

  private static class GridPointEncoder implements Encoder<GridPoint> {

    public GridPointEncoder(int h, int w) {
      this.h = h;
      this.w = w;
    }

    private final int h;
    private final int w;

    @Override
    public int num() {
      return w * h;
    }

    @Override
    public int encode(GridPoint from) {
      return w * from.r + from.c;
    }

    @Override
    public GridPoint decode(int from) {
      return new GridPoint(from / w, from % w);
    }
  }

  private static class GridPoint implements Encoded<GridPoint> {

    private final int r;
    private final int c;

    public GridPoint(int r, int c) {
      this.r = r;
      this.c = c;
    }

    @Override
    public boolean equals(Object o) {
      if (this == o) {
        return true;
      }
      if (o == null || getClass() != o.getClass()) {
        return false;
      }
      GridPoint gridPoint = (GridPoint) o;
      return r == gridPoint.r &&
          c == gridPoint.c;
    }

    @Override
    public int hashCode() {
      return Objects.hash(r, c);
    }
  }

  private static class VerticalHorizontalNeighborGridPointGraph extends GridPointGraph {

    private final boolean[][] grid;

    private final Predicate<GridPoint> predicate = this::predicate;
    private int[][] moves = {{-1, 0}, {1, 0}, {0, -1}, {0, 1}};

    public VerticalHorizontalNeighborGridPointGraph(int h, int w, boolean[][] grid) {
      super(h, w);
      this.grid = grid;
    }

    private boolean predicate(GridPoint p) {
      if (p.r < 0 || h <= p.r) {
        return false;
      }
      if (p.c < 0 || w <= p.c) {
        return false;
      }
      return grid[p.r][p.c];
    }

    private Set<Edge<GridPoint>> neighbors(GridPoint from) {

      return Arrays.stream(moves)
          .map(move -> new GridPoint(from.r + move[0], from.c + move[1]))
          .filter(predicate)
          .map(to -> new ConstantWeightEdge<>(from, to))
          .collect(Collectors.toSet());
    }

    @Override
    public Set<Edge<GridPoint>> getEdges(GridPoint from) {
      if (predicate.test(from)) {
        return neighbors(from);
      } else {
        return Collections.emptySet();
      }
    }
  }

  private static class AdjacencyMatrixGridPointGraph extends IntVertexGraph {

    private final Set<Integer> vertexes = new HashSet<>();

    private final int n;
    private final int[][] matrix;

    public AdjacencyMatrixGridPointGraph(int n, int[][] matrix) {
      this.n = n;
      this.matrix = matrix;
      IntStream.range(0, n).forEach(vertexes::add);
    }

    @Override
    public Set<Integer> getVertexes() {
      return vertexes;
    }

    @Override
    public Set<Edge<Integer>> getEdges(Integer from) {
      return IntStream.range(0, n).mapToObj(to -> new Edge<>(from, to, matrix[from][to]))
          .collect(Collectors.toSet());
    }
  }

  private static class AdjacencyListGridPointGraph extends IntVertexGraph {

    private final Set<Integer> vertexes = new HashSet<>();

    private final int n;
    private final List<Set<Edge<Integer>>> edges;

    public AdjacencyListGridPointGraph(int n, List<Set<Edge<Integer>>> edges) {
      this.n = n;
      this.edges = edges;
      // IntStream.range(0, n).forEach(vertexes::add);
    }

    @Override
    public Set<Integer> getVertexes() {
      return vertexes;
    }

    @Override
    public Set<Edge<Integer>> getEdges(Integer from) {
      return edges.get(from);
    }
  }


  private static abstract class GridPointGraph extends EncodedGraph<GridPoint> {

    public GridPointGraph(int h, int w) {
      super(new GridPointEncoder(h, w));
      this.h = h;
      this.w = w;

      for (int r = 0; r < h; r++) {
        for (int c = 0; c < w; c++) {
          nodes.add(new GridPoint(r, c));
        }
      }
    }

    protected final int h;
    protected final int w;

    protected final Set<GridPoint> nodes = new HashSet<>();

    @Override
    public Set<GridPoint> getVertexes() {
      return nodes;
    }
  }

  private static interface Encoded<E extends Encoded> {

  }

  private static interface Encoder<E extends Encoded> {

    int num();

    int encode(E from);

