Number of Ways to Arrive at Destination

We need to find the number of ways we can travel from intersection 0 to intersection ( n-1 ) in the shortest amount of time. We’ll use Dijkstra’s algorithm to find the shortest time to reach each intersection and simultaneously count the number of ways to get to that intersection.

Approach

Initialize Data Structures: Create a list time of length ( n ) to store the shortest time to reach each intersection. Initialize all values to infinity except time[0] which is 0. Create a list ways of length ( n ) to store the number of ways to reach each intersection. Initialize all values to 0 except ways[0] which is 1. Use a min-heap heap to keep track of intersections to explore.
Dijkstra’s Algorithm: Use a min-heap to implement Dijkstra’s algorithm. Push (0, 0) to the heap to start with intersection 0. Then, while the heap is not empty, do the following: a. Pop the intersection with the smallest time from the heap. b. For each road connected to this intersection, calculate the time to reach the next intersection using this road. If this time is less than the currently stored time for that intersection, update the time and the number of ways to reach that intersection. If this time is equal to the currently stored time, increment the number of ways. c. Push the next intersection and the updated time to the heap.
Return Result: Return the value ways[n-1] modulo ( 10^9 + 7 ).

Code

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import heapq

class Solution:
    def countPaths(self, n: int, roads: List[List[int]]) -> int:
        MOD = 10**9 + 7
        graph = [[] for _ in range(n)]
        for u, v, time in roads:
            graph[u].append((v, time))
            graph[v].append((u, time))

        time = [float('inf')] * n
        ways = [0] * n
        time[0] = 0
        ways[0] = 1

        heap = [(0, 0)]
        while heap:
            cur_time, u = heapq.heappop(heap)
            if cur_time > time[u]:
                continue
            for v, t in graph[u]:
                new_time = cur_time + t
                if new_time < time[v]:
                    time[v] = new_time
                    ways[v] = ways[u]
                    heapq.heappush(heap, (new_time, v))
                elif new_time == time[v]:
                    ways[v] = (ways[v] + ways[u]) % MOD

        return ways[n-1]

Complexity

The time complexity of this solution is ( O(E + V \log V) ), where ( E ) is the number of roads and ( V ) is the number of intersections. The space complexity is ( O(E + V) ).

Summary

This solution finds the shortest time to reach intersection ( n-1 ) from intersection 0 and simultaneously counts the number of ways to do so. The approach builds on Dijkstra’s algorithm by tracking both time and the number of ways. The final result is the value at ways[n-1], which represents the number of ways to reach the destination in the shortest time.

10 Prerequisite LeetCode Problems

For “1976. Number of Ways to Arrive at Destination”, the following are a good preparation:

“743. Network Delay Time”: This problem introduces the concept of finding shortest paths in a weighted graph, which is an important precursor to solving the target problem.
“787. Cheapest Flights Within K Stops”: Similar to the above problem, but adds an additional constraint of maximum number of stops. This can help in understanding different scenarios in path finding problems.
“Bellman-Ford Algorithm Problems”: Problems like “1510. Stone Game IV” require understanding of this algorithm, which is often used to find shortest paths in a weighted graph.
“207. Course Schedule”: This problem is about finding a path in a directed graph, which can help understand the basics of graph traversal.
“210. Course Schedule II”: An extension to the previous problem, where we need to find an order to finish all courses.
“1135. Connecting Cities With Minimum Cost”: This problem requires finding the minimum cost to connect all cities, which is another way of saying finding the shortest paths connecting all nodes in a graph.
“1197. Minimum Knight Moves”: This problem is about finding shortest paths in an unweighted graph, a useful introduction to the concept of shortest paths.
“Bellman-Ford Algorithm Problems”: Problems like “1510. Stone Game IV” require understanding of this algorithm, which is often used to find shortest paths in a weighted graph.
“126. Word Ladder II”: This problem requires finding all shortest transformation sequences from beginWord to endWord, which is similar to the target problem but with words instead of cities.
“127. Word Ladder”: A simpler version of the above problem, but still requires finding the shortest transformation sequence.

These cover path finding algorithms and techniques, which are necessary for solving the target problem. Understanding how to traverse a graph and how to find shortest paths in it are the key skills needed for “1976. Number of Ways to Arrive at Destination”.

Problem Classification

Problem Statement:You are in a city that consists of n intersections numbered from 0 to n - 1 with bi-directional roads between some intersections. The inputs are generated such that you can reach any intersection from any other intersection and that there is at most one road between any two intersections.

You are given an integer n and a 2D integer array roads where roads[i] = [ui, vi, timei] means that there is a road between intersections ui and vi that takes timei minutes to travel. You want to know in how many ways you can travel from intersection 0 to intersection n - 1 in the shortest amount of time.

Return the number of ways you can arrive at your destination in the shortest amount of time. Since the answer may be large, return it modulo 109 + 7.

