I have been working on this problem for quite a few days at this point and am making no progress. I haven’t been able to make any progress and would be the happiest man alive if I was given some advice on how to move forward or even the solution to this problem.

I have given the problem here and my current attempt at a solution below:

Problem – Here is an incorrect implementation of Floyd-Warshall.

floyd_warshall(dist, n):

Assume dist[i][j] is positive infinity if there is no edge between them

for i ranging from 1 to n:
for j ranging from 1 to n:
for k ranging from 1 to n:
dist[i][j] = min(dist[i][j], dist[i][k] + dist[k][j])
Here is an attempt at patching it.

floyd_warshall_patch1(dist, n, k):

dist[i][i] is zero

dist[i][j] is otherwise the weighted of the directed edge from i to j if it exists

dist[i][j] is otherwise positive infinity

for i ranging from 1 to k:
floyd_warshall(dist, n)
Here is another attempt at patching it.

floyd_warshall_patch2(dist, n)

dist[i][i] is zero

dist[i][j] is otherwise the weighted of the directed edge from i to j if it exists

dist[i][j] is otherwise positive infinity

for i ranging from 1 to n:
for j ranging from 1 to n:
for k ranging from 1 to n:
dist[j][k] = min(dist[j][k], dist[j][i] + dist[i][k])
Your job is to construct a directed graph with N vertices and M edges such that, given a parameter K, the output of floyd_warshall_patch1 when given matches the output of floyd_warshall_patch2 on the given graph, but the output of floyd_warshall_patch1 when given K−1 does not.

Additional Constraints -(https://i.sstatic.net/gw1gVpkI.png)

My solution – def floyd_warshall_patch1(dist, n, k):
for i in range(k):
dist[i][i] = 0
for k in range(n):
for i in range(n):
for j in range(n):
dist[i][j] = min(dist[i][j], dist[i][k] + dist[k][j])

def floyd_warshall_patch2(dist, n):
for i in range(n):
dist[i][i] = 0
for k in range(n):
for i in range(n):
for j in range(n):
dist[i][j] = min(dist[i][j], dist[i][k] + dist[k][j])

def create_graph(N, M, K):

if K >= N or K == N - 1 or M < N - 1:
    return "NO", []

edges = []
# Create a path from 1 to N
for i in range(1, N):
    edges.append((i, i + 1, 1))

# Add a back edge to form a cycle
if M > N - 1:
    edges.append((N, 1, 10))  # Adding a cycle with a reasonable weight

# Optionally add more complexity with additional edges
additional_edges = M - len(edges)
while additional_edges > 0:
    from_node = additional_edges % N + 1
    to_node = (additional_edges * 2) % N + 1
    if from_node != to_node:
        edges.append((from_node, to_node, additional_edges + 5))  # Randomized weight for complexity
    additional_edges -= 1

return "YES", edges

N, M, K = map(int, input().split())

result, edges = create_graph(N, M, K)

print(result)
if result == “YES”:
for edge in edges:
print(” “.join(map(str, edge)))

I just have no idea what to do.