**π‘ Problem Formulation:** Imagine you are given a set of note exchange rules structured as pairs of (note_i, note_j, time) meaning you can exchange note_i for note_j after a certain time has elapsed. The goal is to find the minimum time required to exchange a specific note for another if possible. For example, given input rules [(1, 2, 5), (2, 3, 5)] and a request to exchange note 1 for note 3, the desired output is 10, representing the minimum cumulative time for both exchanges.

## Method 1: Using Floyd-Warshall Algorithm

The Floyd-Warshall algorithm is a classic way to find the shortest paths in a weighted graph with positive or negative edge weights (but with no negative cycles). In the context of exchanging notes, think of each note as a vertex in the graph, and the time to exchange as the edge weight. The algorithm will compute the shortest time to exchange each note with every other note.

Here’s an example:

def floyd_warshall(notes, times): # Initialize distance table dist = {n: {m: float('inf') for m in notes} for n in notes} for n in dist: dist[n][n] = 0 for n1, n2, t in times: dist[n1][n2] = t # Floyd-Warshall algorithm for k in notes: for i in notes: for j in notes: dist[i][j] = min(dist[i][j], dist[i][k] + dist[k][j]) return dist notes = [1, 2, 3] times = [(1, 2, 5), (2, 3, 5)] print(floyd_warshall(notes, times)[1][3])

The output of this code snippet:

`10`

This snippet defines a function `floyd_warshall()`

that takes a list of notes and a list of tuples representing time to exchange. It initializes a distance table, updates direct exchanges, and applies the Floyd-Warshall algorithm to find the shortest exchange pathways. The final line prints out the minimum time to exchange note 1 for note 3, which is 10.

## Method 2: Dijkstra’s Algorithm

Dijkstra’s algorithm is perfect for finding the shortest path from a single source to all other vertices in a graph with non-negative edge weights. In our case, we treat each note as a vertex and the time required for an exchange as the weight of that edge. Starting from the source note, the algorithm keeps track of the minimum time to reach each note until it processes all reachable notes.

Here’s an example:

import heapq def dijkstra(notes, source, target, times): # Initialize distances and priority queue distances = {note: float('inf') for note in notes} distances[source] = 0 queue = [(0, source)] # Process the queue while queue: current_time, current_note = heapq.heappop(queue) if current_note == target: return current_time for n1, n2, t in filter(lambda x: x[0] == current_note, times): next_time = current_time + t if next_time < distances[n2]: distances[n2] = next_time heapq.heappush(queue, (next_time, n2)) return distances[target] notes = [1, 2, 3] source = 1 target = 3 times = [(1, 2, 5), (2, 3, 5)] print(dijkstra(notes, source, target, times))

The output of this code snippet:

`10`

In this example, the function `dijkstra()`

is used to find the minimum time between a source and target note. It uses a priority queue to process each note with the lowest recorded time first. This ensures efficiency and that it will find the minimum time required to exchange from note 1 to note 3, which is shown when we print out the result, yielding a value of 10.

## Method 3: Bellman-Ford Algorithm

The Bellman-Ford algorithm is another single-source shortest path algorithm that is capable of handling graphs with negative weight edges without cycles. This algorithm iterates over all edges and relaxes the cost to reach all vertices. For note exchange with only positive times, it can also be used efficiently.

Here’s an example:

def bellman_ford(notes, source, times): # Initialize distances distance = {note: float('inf') for note in notes} distance[source] = 0 # Relax edges repeatedly for _ in range(len(notes) - 1): for n1, n2, t in times: if distance[n1] != float('inf') and distance[n1] + t < distance[n2]: distance[n2] = distance[n1] + t return distance notes = [1, 2, 3] source = 1 times = [(1, 2, 5), (2, 3, 5)] print(bellman_ford(notes, source, times)[3])

The output of this code snippet:

`10`

The provided example defines a `bellman_ford()`

function which computes the shortest time to exchange each note pair starting from a given source note. After the relaxation steps are done, the minimum time to exchange notes 1 for 3 is found and displayed, which is 10 in this case.

## Method 4: Custom Recursive Searching

A custom recursive search can be implemented to explore all possible exchanges in a depth-first manner. This method can be useful if we’re not dealing with a large number of nodes and the graph is sparsely connected.

Here’s an example:

def recursive_search(source, target, times, visited=None, time=0): if visited is None: visited = set() if source == target: return time visited.add(source) min_time = float('inf') for n1, n2, t in filter(lambda x: x[0] == source and x[1] not in visited, times): next_time = recursive_search(n2, target, times, visited, time + t) if next_time: min_time = min(min_time, next_time) visited.remove(source) return min_time if min_time != float('inf') else None notes = [1, 2, 3] source = 1 target = 3 times = [(1, 2, 5), (2, 3, 5)] print(recursive_search(source, target, times))

The output of this code snippet:

`10`

This example showcases a custom depth-first search approach where a function `recursive_search()`

is used to explore all possible note exchanges. It tries each possible path, adding the time elapsed, and tracks the smallest time taken to reach the target note. The result is the minimum exchange time from note 1 to note 3, which is also 10 here.

## Bonus One-Liner Method 5: Leveraging NetworkX Library

NetworkX is a Python library designed to handle complex networks with relative ease. Using its built-in algorithms, we can quickly find the shortest path and minimum exchange time in our note exchange system.

Here’s an example:

import networkx as nx def networkx_shortest_path(notes, times, source, target): G = nx.DiGraph() G.add_weighted_edges_from(times) return nx.dijkstra_path_length(G, source, target) notes = [1, 2, 3] times = [(1, 2, 5), (2, 3, 5)] print(networkx_shortest_path(notes, times, 1, 3))

The output of this code snippet:

`10`

This code snippet demonstrates the use of the `networkx`

library. By creating a directed graph and adding the given exchange times as weighted edges, the function `networkx_shortest_path()`

uses Dijkstra’s algorithm to find the shortest path. The minimum time to exchange note 1 for note 3 is quickly found to be 10.

## Summary/Discussion

**Method 1:**Floyd-Warshall Algorithm. This method finds shortest paths between all pairs of notes, making it great for repeated queries. It has a relatively high time complexity, which can be a drawback for very large sets of notes.**Method 2:**Dijkstra’s Algorithm. Efficiently finds the shortest path from a single source to a target note. It has better performance on sparse and non-negative weighted graphs, but it’s not suitable for negative weight cycles.**Method 3:**Bellman-Ford Algorithm. Handles negative weight edges and can detect negative cycles, though it has a slower runtime compared to Dijkstra’s for graphs without negative weights.**Method 4:**Custom Recursive Searching. This method allows customization and is easy to understand. It’s not as efficient on dense graphs or with a large number of notes.**Bonus Method 5:**Leveraging NetworkX Library. A high-level, quick method suitable for straightforward use-cases. It abstracts away the underlying complexity but adds a dependency to the project.