# Path with minimum XOR sum of edges in a directed graph

Given a directed graph with N nodes and E edges, a source S and a destination D nodes. The task is to find the path with the minimum XOR sum 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}, 5}, {{1, 3}, 9}, {{2, 3}, 1}}, S = 1, and D = 3
Output: 4
The path with smallest XOR of edges weight will be 1->2->3
with XOR sum as 5^1 = 4.

Input: N = 3, E = 3, Edges = {{{3, 2}, 5}, {{3, 3}, 9}, {{3, 3}, 1}}, S = 1, and D = 3
Output: -1

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

Approach: The idea is to use Dijkstra’s shortest path algorithm with a slight variation. Below is the step-wise approach for the problem:

• Base Case: If the source node is equal to the destination then return 0.
• Initialise a priority-queue with source node and its weight as 0 and a visited array.
• While priority queue is not empty:
1. Pop the top-most element from priority queue. Let’s call it as current node.
2. Check if the current node is already visited with the help of the visited array, If yes then continue.
3. If the current node is the destination node then return the XOR sum distance of the current node from the source node.
4. Iterate all the nodes adjacent to current node and push into priority queue and their distance as XOR sum with the current distance and edge weight.
• Otherwise there is no path from source to destination. Therefore, return -1

Below is the implementation of the above approach:

## C++

 `// C++ implementation of the approach ` ` `  `#include ` `using` `namespace` `std; ` ` `  `// Function to return the smallest ` `// xor sum of edges ` `double` `minXorSumOfEdges( ` `    ``int` `s, ``int` `d, ` `    ``vector > > gr) ` `{ ` `    ``// If the source is equal ` `    ``// to the destination ` `    ``if` `(s == d) ` `        ``return` `0; ` ` `  `    ``// Initialise the priority queue ` `    ``set > pq; ` `    ``pq.insert({ 0, s }); ` ` `  `    ``// Visited array ` `    ``bool` `v[gr.size()] = { 0 }; ` ` `  `    ``// While the priority-queue ` `    ``// is not empty ` `    ``while` `(pq.size()) { ` ` `  `        ``// Current node ` `        ``int` `curr = pq.begin()->second; ` ` `  `        ``// Current xor sum of distance ` `        ``int` `dist = pq.begin()->first; ` ` `  `        ``// Popping the top-most element ` `        ``pq.erase(pq.begin()); ` ` `  `        ``// If already visited continue ` `        ``if` `(v[curr]) ` `            ``continue``; ` ` `  `        ``// Marking the node as visited ` `        ``v[curr] = 1; ` ` `  `        ``// If it is a destination node ` `        ``if` `(curr == d) ` `            ``return` `dist; ` ` `  `        ``// Traversing the current node ` `        ``for` `(``auto` `it : gr[curr]) ` `            ``pq.insert({ dist ^ it.second, ` `                        ``it.first }); ` `    ``} ` ` `  `    ``// If no path exists ` `    ``return` `-1; ` `} ` ` `  `// Driver code ` `int` `main() ` `{ ` `    ``int` `n = 3; ` ` `  `    ``// Graph as adjacency matrix ` `    ``vector > > ` `        ``gr(n + 1); ` ` `  `    ``// Input edges ` `    ``gr.push_back({ 3, 9 }); ` `    ``gr.push_back({ 3, 1 }); ` `    ``gr.push_back({ 2, 5 }); ` ` `  `    ``// Source and destination ` `    ``int` `s = 1, d = 3; ` ` `  `    ``cout << minXorSumOfEdges(s, d, gr); ` ` `  `    ``return` `0; ` `} `

## Python3

 `# Python3 implementation of the approach ` `from` `collections ``import` `deque ` ` `  `# Function to return the smallest ` `# xor sum of edges ` `def` `minXorSumOfEdges(s, d, gr): ` `     `  `    ``# If the source is equal ` `    ``# to the destination ` `    ``if` `(s ``=``=` `d): ` `        ``return` `0` ` `  `    ``# Initialise the priority queue ` `    ``pq ``=` `[] ` `    ``pq.append((``0``, s)) ` ` `  `    ``# Visited array ` `    ``v ``=` `[``0``] ``*` `len``(gr) ` ` `  `    ``# While the priority-queue ` `    ``# is not empty ` `    ``while` `(``len``(pq) > ``0``): ` `        ``pq ``=` `sorted``(pq) ` ` `  `        ``# Current node ` `        ``curr ``=` `pq[``0``][``1``] ` ` `  `        ``# Current xor sum of distance ` `        ``dist ``=` `pq[``0``][``0``] ` ` `  `        ``# Popping the top-most element ` `        ``del` `pq[``0``] ` ` `  `        ``# If already visited continue ` `        ``if` `(v[curr]): ` `            ``continue` ` `  `        ``# Marking the node as visited ` `        ``v[curr] ``=` `1` ` `  `        ``# If it is a destination node ` `        ``if` `(curr ``=``=` `d): ` `            ``return` `dist ` ` `  `        ``# Traversing the current node ` `        ``for` `it ``in` `gr[curr]: ` `            ``pq.append((dist ^ it[``1``], ` `                              ``it[``0``])) ` `    ``# If no path exists ` `    ``return` `-``1` ` `  `# Driver code ` `if` `__name__ ``=``=` `'__main__'``: ` `     `  `    ``n ``=` `3` `     `  `    ``# Graph as adjacency matrix ` `    ``gr ``=` `[[] ``for` `i ``in` `range``(n ``+` `1``)] ` ` `  `    ``# Input edges ` `    ``gr[``1``].append([ ``3``, ``9` `]) ` `    ``gr[``2``].append([ ``3``, ``1` `]) ` `    ``gr[``1``].append([ ``2``, ``5` `]) ` ` `  `    ``# Source and destination ` `    ``s ``=` `1` `    ``d ``=` `3` ` `  `    ``print``(minXorSumOfEdges(s, d, gr)) ` `     `  `# This code is contributed by mohit kumar 29 `

Output:

```4
```

Time complexity: O((E + V) logV) My Personal Notes arrow_drop_up Check out this Author's contributed articles.

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Improved By : mohit kumar 29