Friday, October 21, 2016

Longest path between any pair of vertices - GeeksforGeeks

Longest path between any pair of vertices - GeeksforGeeks
We are given a map of cities connected with each other via cable lines such that there is no cycle between any two cities. We need to find the maximum length of cable between any two cities for given city map.

```Input : n = 6
1 2 3  // Cable length from 1 to 2 (or 2 to 1) is 3
2 3 4
2 6 2
6 4 6
6 5 5
Output: maximum length of cable = 12```

We can solve above problem in O(V+E) time if the given graph is directed instead of undirected graph. Below are steps.
1) Create a distance array dist[] and initialize all entries of it as minus infinite
2) Order all vertices in toplogical order.
3) Do following for every vertex u in topological order.
…..Do following for every adjacent vertex v of u
………..if (dist[v] < dist[u] + weight(u, v))
……………dist[v] = dist[u] + weight(u, v)
4) Return maximum value from dist[]

Since there is no negative weight, processing vertices in topological order would always produce an array of longest paths dist[] such that dist[u] indicates longest path ending at vertex ‘u’.
The implementation of above approach can be easily adopted from here. The differences here are, there are no negative weight edges and we need overall longest path (not longest paths from a source vertex). Finally we return maximum of all values in dist[].
Time Complexity : O(V + E)

We create undirected graph for given city map and do DFS from every city to find maximum length of cable. While traversing, we look for total cable length to reach the current city and if it’s adjacent city is not visited then call DFS for it but if all adjacent cities are visited for current node, then update the value of max_length if previous value of max_length is less than current value of total cable length.
Time Complexity : O(V * (V + E))
`// visited[] array to make nodes visited`
`// src is starting node for DFS traversal`
`// prev_len is sum of cable length till current node`
`// max_len is pointer which stores the maximum length`
`// of cable value after DFS traversal`
`void` `DFS(vector< pair<``int``,``int``> > graph[], ``int` `src,`
`         ``int` `prev_len, ``int` `*max_len,`
`         ``vector <``bool``> &visited)`
`{`
`    ``// Mark the src node visited`
`    ``visited[src] = 1;`

`    ``// curr_len is for length of cable from src`
`    ``// city to its adjacent city`
`    ``int` `curr_len = 0;`

`    ``// Adjacent is pair type which stores`
`    ``// destination city and cable length`
`    ``pair < ``int``, ``int` `> adjacent;`

`    ``// Traverse all adjacent`
`    ``for` `(``int` `i=0; i<graph[src].size(); i++)`
`    ``{`
`        ``// Adjacent element`
`        ``adjacent = graph[src][i];`

`        ``// If node or city is not visited`
`        ``if` `(!visited[adjacent.first])`
`        ``{`
`            ``// Total length of cable from src city`
`            ``// to its adjacent`
`            ``curr_len = prev_len + adjacent.second;`

`            ``// Call DFS for adjacent city`
`            ``DFS(graph, adjacent.first, curr_len,`
`                ``max_len,  visited);`
`        ``}`

`        ``// If total cable length till now greater than`
`        ``// previous length then update it`
`        ``if` `((*max_len) < curr_len)`
`            ``*max_len = curr_len;`

`        ``// make curr_len = 0 for next adjacent`
`        ``curr_len = 0;`
`    ``}`
`}`

`// n is number of cities or nodes in graph`
`// cable_lines is total cable_lines among the cities`
`// or edges in graph`
`int` `longestCable(vector<pair<``int``,``int``> > graph[],`
`                                          ``int` `n)`
`{`
`    ``// maximum length of cable among the connected`
`    ``// cities`
`    ``int` `max_len = INT_MIN;`

`    ``// call DFS for each city to find maximum`
`    ``// length of cable`
`    ``for` `(``int` `i=1; i<=n; i++)`
`    ``{`
`        ``// initialize visited array with 0`
`        ``vector< ``bool` `> visited(n+1, ``false``);`

`        ``// Call DFS for src vertex i`
`        ``DFS(graph, i, 0, &max_len, visited);`
`    ``}`

`    ``return` `max_len;`
`}`