Find Shortest Path with K Hops

The problem is about finding the shortest path from source node ( s ) to destination node ( d ) in a graph, where you can set the weight of at most ( k ) edges to 0. Here’s an approach to solve the problem.

Approach

Build the Graph: First, create an adjacency list to represent the graph.
Apply Dijkstra’s Algorithm with Modification: Normally, Dijkstra’s algorithm helps us find the shortest path in a weighted graph. However, in this problem, we have the ability to make ( k ) edges weight 0, which requires a slight modification to the algorithm.
Multi-State Dijkstra: Use a 2D distance array distance[n][k+1], where distance[i][j] represents the minimum distance from source s to node i, using exactly j edge weight reductions.
Initialization: Initialize distance[s][0] = 0 and distance[i][j] = INF for other nodes.
Priority Queue: Use a priority queue (min-heap) to efficiently find the node with the minimum distance. Each element in the queue will be a tuple (dist, node, hops), where dist is the distance from source, node is the current node, and hops is the number of edge weight reductions used.
Processing: For each edge (neighbor, weight) of the current node, try to relax the edge by considering each possible number of edge weight reductions (from 0 to ( k )) and update the distance array accordingly.
Return the Result: The answer will be the minimum value in distance[d][0...k].

Code

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import heapq

class Solution:
    def shortestPathWithHops(self, n: int, edges: List[List[int]], s: int, d: int, k: int) -> int:
        # Building the graph
        graph = [[] for _ in range(n)]
        for u, v, w in edges:
            graph[u].append((v, w))
            graph[v].append((u, w)) # Since it's an undirected graph

        # Initialize distance array
        distance = [[float('inf')] * (k + 1) for _ in range(n)]
        distance[s][0] = 0

        # Priority queue for Dijkstra
        pq = [(0, s, 0)] # (dist, node, hops)

        while pq:
            dist, node, hops = heapq.heappop(pq)
            # If it's destination and hops are <= k, then return the distance
            if node == d and hops <= k:
                return dist

            # If the state has already been visited, continue
            if dist > distance[node][hops]:
                continue

            # Explore neighbors
            for neighbor, weight in graph[node]:
                for h in range(hops, k + 1):
                    # If hops are used, the weight becomes 0
                    new_dist = dist + (0 if h > hops else weight)
                    # If a better path is found, update distance and push to queue
                    if new_dist < distance[neighbor][h]:
                        distance[neighbor][h] = new_dist
                        heapq.heappush(pq, (new_dist, neighbor, h))

        return -1 # Return -1 if no path found

Complexity

Time Complexity: (O(E \cdot K \log N)), where (E) is the number of edges, (K) is the allowed edge weight reductions, and (N) is the number of nodes.
Space Complexity: (O(N \cdot K)), for storing the distance array.

Key Takeaways

Utilize the classic Dijkstra’s algorithm but with a multi-state approach to handle the unique condition of changing edge weights.
Use a 2D distance array to keep track of the different states.

Identifying Problem Isomorphism

“Find Shortest Path with K Hops” can be approximately mapped to “Cheapest Flights Within K Stops”.

In both problems, we are dealing with a graph where nodes are connected by weighted edges, and we need to find a path between two given nodes.

In “Find Shortest Path with K Hops”, we need to find the shortest path from ’s’ to ’d’ with at most ‘k’ hops (where a hop is defined as making the weight of an edge 0). The problem involves selecting which edges to “hop” over to minimize the overall path length.

In “Cheapest Flights Within K Stops”, the objective is to find the cheapest price to get from ‘src’ to ‘dst’ with at most ‘K’ stops. Here, a stop is akin to a hop in the previous problem.

However, while both problems require finding an optimal path in a graph with a restriction on the number of steps (hops or stops), they differ in how they define a step and the optimization strategy. In “Find Shortest Path with K Hops”, we change the weight of the edge to 0 for a hop, whereas in “Cheapest Flights Within K Stops”, we just count the number of stops without changing the edge weight.

These problems share enough similarities to be approximately isomorphic. The “Cheapest Flights Within K Stops” is simpler as it doesn’t involve changing the edge weights.

