Solution - #1976 - Mridul/Edited - 23/03/2025

Author: mrid88

Problems/1976 - Number of Ways to Arrive at Destination - Medium/Explanation.md

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+# Explanation of the Program
+
+### **Objective**
+The goal of the program is to count the number of distinct shortest paths from node `0` to node `n-1` in a weighted, undirected graph. The result is returned modulo \( 10^9 + 7 \).
+
+---
+
+### **Key Components**
+1. **Class `Pair`:**  
+   - Represents a pair of values (distance and node index).
+   - Used for storing the distance of a node from the source and the index of the node.
+
+2. **Adjacency List (`adj`):**  
+   - The graph is represented as an adjacency list where each node's neighbors and edge weights are stored.
+   - The adjacency list is built using input data from the `roads` array.
+
+3. **Priority Queue (`pq`):**  
+   - Implements a min-heap to efficiently get the node with the smallest distance.
+   - Helps in applying Dijkstra's algorithm.
+
+4. **Arrays `dist` and `ways`:**  
+   - `dist[i]`: Stores the shortest distance from the source node (node `0`) to node `i`.
+   - `ways[i]`: Stores the number of distinct ways to reach node `i` with the shortest distance.
+
+---
+
+### **Step-by-Step Process**
+1. **Input Parsing and Graph Construction:**
+   - The graph is built using the `roads` array, where each road connects two nodes (`u` and `v`) with a specific weight (`w`).
+   - Two directed edges are created for each road since the graph is undirected.
+
+2. **Initialization:**
+   - Initialize the `dist` array with `Long.MAX_VALUE` to signify unreachable nodes.
+   - Initialize the `ways` array to `0` (initially no paths to any node).
+   - Set `dist[0] = 0` (distance from node `0` to itself is `0`).
+   - Set `ways[0] = 1` (only one way to stay at the source).
+
+3. **Dijkstra's Algorithm:**
+   - Use the priority queue to process the node with the smallest distance.
+   - For each neighbor of the current node:
+     - If a shorter path is found, update the distance and reset the number of ways.
+     - If a path with the same shortest distance is found, add the number of ways to reach the current node to the neighbor's path count.
+
+4. **Return Result:**
+   - The number of ways to reach the destination node (`n-1`) is returned modulo \( 10^9 + 7 \).
+
+---
+
+### **Code Explanation**
+#### 1. **Graph Representation**
+```java
+ArrayList<ArrayList<Pair>> adj = new ArrayList<>();
+for(int i = 0; i < n; i++) {
+    adj.add(new ArrayList<>());
+}
+for(int i = 0; i < m; i++) {
+    adj.get(roads[i][0]).add(new Pair(roads[i][2], roads[i][1]));
+    adj.get(roads[i][1]).add(new Pair(roads[i][2], roads[i][0]));
+}
+
+Time Complexity: O((n+m)log(n)) 
+The program processes all m roads to build the adjacency list.
+This operation takes O(m) time.
+Dijkstra’s Algorithm:
+Each node and its neighbors are processed through the priority queue.
+Popping a node from the priority queue takes O(log n) time.
+Over all m edges, the operations amount to O((n + m) log n) time.
+Overall Time Complexity:
+The dominant factor is Dijkstra’s algorithm, so the time complexity is O((n + m) log n).
+
+Space Complexity: O(n)
+The adjacency list stores all m edges, requiring O(m) space.
+The dist and ways arrays require O(n) space each.
+The priority queue stores up to O(n) nodes.
+Overall Space Complexity:
+The total space complexity is O(n + m)

Problems/1976 - Number of Ways to Arrive at Destination - Medium/Solution.java

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+class Pair{
+    long first;
+    long second;
+    public Pair(long first,long second) {
+        this.first = first;
+        this.second = second;
+    }
+}
+class Solution {
+    public int countPaths(int n, int[][] roads) {
+        ArrayList<ArrayList<Pair>> adj = new ArrayList<>();
+        for(int i = 0;i < n; i++){
+            adj.add(new ArrayList<>());
+        }
+        int m = roads.length;
+        for(int i = 0; i<m; i++){
+            adj.get(roads[i][0]).add(new Pair(roads[i][2], roads[i][1]));
+            adj.get(roads[i][1]).add(new Pair(roads[i][2], roads[i][0]));
+        }
+        PriorityQueue<Pair> pq = new PriorityQueue<>((x,y)->Long.compare(x.first, y.first));
+        long dist[] = new long[n];
+        long ways[] = new long[n];
+      
+
+        for(int i = 0;i < n; i++){
+            dist[i] = Long.MAX_VALUE;
+            ways[i] = 0;
+        }
+        pq.add(new Pair(0,0));
+        dist[0] = 0;
+        ways[0] = 1;
+        int mod = (int)(1e9 + 7);
+        while(!pq.isEmpty()){
+            long dis = pq.peek().first;
+            int node = (int)(pq.peek().second);
+            pq.remove();
+            for(Pair it : adj.get(node)){
+                long newDis = it.first;
+                int adjNode = (int)it.second;
+                 if(dis + newDis < dist[adjNode]){
+                    dist[adjNode] = dis + newDis;
+                    pq.add(new Pair(dis + newDis, adjNode));
+                    ways[adjNode]  = ways[node];
+                }
+                else if(dis + newDis == dist[adjNode]){
+                    ways[adjNode] = (ways[adjNode] + ways[node])%mod;
+                }
+            }
+        }
+        return (int)(ways[n-1]%mod);
+    }
+}