Example 1:

Input: n = 7, roads = [[0,6,7],[0,1,2],[1,2,3],[1,3,3],[6,3,3],[3,5,1],[6,5,1],[2,5,1],[0,4,5],[4,6,2]] Output: 4 Explanation: The shortest amount of time it takes to go from intersection 0 to intersection 6 is 7 minutes. The four ways to get there in 7 minutes are:

0 ➝ 6
0 ➝ 4 ➝ 6
0 ➝ 1 ➝ 2 ➝ 5 ➝ 6
0 ➝ 1 ➝ 3 ➝ 5 ➝ 6

Example 2:

Input: n = 2, roads = [[1,0,10]] Output: 1 Explanation: There is only one way to go from intersection 0 to intersection 1, and it takes 10 minutes.

Constraints:

1 <= n <= 200 n - 1 <= roads.length <= n * (n - 1) / 2 roads[i].length == 3 0 <= ui, vi <= n - 1 1 <= timei <= 109 ui != vi There is at most one road connecting any two intersections. You can reach any intersection from any other intersection.

Analyze the provided problem statement. Categorize it based on its domain, ignoring ‘How’ it might be solved. Identify and list out the ‘What’ components. Based on these, further classify the problem. Explain your categorizations.

Clarification Questions

What are the clarification questions we can ask about this problem?

Problem Analysis and Key Insights

What are the key insights from analyzing the problem statement?

Problem Boundary

What is the scope of this problem?

How to establish the boundary of this problem?

Distilling the Problem to Its Core Elements

Can you identify the fundamental concept or principle this problem is based upon? Please explain. What is the simplest way you would describe this problem to someone unfamiliar with the subject? What is the core problem we are trying to solve? Can we simplify the problem statement? Can you break down the problem into its key components? What is the minimal set of operations we need to perform to solve this problem?

Visual Model of the Problem

How to visualize the problem statement for this problem?

Problem Restatement

Could you start by paraphrasing the problem statement in your own words? Try to distill the problem into its essential elements and make sure to clarify the requirements and constraints. This exercise should aid in understanding the problem better and aligning our thought process before jumping into solving it.

Abstract Representation of the Problem

Could you help me formulate an abstract representation of this problem?

Given this problem, how can we describe it in an abstract way that emphasizes the structure and key elements, without the specific real-world details?

Terminology

Are there any specialized terms, jargon, or technical concepts that are crucial to understanding this problem or solution? Could you define them and explain their role within the context of this problem?

Problem Simplification and Explanation

Could you please break down this problem into simpler terms? What are the key concepts involved and how do they interact? Can you also provide a metaphor or analogy to help me understand the problem better?

Constraints

Given the problem statement and the constraints provided, identify specific characteristics or conditions that can be exploited to our advantage in finding an efficient solution. Look for patterns or specific numerical ranges that could be useful in manipulating or interpreting the data.

What are the key insights from analyzing the constraints?

Case Analysis

Could you please provide additional examples or test cases that cover a wider range of the input space, including edge and boundary conditions? In doing so, could you also analyze each example to highlight different aspects of the problem, key constraints and potential pitfalls, as well as the reasoning behind the expected output for each case? This should help in generating key insights about the problem and ensuring the solution is robust and handles all possible scenarios.

Provide names by categorizing these cases

What are the edge cases?

How to visualize these cases?

What are the key insights from analyzing the different cases?

Identification of Applicable Theoretical Concepts

Can you identify any mathematical or algorithmic concepts or properties that can be applied to simplify the problem or make it more manageable? Think about the nature of the operations or manipulations required by the problem statement. Are there existing theories, metrics, or methodologies in mathematics, computer science, or related fields that can be applied to calculate, measure, or perform these operations more effectively or efficiently?

Simple Explanation

Can you explain this problem in simple terms or like you would explain to a non-technical person? Imagine you’re explaining this problem to someone without a background in programming. How would you describe it? If you had to explain this problem to a child or someone who doesn’t know anything about coding, how would you do it? In layman’s terms, how would you explain the concept of this problem? Could you provide a metaphor or everyday example to explain the idea of this problem?

Problem Breakdown and Solution Methodology

Given the problem statement, can you explain in detail how you would approach solving it? Please break down the process into smaller steps, illustrating how each step contributes to the overall solution. If applicable, consider using metaphors, analogies, or visual representations to make your explanation more intuitive. After explaining the process, can you also discuss how specific operations or changes in the problem’s parameters would affect the solution? Lastly, demonstrate the workings of your approach using one or more example cases.

Inference of Problem-Solving Approach from the Problem Statement

Can you identify the key terms or concepts in this problem and explain how they inform your approach to solving it? Please list each keyword and how it guides you towards using a specific strategy or method. How can I recognize these properties by drawing tables or diagrams?

How did you infer from the problem statement that this problem can be solved using ?

Simple Explanation of the Proof

I’m having trouble understanding the proof of this algorithm. Could you explain it in a way that’s easy to understand?

Could you please provide a stepwise refinement of our approach to solving this problem?
How can we take the high-level solution approach and distill it into more granular, actionable steps?
Could you identify any parts of the problem that can be solved independently?
Are there any repeatable patterns within our solution?

Solution Approach and Analysis

Identify Invariant

What is the invariant in this problem?

Identify Loop Invariant

What is the loop invariant in this problem?

Is invariant and loop invariant the same for this problem?

