# Program to find out the path between two vertices in a graph that has the minimum penalty (Python)

PythonServer Side ProgrammingProgramming

Suppose we are given an undirected, weighted graph and are asked to find out the path with the minimum possible penalty from node a to node b. The penalty of a path is the bitwise OR of the weights of all the edges in the path. So, we must find out such a 'minimum penalty' path, and if there exists no path between the two nodes, we return -1.

So, if the input is like start (s) = 1, end (e) = 3; then the output will be 15.

There exist two paths between vertices 1 and 3. The optimal path is 1->2->3, the cost of the path is (10 OR 5) = 15.

To solve this, we will follow these steps −

• Define a function helper() . This will take G, s, e
• v := a new set
• c := a new list of size n initialized with value infinity
• heap := a new heap containing pair (0, s)
• while size of heap > 0, do
• cst := pop the smallest item from heap
• cur := pop the smallest item from heap
• c[cur] := minimum of (cst, c[cur])
• if (cst, cur) is present in v, then
• go for the next iteration
• if cur is same as e, then
• return c[cur]
• add pair (cst, cur) to v
• for each neighbor, n_cost in G[cur], do
• push values ((n_cost OR cst), neighbor) to heap
• return c[e]
• G := [a new list containing n + 1 emoty lists]
• for each item in edges, do
• u := item
• v := item
• w := item
• insert pair (v, w) at the end of G[u]
• insert pair (u, w) at the end of G[v]
• ans := helper(G, s, e)
• return -1 if ans is same as inf otherwise return ans

## Example

Let us see the following implementation to get better understanding −

import heapq
from math import inf

def helper(G, s, e):
v = set()
c = [inf] * len(G)
heap = [(0, s)]
while len(heap) > 0:
cst, cur = heapq.heappop(heap)
c[cur] = min(cst, c[cur])
if (cst, cur) in v:
continue
if cur == e:
return c[cur]
for neighbor, n_cost in G[cur]:
heapq.heappush(heap, (n_cost | cst, neighbor))
return c[e]

def solve(n, edges, s, e):
G = [[] for _ in range(n + 1)]
for item in edges:
u, v, w = map(int, item)
G[u].append((v, w))
G[v].append((u, w))
ans = helper(G, s, e)
return -1 if ans == inf else ans

print(solve(4, [(1, 2, 10), (2, 3, 5), (2, 4, 15), (1, 4, 20)], 1, 3))

## Input

4, [(1, 2, 10), (2, 3, 5), (2, 4, 15), (1, 4, 20)], 1, 3

## Output

15