Please subscribe to the official Codeforces channel in Telegram via the link https://t.me/codeforces_official. ×

Revise Complete Graphs For Interview | C++

Revision en18, by harshiscoding, 2021-09-30 13:26:59

Vertices,edges,undirected and directed graph,cyclic & acyclic graph,degree indegree & outdegree and weights. In an undirected graph, the sum of degrees of all vertices is double the vertices (We consider degree=indegree=outdegree in an undirected graph).

Graph representation in C++

Let us say there are n nodes and m edges. And adj[i][j] represents there is an edge from node i to j.

There are two ways to represent a graph

1. Adjacency matrix:

code

Time Complexity:O(m)**** Space Complexity:**O(n*n)**

2. Adjacency List

code

Time Complexity:**O(m)** Space Complexity:**O(n*n)**

If there are also weights in edges, we create the adjacency list as ' vector <vector < pair <int,int> > > ' where first value is node(v) and second value represents weight b/w the nodes(u and v).

Since there can be multiple disconnected components in a graph, while graph traversal we have to call BFS/DFS from every unvisited node. To avoid repetition we store information if a node is visited or not by creating a boolean array where is_visited[i]==true represents node 'i' is visited.

for(int i=1i<=n;i++)
{
if(!is_visited[i]) BFS(i);
}

BFS

code

Time Complexity:**O(n+e)** ( O(n+e): n time taken for visiting n nodes and e time taken for traversing through adjacent nodes)

Space Complexity:**O(n)** ( O(n) for visiting array and O(n) for queue )

DFS

code

Time Complexity:**O(n+e)** ( O(n+e): n time taken for visiting n nodes and e time taken for traversing through adjacent nodes)

Space Complexity:**O(n)** ( O(n) for visiting array and O(n) for stack space )

Cycle Detection in Graphs

Case 1: Cycle detection in undirected graph using dfs

If a node adjacent to a node ( that is not its parent node ) is already visited, the component contains a cycle.

code

Case 2:Cycle detection in the undirected graph using BFS


code

Case 3:Cycle detection in the Directed graph using DFS-

code

Case 3:Cycle detection in the Directed graph using BFS

Assume the directed graph to be acyclic i.e. DAG and find its topological order.If we can do topological sorting i.e. pushing all nodes in queue , the graph is acyclic,other wise it is cyclic.

We increase our count by 1 (starting from 0) each time we push a node in queue. And if final value of count==no. of nodes in graph, the graph is a DAG.

code

Bipartite graph:

A graph will not be a bipartite iff it has a cycle with an odd no. of nodes.

Check whether a graph is bipartite or not using BFS

code

Check whether a graph is bipartite or not using DFS

code

Topological Sorting:

Only possible in DAG. It is a linear order of vertices such that for every directed edge u-->v, vertex u comes before v in that order. A DAG has at least one node with indegree=0.

Topological sorting using BFS

First, calculate indegree of all nodes and store it in vector.Then push all nodes with degree==0 in a queue.Take each node one by one out of queue and for all its child ,decrease their indegree by 1. After this if any child has indegree==0, push them in the queue.

code

Topological sorting using DFS

Topological order is the order of nodes in decreasing order of finishing time i.e. the node that gets finished at last comes first in order.

code

Shortest path algorithms:

Shortest path of all nodes from a node in a uni-weighted undirected graph.

Use BFS for this because BFS visits nodes in a sequential manner. That is nodes at the same level are visited simultaneously. Run BFS and equate dist[node] = 1+dist[parent].

code

Shortest path of all nodes from a node in a weighted DAG

The weights can be -ve.

Minimum sum to reach a node 'v' i.e. dist[v] is minimum of all (dist[u]+weight(u-->v)),where u is node from all its parents. Similarly, dist[u] is calculated with the help of its parents.We can observe that ultimately we have to start calculating dist[] from source node and in topological order ,we have to visit the nodes

We visit nodes in topological order and relax all its children. In this way, each node is relaxed by all its parents.

code

Shortest path in +ve weighted graph.

Dijkstra's algorithm is basically an efficient version of breadth-first search on a different graph.

Given a graph G with positive integer edge-weights, one could construct a new unweighted graph H, by replacing each weighted edge with a number of edges equivalent to its weight. So, for example, if you had an edge (u,v) with weight 10 in G, you'd replace this edge with a series of 10 edges between u and v.