Thought Process

Can you explain the basic thought process and steps involved in solving this type of problem?

Explain the thought process by thinking step by step to solve this problem from the problem statement and code the final solution. Write code in Python3. What are the cues in the problem statement? What direction does it suggest in the approach to the problem? Generate insights about the problem statement.

Establishing Preconditions and Postconditions

Parameters:
- What are the inputs to the method?
- What types are these parameters?
- What do these parameters represent in the context of the problem?
Preconditions:
- Before this method is called, what must be true about the state of the program or the values of the parameters?
- Are there any constraints on the input parameters?
- Is there a specific state that the program or some part of it must be in?
Method Functionality:
- What is this method expected to do?
- How does it interact with the inputs and the current state of the program?
Postconditions:
- After the method has been called and has returned, what is now true about the state of the program or the values of the parameters?
- What does the return value represent or indicate?
- What side effects, if any, does the method have?
Error Handling:
- How does the method respond if the preconditions are not met?
- Does it throw an exception, return a special value, or do something else?

Problem Decomposition

Problem Understanding:
- Can you explain the problem in your own words? What are the key components and requirements?
Initial Breakdown:
- Start by identifying the major parts or stages of the problem. How can you break the problem into several broad subproblems?
Subproblem Refinement:
- For each subproblem identified, ask yourself if it can be further broken down. What are the smaller tasks that need to be done to solve each subproblem?
Task Identification:
- Within these smaller tasks, are there any that are repeated or very similar? Could these be generalized into a single, reusable task?
Task Abstraction:
- For each task you’ve identified, is it abstracted enough to be clear and reusable, but still makes sense in the context of the problem?
Method Naming:
- Can you give each task a simple, descriptive name that makes its purpose clear?
Subproblem Interactions:
- How do these subproblems or tasks interact with each other? In what order do they need to be performed? Are there any dependencies?

From Brute Force to Optimal Solution

Could you please begin by illustrating a brute force solution for this problem? After detailing and discussing the inefficiencies of the brute force approach, could you then guide us through the process of optimizing this solution? Please explain each step towards optimization, discussing the reasoning behind each decision made, and how it improves upon the previous solution. Also, could you show how these optimizations impact the time and space complexity of our solution?

Code Explanation and Design Decisions

Identify the initial parameters and explain their significance in the context of the problem statement or the solution domain.
Discuss the primary loop or iteration over the input data. What does each iteration represent in terms of the problem you’re trying to solve? How does the iteration advance or contribute to the solution?
If there are conditions or branches within the loop, what do these conditions signify? Explain the logical reasoning behind the branching in the context of the problem’s constraints or requirements.
If there are updates or modifications to parameters within the loop, clarify why these changes are necessary. How do these modifications reflect changes in the state of the solution or the constraints of the problem?
Describe any invariant that’s maintained throughout the code, and explain how it helps meet the problem’s constraints or objectives.
Discuss the significance of the final output in relation to the problem statement or solution domain. What does it represent and how does it satisfy the problem’s requirements?

Remember, the focus here is not to explain what the code does on a syntactic level, but to communicate the intent and rationale behind the code in the context of the problem being solved.

Coding Constructs

Consider the code for the solution of this problem.

What are the high-level problem-solving strategies or techniques being used by this code?
If you had to explain the purpose of this code to a non-programmer, what would you say?
Can you identify the logical elements or constructs used in this code, independent of any programming language?
Could you describe the algorithmic approach used by this code in plain English?
What are the key steps or operations this code is performing on the input data, and why?
Can you identify the algorithmic patterns or strategies used by this code, irrespective of the specific programming language syntax?

Language Agnostic Coding Drills

Your mission is to deconstruct this code into the smallest possible learning units, each corresponding to a separate coding concept. Consider these concepts as unique coding drills that can be individually implemented and later assembled into the final solution.

Dissect the code and identify each distinct concept it contains. Remember, this process should be language-agnostic and generally applicable to most modern programming languages.
Once you’ve identified these coding concepts or drills, list them out in order of increasing difficulty. Provide a brief description of each concept and why it is classified at its particular difficulty level.
Next, describe the problem-solving approach that would lead from the problem statement to the final solution. Think about how each of these coding drills contributes to the overall solution. Elucidate the step-by-step process involved in using these drills to solve the problem. Please refrain from writing any actual code; we’re focusing on understanding the process and strategy.

Targeted Drills in Python

Now that you’ve identified and ordered the coding concepts from a complex software code in the previous exercise, let’s focus on creating Python-based coding drills for each of those concepts.

Begin by writing a separate piece of Python code that encapsulates each identified concept. These individual drills should illustrate how to implement each concept in Python. Please ensure that these are suitable even for those with a basic understanding of Python.
In addition to the general concepts, identify and write coding drills for any problem-specific concepts that might be needed to create a solution. Describe why these drills are essential for our problem.
Once all drills have been coded, describe how these pieces can be integrated together in the right order to solve the initial problem. Each drill should contribute to building up to the final solution.

Remember, the goal is to not only to write these drills but also to ensure that they can be cohesively assembled into one comprehensive solution.