    E decode(int from);
  }

  private static abstract class EncodedGraph<V extends Encoded<V>> implements Graph<V> {

    public EncodedGraph(Encoder<V> encoder) {
      this.encoder = encoder;
      this.delegate = new IntVertexGraph() {
        @Override
        public Set<Integer> getVertexes() {
          return EncodedGraph.this.getVertexes().stream()
              .map(encoder::encode)
              .collect(Collectors.toSet());
        }

        @Override
        public Set<Edge<Integer>> getEdges(Integer from) {
          return EncodedGraph.this.getEdges(encoder.decode(from)).stream()
              .map(e -> new Edge<>(encoder.encode(e.getFrom()), encoder.encode(e.getTo()),
                  e.getWeight()))
              .collect(Collectors.toSet());
        }
      };
    }

    private final Encoder<V> encoder;
    private final IntVertexGraph delegate;

    public PathQuery<V> pathQuery(Function<Graph<Integer>, PathQuery<Integer>> delegate) {
      return new EncodedPathQuery<>(delegate, this);
    }

    private static class EncodedPathQuery<V extends Encoded<V>> implements PathQuery<V> {

      EncodedGraph<V> graph;
      PathQuery<Integer> delegate;

      private EncodedPathQuery(Function<Graph<Integer>, PathQuery<Integer>> delegate,
          EncodedGraph<V> graph) {
        this.graph = graph;
        this.delegate = delegate.apply(graph.delegate);
      }

      @Override
      public VertexPath<V> path(V begin, V end) {
        VertexPath<Integer> path = delegate
            .path(graph.encoder.encode(begin), graph.encoder.encode(end));
        if (path == null) {
          return null;
        }
        return new EfficientVertexPath<>(graph.encoder.decode(path.getBegin()),
            graph.encoder.decode(path.getEnd()), path.getWeight());
      }
    }
  }

  private static interface PathQuery<V> {

    Graph.VertexPath<V> path(V begin, V end);
  }

  private static abstract class QueuedPathQuery implements PathQuery<Integer> {

    protected final Graph<Integer> graph;
    private final Queue<Graph.VertexPath<Integer>> queue;

    public QueuedPathQuery(Graph<Integer> graph, Queue<Graph.VertexPath<Integer>> queue) {
      this.graph = graph;
      this.queue = queue;
    }

    public Graph.VertexPath<Integer> path(Integer begin, Integer end) {

      prepare(begin, end);

      queue.add(new Graph.EfficientVertexPath<>(begin));

      while (!queue.isEmpty()) {
        Graph.VertexPath<Integer> path = queue.remove();
        Integer head = path.getEnd();

        if (head.equals(end)) {
          return path;
        }
        for (Graph.Edge<Integer> e : graph.getEdges(head)) {
          if (predicate(e)) {
            Graph.VertexPath<Integer> p = path.append(e);
            queue.add(p);
            mark(p);
          }
        }
      }
      return null;
    }

    abstract void prepare(Integer begin, Integer end);

    abstract boolean predicate(Graph.Edge<Integer> edge);

    abstract void mark(Graph.VertexPath<Integer> path);
  }

  private static class DijkstraPathQuery extends QueuedPathQuery {

    private final long[] distance;

    public DijkstraPathQuery(Graph<Integer> graph) {
      super(graph, new PriorityQueue<>(
          Comparator.comparingLong(Graph.VertexPath::getWeight)));
      distance = new long[graph.getVertexes().size()];
    }


    @Override
    void prepare(Integer begin, Integer end) {
      IntStream.range(0, graph.getVertexes().size())
          .forEach(i -> distance[i] = Integer.MAX_VALUE);
      distance[begin] = 0;
    }

    @Override
    boolean predicate(Graph.Edge<Integer> edge) {
      return distance[edge.getFrom()] + edge.getWeight() < distance[edge.getTo()];
    }

    @Override
    void mark(Graph.VertexPath<Integer> path) {
      distance[path.getEnd()] = path.getWeight();
    }
  }

  private static class BfsPathQuery extends QueuedPathQuery {

    private final Set<Integer> visited = new HashSet<>();

    public BfsPathQuery(Graph<Integer> graph) {
      super(graph, new ArrayDeque<>());
    }

    @Override
    void prepare(Integer begin, Integer end) {
      visited.add(begin);
    }

    @Override
    boolean predicate(Graph.Edge<Integer> edge) {
      return !visited.contains(edge.getTo());
    }

    @Override
    void mark(Graph.VertexPath<Integer> path) {
      visited.add(path.getEnd());
    }
  }

  private static abstract class MemoDfsPathQuery extends DfsPathQuery {

    public MemoDfsPathQuery(Graph<Integer> graph) {
      super(graph);
    }

    abstract Graph.VertexPath<Integer> fetch(Integer begin, Integer end);

    abstract void memo(Graph.VertexPath<Integer> path);