10 Prerequisite LeetCode Problems

“2714. Find Shortest Path with K Hops” involves graph theory, shortest path finding (Dijkstra’s algorithm for example), and manipulation of edge weights. Here are some problems to prepare for this:

743. Network Delay Time: This problem involves finding shortest paths in a network, which is directly applicable to the given problem.
787. Cheapest Flights Within K Stops: Similar to the given problem but with an additional constraint on the number of stops.
1514. Path with Maximum Probability: This problem also involves finding paths in a graph, but with an additional twist.
1202. Smallest String With Swaps: This problem can be seen as a graph problem, where each connected component in the graph can be independently sorted.
133. Clone Graph: This problem requires a good understanding of graph traversal algorithms.
399. Evaluate Division: This problem can be solved with graph theory, where each equation is an edge in the graph.
207. Course Schedule: This problem is a classic graph theory problem that involves finding cycles in a graph.
210. Course Schedule II: An extension of the Course Schedule problem, but this time we’re also asked to return a valid ordering of courses.
684. Redundant Connection: It is another problem about manipulating and understanding graphs.
1135. Connecting Cities With Minimum Cost: It is a problem about constructing a minimal spanning tree (MST) in a graph, which could be helpful to understand different properties and algorithms related to graphs.

It’s also important to have a good grasp of graph theory and related algorithms.

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?

Problem Classification

Problem Statement:

You are given a positive integer n which is the number of nodes of a 0-indexed undirected weighted connected graph and a 0-indexed 2D array edges where edges[i] = [ui, vi, wi] indicates that there is an edge between nodes ui and vi with weight wi.

You are also given two nodes s and d, and a positive integer k, your task is to find the shortest path from s to d, but you can hop over at most k edges. In other words, make the weight of at most k edges 0 and then find the shortest path from s to d.

Return the length of the shortest path from s to d with the given condition.

Example 1:

Input: n = 4, edges = [[0,1,4],[0,2,2],[2,3,6]], s = 1, d = 3, k = 2 Output: 2 Explanation: In this example there is only one path from node 1 (the green node) to node 3 (the red node), which is (1->0->2->3) and the length of it is 4 + 2 + 6 = 12. Now we can make weight of two edges 0, we make weight of the blue edges 0, then we have 0 + 2 + 0 = 2. It can be shown that 2 is the minimum length of a path we can achieve with the given condition.

Example 2:

Input: n = 7, edges = [[3,1,9],[3,2,4],[4,0,9],[0,5,6],[3,6,2],[6,0,4],[1,2,4]], s = 4, d = 1, k = 2 Output: 6 Explanation: In this example there are 2 paths from node 4 (the green node) to node 1 (the red node), which are (4->0->6->3->2->1) and (4->0->6->3->1). The first one has the length 9 + 4 + 2 + 4 + 4 = 23, and the second one has the length 9 + 4 + 2 + 9 = 24. Now if we make weight of the blue edges 0, we get the shortest path with the length 0 + 4 + 2 + 0 = 6. It can be shown that 6 is the minimum length of a path we can achieve with the given condition.

Example 3:

Input: n = 5, edges = [[0,4,2],[0,1,3],[0,2,1],[2,1,4],[1,3,4],[3,4,7]], s = 2, d = 3, k = 1 Output: 3 Explanation: In this example there are 4 paths from node 2 (the green node) to node 3 (the red node), which are (2->1->3), (2->0->1->3), (2->1->0->4->3) and (2->0->4->3). The first two have the length 4 + 4 = 1 + 3 + 4 = 8, the third one has the length 4 + 3 + 2 + 7 = 16 and the last one has the length 1 + 2 + 7 = 10. Now if we make weight of the blue edge 0, we get the shortest path with the length 1 + 2 + 0 = 3. It can be shown that 3 is the minimum length of a path we can achieve with the given condition.

Constraints:

2 <= n <= 500 n - 1 <= edges.length <= min(104, n * (n - 1) / 2) edges[i].length = 3 0 <= edges[i][0], edges[i][1] <= n - 1 1 <= edges[i][2] <= 106 0 <= s, d, k <= n - 1 s != d The input is generated such that the graph is connected and has no repeated edges or self-loops

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.

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?

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 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?

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 following piece of complex software code.

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.