Problem: Number of Restricted Paths From First to Last Node

There is an undirected weighted connected graph. You are given a positive
integer `n` which denotes that the graph has `n` nodes labeled from `1` to
`n`, and an array `edges` where each `edges[i] = [ui, vi, weighti]` denotes
that there is an edge between nodes `ui` and `vi` with weight equal to
`weighti`.

A path from node `start` to node `end` is a sequence of nodes `[z0, z1, z2,
..., zk]` such that `z0 = start` and `zk = end` and there is an edge between
`zi` and `zi+1` where `0 <= i <= k-1`.

The distance of a path is the sum of the weights on the edges of the path. Let
`distanceToLastNode(x)` denote the shortest distance of a path between node
`n` and node `x`. A restricted path is a path that also satisfies that
`distanceToLastNode(zi) > distanceToLastNode(zi+1)` where `0 <= i <= k-1`.

Return the number of restricted paths from node `1` to node `n`. Since that
number may be too large, return it modulo `109 + 7`.


Python solution:
import math
import itertools
import bisect
import collections
import string
import heapq
import functools
import sortedcontainers
from typing import List, Dict, Tuple, Iterator

class Solution:
  def countRestrictedPaths(self, n: int, edges: List[List[int]]) -> int:
    graph = [[] for _ in range(n)]

    for u, v, w in edges:
      graph[u - 1].append((v - 1, w))
      graph[v - 1].append((u - 1, w))

    return self._dijkstra(graph, 0, n - 1)

  def _dijkstra(self, graph: List[List[Tuple[int, int]]], src: int, dst: int) -> int:
    kMod = 10**9 + 7
    ways = [0] * len(graph)
    dist = [math.inf] * len(graph)

    ways[dst] = 1
    dist[dst] = 0
    minHeap = [(dist[dst], dst)]

    while minHeap:
      d, u = heapq.heappop(minHeap)
      if d > dist[u]:
        continue
      for v, w in graph[u]:
        if d + w < dist[v]:
          dist[v] = d + w
          heapq.heappush(minHeap, (dist[v], v))
        if dist[v] < dist[u]:
          ways[u] += ways[v]
          ways[u] %= kMod

    return ways[src]


Java solution:
class Solution {
  public int countRestrictedPaths(int n, int[][] edges) {
    List<Pair<Integer, Integer>>[] graph = new List[n];
    Arrays.setAll(graph, i -> new ArrayList<>());

    for (int[] edge : edges) {
      final int u = edge[0] - 1;
      final int v = edge[1] - 1;
      final int w = edge[2];
      graph[u].add(new Pair<>(v, w));
      graph[v].add(new Pair<>(u, w));
    }

    return dijkstra(graph, 0, n - 1);
  }

  private int dijkstra(List<Pair<Integer, Integer>>[] graph, int src, int dst) {
    final int MOD = 1_000_000_007;
    // ways[i] := the number of restricted path from i to n
    long[] ways = new long[graph.length];
    // dist[i] := the distance to the last node of i
    long[] dist = new long[graph.length];
    Arrays.fill(dist, Long.MAX_VALUE);

    ways[dst] = 1;
    dist[dst] = 0;
    // (d, u)
    Queue<Pair<Long, Integer>> minHeap =
        new PriorityQueue<>(Comparator.comparingLong(Pair::getKey));
    minHeap.offer(new Pair<>(dist[dst], dst));

    while (!minHeap.isEmpty()) {
      final long d = minHeap.peek().getKey();
      final int u = minHeap.poll().getValue();
      if (d > dist[u])
        continue;
      for (Pair<Integer, Integer> pair : graph[u]) {
        final int v = pair.getKey();
        final int w = pair.getValue();
        if (d + w < dist[v]) {
          dist[v] = d + w;
          minHeap.offer(new Pair<>(dist[v], v));
        }
        if (dist[v] < dist[u]) {
          ways[u] += ways[v];
          ways[u] %= MOD;
        }
      }
    }

    return (int) ways[src];
  }
}


C++ solution:
class Solution {
 public:
  int countRestrictedPaths(int n, vector<vector<int>>& edges) {
    vector<vector<pair<int, int>>> graph(n);

    for (const vector<int>& edge : edges) {
      const int u = edge[0] - 1;
      const int v = edge[1] - 1;
      const int w = edge[2];
      graph[u].emplace_back(v, w);
      graph[v].emplace_back(u, w);
    }

    return dijkstra(graph, 0, n - 1);
  }

 private:
  int dijkstra(const vector<vector<pair<int, int>>>& graph, int src, int dst) {
    constexpr int kMod = 1'000'000'007;
    // ways[i] := the number of restricted path from i to n
    vector<long> ways(graph.size());
    // dist[i] := the distance to the last node of i
    vector<long> dist(graph.size(), LONG_MAX);

    ways[dst] = 1;
    dist[dst] = 0;
    using P = pair<long, int>;  // (d, u)
    priority_queue<P, vector<P>, greater<>> minHeap;
    minHeap.emplace(dist[dst], dst);

    while (!minHeap.empty()) {
      const auto [d, u] = minHeap.top();
      minHeap.pop();
      if (d > dist[u])
        continue;
      for (const auto& [v, w] : graph[u]) {
        if (d + w < dist[v]) {
          dist[v] = d + w;
          minHeap.emplace(dist[v], v);
        }
        if (dist[v] < dist[u]) {
          ways[u] += ways[v];
          ways[u] %= kMod;
        }
      }
    }

    return ways[src];
  }
};