    @Override
    public Graph.VertexPath<Integer> path(Graph.VertexPath<Integer> path, Integer parent,
        Integer end) {
      Graph.VertexPath<Integer> memo = fetch(path.getBegin(), end);
      if (memo != null) {
        return memo;
      }
      memo(path);
      return super.path(path, parent, end);
    }
  }

  private static class DfsPathQuery implements PathQuery<Integer> {

    protected final Graph<Integer> graph;

    public DfsPathQuery(Graph<Integer> graph) {
      this.graph = graph;
    }

    @Override
    public Graph.VertexPath<Integer> path(Integer begin, Integer end) {
      return path(new Graph.EfficientVertexPath<>(begin), null, end);
    }

    protected Graph.VertexPath<Integer> path(Graph.VertexPath<Integer> path, Integer parent,
        Integer end) {
      Integer head = path.getEnd();
      if (path.getEnd().equals(end)) {
        return path;
      }
      for (Graph.Edge<Integer> e : graph.getEdges(head)) {
        if (!e.getTo().equals(parent)) {
          Graph.VertexPath<Integer> next = path(path.append(e), head, end);
          if (next != null) {
            return next;
          }
        }
      }
      return null;
    }
  }

  private static class WarshallFloydQuery implements PathQuery<Integer> {

    private Graph.VertexPath<Integer>[][] shortest;

    private WarshallFloydQuery(Graph<Integer> graph) {
      Set<Integer> nodes = graph.getVertexes();

      shortest = new Graph.VertexPath[nodes.size()][nodes.size()];

      for (int from : nodes) {
        Set<Graph.Edge<Integer>> edges = graph.getEdges(from);
        for (Graph.Edge<Integer> e : edges) {
          shortest[e.getFrom()][e.getTo()] = new Graph.EfficientVertexPath<>(e);
        }
        shortest[from][from] = new Graph.EfficientVertexPath<>(from);
      }

      for (int relay : nodes) {
        for (int from : nodes) {
          for (int dest : nodes) {
            Graph.VertexPath<Integer> pathA = shortest[from][relay];
            Graph.VertexPath<Integer> pathB = shortest[relay][dest];
            if (pathA != null && pathB != null) {
              if (shortest[from][dest] == null
                  || pathA.getWeight() + pathB.getWeight() < shortest[from][dest]
                  .getWeight()) {
                shortest[from][dest] = pathA.append(pathB);
              }
            }
          }
        }
      }
    }

    @Override
    public Graph.VertexPath<Integer> path(Integer begin, Integer end) {
      return shortest[begin][end];
    }
  }

  private static abstract class IntVertexGraph implements Graph<Integer> {

  }

  private static interface Graph<V> {

    static class Edge<V> {

      public Edge(V from, V to, long weight) {
        this.from = from;
        this.to = to;
        this.weight = weight;
      }

      private final V from;

      private final V to;
      private final long weight;


      V getFrom() {
        return from;
      }

      V getTo() {
        return to;
      }

      long getWeight() {
        return weight;
      }
    }

    static class ConstantWeightEdge<V> extends Edge<V> {

      public ConstantWeightEdge(V from, V to) {
        super(from, to, 1);
      }
    }

    Set<V> getVertexes();

    Set<Edge<V>> getEdges(V from);

    static interface VertexPath<V> {

      V getBegin();

      V getEnd();

      long getWeight();

      VertexPath<V> append(VertexPath<V> other);

      VertexPath<V> append(Edge<V> edge);
    }

    static class EfficientVertexPath<V> implements VertexPath<V> {

      final V begin;
      final V end;
      final long weight;

      public EfficientVertexPath(V begin) {
        this.begin = begin;
        this.end = begin;
        this.weight = 0;
      }

      public EfficientVertexPath(V begin, V end, long weight) {
        this.begin = begin;
        this.end = end;
        this.weight = weight;
      }

      public EfficientVertexPath(Edge<V> edge) {
        this.begin = edge.getFrom();
        this.end = edge.getTo();
        this.weight = edge.getWeight();
      }

      private EfficientVertexPath(VertexPath<V> path, Edge<V> append) {
        this.begin = path.getBegin();
        if (!path.getEnd().equals(append.getFrom())) {
          throw new IllegalStateException("not correct edge.");
        }
        this.end = append.getTo();
        this.weight = path.getWeight() + append.getWeight();
      }

      private EfficientVertexPath(VertexPath<V> pathA, VertexPath<V> pathB) {
        this.begin = pathA.getBegin();
        if (!pathA.getEnd().equals(pathB.getBegin())) {
          throw new IllegalStateException("not correct edge.");
        }
        this.end = pathB.getEnd();
        this.weight = pathA.getWeight() + pathB.getWeight();
      }

