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Number Of Ways To Arrive At Destination Leetcode

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Number Of Ways To Arrive At Destination Leetcode

Exploring the Number of Ways to Arrive at Destination on LeetCodeOne of the most frequently discussed algorithm problems on LeetCode is calculating the number of ways to arrive at a destination. This type of problem tests your understanding of dynamic programming, modular arithmetic, and efficient pathfinding. Whether you are a beginner preparing for coding interviews or an experienced developer refining your algorithmic skills, this is a topic worth mastering.

In this topic, we’ll explore the logic behind the problem, break down common approaches, and look at optimization techniques to help you solve it effectively.

Understanding the Problem

The typical version of this problem presents a directed weighted graph, where you are given

  • A total number of nodes labeled from 0 to n - 1.

  • A list of roads where each road connects two nodes and has a travel time.

  • A start node (often node 0) and a destination node (usually node n - 1).

Your task is to count how many different shortest paths you can take from the start to the destination. The catch is you must only consider paths that take the minimum possible time.

Key Concepts

Before jumping into code or formulas, let’s review a few important concepts

1. Dijkstra’s Algorithm

To find the shortest paths efficiently in a graph with positive weights, Dijkstra’s algorithm is often used. This helps determine the minimum time required to reach each node from the start.

2. Dynamic Programming on Graphs

Once you know the shortest distance to each node, you can use dynamic programming (DP) to count how many ways each node can be reached using paths of that exact shortest length.

3. Modulo Arithmetic

Since the number of valid paths can be very large, most implementations require the result to be returned modulo 1e9+7 to avoid integer overflow.

A Simple Example

Imagine the following roads

roads = [[0, 1, 1],[0, 2, 1],[1, 3, 1],[2, 3, 1]]

You want to go from node 0 to node 3. There are two valid paths

  • 0 → 1 → 3

  • 0 → 2 → 3

Both take 2 units of time. So the number of ways to arrive at the destination is 2.

Step-by-Step Approach

Let’s break the problem down into steps

1. Build the Graph

Create an adjacency list where each node points to its neighbors and the associated travel time.

from collections import defaultdictgraph = defaultdict(list)for u, v, time in roadsgraph[u].append((v, time))graph[v].append((u, time)) # Assuming undirected graph

2. Use Dijkstra’s Algorithm

Use a min-heap (priority queue) to compute the shortest time to each node.

import heapqdist = [float('inf')] * ndist[0] = 0ways = [0] * nways[0] = 1heap = [(0, 0)]while heaptime, u = heapq.heappop(heap)if time > dist[u]continuefor v, t in graph[u]if time + t < dist[v]dist[v] = time + tways[v] = ways[u]heapq.heappush(heap, (dist[v], v))elif time + t == dist[v]ways[v] = (ways[v] + ways[u]) % (10**9 + 7)

3. Return the Result

return ways[n - 1]

This tells you how many ways you can reach the final node using paths of minimum time.

Common Pitfalls

  • Not using a min-heap Dijkstra’s algorithm requires a priority queue for optimal efficiency.

  • Ignoring modulo operations Without % (10**9 + 7), you risk integer overflow.

  • Not updating the path count correctly You must reset the number of ways if a shorter path is found and only add when an equally short path is found.

Time and Space Complexity

  • Time Complexity O((N + E) log N), where N is the number of nodes and E is the number of edges.

  • Space Complexity O(N + E) for storing the graph and auxiliary arrays.

This is efficient even for larger graphs (up to 10^5 nodes), as required in many LeetCode problems.

Variants of the Problem

On LeetCode and similar platforms, you may encounter different versions of this problem

1. Unique Paths in a Grid

Instead of a graph, you move on a 2D grid with constraints on movement (e.g., only right or down). In this case, dynamic programming based on grid positions is used.

2. Weighted Paths with Cost Limit

You might be required to count all paths under a certain cost threshold. In such cases, Dijkstra may not be sufficient, and DFS + memoization or BFS may be more appropriate.

3. Counting Paths Without Shortest Constraint

Here, you count all possible paths from source to destination, regardless of the total cost. Be cautious of infinite loops or exponential time in such problems.

When to Use Dijkstra + DP

  • When all edge weights are non-negative.

  • When you are asked to count only the shortest paths.

  • When the graph has many nodes and you need efficiency.

Why It Matters

Understanding how to solve the ‘number of ways to arrive at destination’ problem builds a strong foundation in

  • Graph theory

  • Priority queues

  • Modular arithmetic

  • Dynamic programming

It’s an excellent example of how multiple algorithmic techniques come together in one challenge. Whether you’re tackling interviews or refining your LeetCode skills, mastering this problem equips you with tools useful across a wide range of coding scenarios.

Solving the number of ways to arrive at destination problem on LeetCode is more than just an exercise it’s a chance to apply and deepen your algorithmic thinking. From building the graph to implementing Dijkstra’s algorithm and applying dynamic programming, each step offers insight into efficient problem-solving.

Keep practicing, experiment with variations, and make sure you understand the logic behind each approach. With time, these types of problems will become second nature.