# Path with smallest product of edges with weight > 0

Given a directed graph with N nodes and E edges where the weight of each edge is > 0, also given a source S and a destination D. The task is to find the path with minimum product of edges from S to D. If there is no path from S to D, then print -1.

Examples:

Input: N = 3, E = 3, Edges = {{{1, 2}, 0.5}, {{1, 3}, 1.9}, {{2, 3}, 3}}, S = 1, and D = 3
Output: 1.5
Explanation:
The shortest path will be 1->2->3
with value 0.5*3 = 1.5

Input: N = 3, E = 3, Edges = {{{1, 2}, 0.5}, {{2, 3}, 0.5}, {{3, 1}, 0.5}}, S = 1, and D = 3
Output: cycle detected

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Approach: The idea is to use bellman ford algorithm. It is because Dijkstra’s algorithm cannot be used here as it works only with non-negative edges. It is because while multiplying values between [0-1), the product keeps decreasing indefinitely and 0 is returned finally.

Moreover, cycles need to be detected because if a cycle exists, the product of this cycle will indefinitely decrease the product to 0 and the product will tend to 0. For, simplicity, we will report such cycles.

The following steps can be followed to compute the result:

1. Initialise an array, dis[] with initial value as ‘inf’ except dis[S] as 1.
2. Run a loop from 1 – N-1. For each edge in the graph:
• dis[edge.second] = min(dis[edge.second], dis[edge.first]*weight(edge))
3. Run another loop for each edge in the graph, if any edge exits with (dis[edge.second] > dis[edge.first]*weight(edge)), then cycle is detected.
4. If dist[d] in infinity, return -1, else return dist[d].

Below is the implementation of the above approach:

## C++

 `// C++ implementation of the approach. ` `#include ` `using` `namespace` `std; ` ` `  `double` `inf = std:: ` `    ``numeric_limits<``double``>::infinity(); ` ` `  `// Function to return the smallest  ` `// product of edges ` `double` `bellman(``int` `s, ``int` `d, ` `               ``vector, ` `                           ``double``> > ` `                   ``ed, ` `               ``int` `n) ` `{ ` `    ``// If the source is equal ` `    ``// to the destination ` `    ``if` `(s == d) ` `        ``return` `0; ` ` `  `    ``// Array to store distances ` `    ``double` `dis[n + 1]; ` ` `  `    ``// Initialising the array ` `    ``for` `(``int` `i = 1; i <= n; i++) ` `        ``dis[i] = inf; ` `    ``dis[s] = 1; ` ` `  `    ``// Bellman ford algorithm ` `    ``for` `(``int` `i = 0; i < n - 1; i++) ` `        ``for` `(``auto` `it : ed) ` `            ``dis[it.first.second] = min(dis[it.first.second], ` `                                       ``dis[it.first.first] ` `                                           ``* it.second); ` ` `  `    ``// Loop to detect cycle ` `    ``for` `(``auto` `it : ed) { ` `        ``if` `(dis[it.first.second] ` `            ``> dis[it.first.first] * it.second) ` `            ``return` `-2; ` `    ``} ` ` `  `    ``// Returning final answer ` `    ``if` `(dis[d] == inf) ` `        ``return` `-1; ` `    ``else` `        ``return` `dis[d]; ` `} ` ` `  `// Driver code ` `int` `main() ` `{ ` ` `  `    ``int` `n = 3; ` `    ``vector, ``double``> > ed; ` ` `  `    ``// Input edges ` `    ``ed = { { { 1, 2 }, 0.5 }, ` `           ``{ { 1, 3 }, 1.9 }, ` `           ``{ { 2, 3 }, 3 } }; ` ` `  `    ``// Source and Destination ` `    ``int` `s = 1, d = 3; ` ` `  `    ``// Bellman ford ` `    ``double` `get = bellman(s, d, ed, n); ` ` `  `    ``if` `(get == -2) ` `        ``cout << ``"Cycle Detected"``; ` `    ``else` `        ``cout << get; ` `} `

## Python3

 `# Python3 implementation of the approach.  ` `import` `sys ` ` `  `inf ``=` `sys.maxsize; ` ` `  `# Function to return the smallest  ` `# product of edges  ` `def` `bellman(s, d, ed, n) :  ` ` `  `    ``# If the source is equal  ` `    ``# to the destination  ` `    ``if` `(s ``=``=` `d) : ` `        ``return` `0``;  ` ` `  `    ``# Array to store distances  ` `    ``dis ``=` `[``0``]``*``(n ``+` `1``);  ` ` `  `    ``# Initialising the array  ` `    ``for` `i ``in` `range``(``1``, n ``+` `1``) : ` `        ``dis[i] ``=` `inf;  ` `         `  `    ``dis[s] ``=` `1``;  ` ` `  `    ``# Bellman ford algorithm  ` `    ``for` `i ``in` `range``(n ``-` `1``) :  ` `        ``for` `it ``in` `ed :  ` `            ``dis[it[``1``]] ``=` `min``(dis[it[``1``]], dis[it[``0``]] ``*` `ed[it]);  ` ` `  `    ``# Loop to detect cycle  ` `    ``for` `it ``in` `ed : ` `        ``if` `(dis[it[``1``]] > dis[it[``0``]] ``*` `ed[it]) : ` `            ``return` `-``2``;  ` ` `  `    ``# Returning final answer  ` `    ``if` `(dis[d] ``=``=` `inf) : ` `        ``return` `-``1``;  ` `    ``else` `: ` `        ``return` `dis[d];  ` ` `  `# Driver code  ` `if` `__name__ ``=``=` `"__main__"` `:  ` ` `  `    ``n ``=` `3``; ` `     `  `    ``# Input edges  ` `    ``ed ``=` `{ ( ``1``, ``2` `) : ``0.5` `,  ` `        ``( ``1``, ``3` `) : ``1.9` `,  ` `        ``( ``2``, ``3` `) : ``3` `};  ` ` `  `    ``# Source and Destination  ` `    ``s ``=` `1``; d ``=` `3``;  ` ` `  `    ``# Bellman ford  ` `    ``get ``=` `bellman(s, d, ed, n);  ` ` `  `    ``if` `(get ``=``=` `-``2``) : ` `        ``print``(``"Cycle Detected"``);  ` `    ``else` `: ` `        ``print``(get);  ` ` `  `# This code is contributed by AnkitRai01 `

Output:

```1.5
```

Time complexity: O(E*V)

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