      @Override
      public V getBegin() {
        return begin;
      }

      @Override
      public V getEnd() {
        return end;
      }

      @Override
      public long getWeight() {
        return weight;
      }

      @Override
      public VertexPath<V> append(VertexPath<V> other) {
        return new EfficientVertexPath<>(this, other);
      }

      @Override
      public VertexPath<V> append(Edge<V> edge) {
        return new EfficientVertexPath<>(this, edge);
      }

      @Override
      public boolean equals(Object o) {
        if (this == o) {
          return true;
        }
        if (o == null || getClass() != o.getClass()) {
          return false;
        }
        EfficientVertexPath<?> that = (EfficientVertexPath<?>) o;
        return weight == that.weight &&
            Objects.equals(begin, that.begin) &&
            Objects.equals(end, that.end);
      }

      @Override
      public int hashCode() {
        return Objects.hash(begin, end, weight);
      }
    }
  }

  private static class FermatCombination {

    private long fact[];
    private long mod;

    public FermatCombination(int size, long mod) {
      this.fact = new long[size + 1];
      this.mod = mod;

      this.fact[0] = 1;

      for (int i = 1; i <= size; i++) {
        fact[i] = (fact[i - 1] * i) % mod;
      }
    }

    private long factorial(int n) {
      return fact[n];
    }

    private long inverse(long n) {
      return pow(n, mod - 2) % mod;
    }

    private long pow(long x, long n) {
      long ans = 1;
      while (n > 0) {
        if ((n & 1) == 1) {
          ans = ans * x % mod;
        }
        x = x * x % mod;
        n >>= 1;
      }
      return ans;
    }

    long combination(int n, int k) {
      long ans = 1;
      ans *= factorial(n);
      ans %= mod;
      ans *= inverse(factorial(n - k));
      ans %= mod;
      ans *= inverse(factorial(k));
      ans %= mod;
      return ans;
    }
  }

  private static class Scanner {

    private final InputStream is;
    private final byte[] buffer = new byte[1024];
    private int ptr = 0;
    private int buflen = 0;

    Scanner(InputStream is) {
      this.is = is;
    }

    private boolean hasNextByte() {
      if (ptr < buflen) {
        return true;
      } else {
        ptr = 0;
        try {
          buflen = is.read(buffer);
        } catch (IOException e) {
          e.printStackTrace();
        }
        if (buflen <= 0) {
          return false;
        }
      }
      return true;
    }

    private int readByte() {
      if (hasNextByte()) {
        return buffer[ptr++];
      } else {
        return -1;
      }
    }

    private static boolean isPrintableChar(int c) {
      return 33 <= c && c <= 126;
    }

    public boolean hasNext() {
      while (hasNextByte() && !isPrintableChar(buffer[ptr])) {
        ptr++;
      }
      return hasNextByte();
    }

    public String next() {
      if (!hasNext()) {
        throw new NoSuchElementException();
      }
      StringBuilder sb = new StringBuilder();
      int b = readByte();
      while (isPrintableChar(b)) {
        sb.appendCodePoint(b);
        b = readByte();
      }
      return sb.toString();
    }

    public long nextLong() {
      if (!hasNext()) {
        throw new NoSuchElementException();
      }
      long n = 0;
      boolean minus = false;
      int b = readByte();
      if (b == '-') {
        minus = true;
        b = readByte();
      }
      if (b < '0' || '9' < b) {
        throw new NumberFormatException();
      }
      while (true) {
        if ('0' <= b && b <= '9') {
          n *= 10;
          n += b - '0';
        } else if (b == -1 || !isPrintableChar(b)) {
          return minus ? -n : n;
        } else {
          throw new NumberFormatException();
        }
        b = readByte();
      }
    }

    public int nextInt() {
      long nl = nextLong();
      if (nl < Integer.MIN_VALUE || nl > Integer.MAX_VALUE) {
        throw new NumberFormatException();
      }
      return (int) nl;
    }

    public double nextDouble() {
      return Double.parseDouble(next());
    }
  }
}