Since BFS visits nodes in increasing order of their level (distance from source node) i.e. at any point of time,if a node n1 level is smaller than node n2, minimum distance from n1 will be calculated earlier than n2.

From this, we get an intuition that at any point of time, calculate minimum distance of children of that node which is nearest to the source node. This is a kind of greedy approach because we are relaxing children of that node which is locally(at a point of time) nearest to the source.

Since min_priority_queue always stores elements in increasing order,we will use it.We can also use set instead of it.

code

Why Dijkstra doesn't work with -ve weights:

If all weights are non-negative, adding an edge can never make a path shorter but this is not valid if edge weight is -ve.Dry run a case to explain it further.

Bellman Ford

Since the negative weighted cycle has sum — infinity, Bellman-Ford works in a directed graph iff the graph has no -ve weighted cycle and works in an undirected graph iff all weights are non -ve.

Relax all edges n-1 times.Why exactly n-1 times.Since in each relaxation,in worst case,only one node will get relaxed and maximum distance of a node from source node is n-1. Consider example 1-->2-->3-->4-->5. First 5 is relaxed,then 4 and so on till 2 ,n-1 times.

code

DSU

It is a data structure used to perform the union of disjoint sets efficiently.Each node in the set has parent.

Representative node:Topmost node of a set whose parent is the node itself.

Naive Implementation:

code

Time complexity:O(n) for make_set and O(d) for find_set() and union_set() where d=maximum depth possible of a node.

To reduce time complexity of find_set() and union_set(),we have to decrease depth.This can be possible if a parent has maximum children possible.

We can do this by doing modification in find_set() and union_set():

For finding representative node a node using find_set(), we will be traversing to its parent, grandfather and so on... . While traversing, we make a representative node,the parent of each node traversed.

While merging two sets, we make the smaller-sized set child of the larger-sized set to ensure minimum depth. (visualize why so, by a diagram). To know the size of the set, we have to maintain an array ,where rnk[i] tells the size of the set treating 'i' as representative node.

code

Time complexity :O(1)

( Using path compression in find_set() alone reduces time complexity to O (log N)approximately.And time Complexity of find_set() and union_set(), when you use both path compression and union by rank : O( α(N) ) where α(N) = Inverse Ackermann Function which is approximately equal to O(1). )

History

 
 
 
 
Revisions
 
 
  Rev. Lang. By When Δ Comment
en43 English harshiscoding 2022-09-03 16:10:01 36
en42 English harshiscoding 2022-09-03 16:08:44 1 Tiny change: 'SU**\n\n1. Finding th' -> 'SU**\n\n1.Finding th'
en41 English harshiscoding 2022-09-03 16:07:56 360
en40 English harshiscoding 2022-09-03 09:01:58 177
en39 English harshiscoding 2022-09-02 09:05:56 14
en38 English harshiscoding 2022-09-02 09:02:37 17
en37 English harshiscoding 2022-09-01 15:49:51 240
en36 English harshiscoding 2022-08-31 14:19:06 57
en35 English harshiscoding 2022-05-29 14:19:28 0 (published)
en34 English harshiscoding 2021-12-12 18:30:08 47
en33 English harshiscoding 2021-12-12 18:28:58 241
en32 English harshiscoding 2021-12-12 12:44:47 1008
en31 English harshiscoding 2021-12-11 22:11:56 57
en30 English harshiscoding 2021-12-11 22:08:01 190
en29 English harshiscoding 2021-12-11 21:17:12 334
en28 English harshiscoding 2021-12-11 18:51:41 18 Tiny change: 'ouble the vertices (We con' -> 'ouble the the no. of edges (We con'
en27 English harshiscoding 2021-10-02 02:07:28 38
en26 English harshiscoding 2021-10-02 02:06:13 776
en25 English harshiscoding 2021-10-02 01:31:17 88
en24 English harshiscoding 2021-10-02 01:29:02 21237
en23 English harshiscoding 2021-10-02 01:28:45 11261 Reverted to en21
en22 English harshiscoding 2021-10-02 00:57:40 11261 Reverted to en16
en21 English harshiscoding 2021-10-02 00:57:20 2367 Reverted to en9
en20 English harshiscoding 2021-10-02 00:56:37 23786 Reverted to en8
en19 English harshiscoding 2021-10-01 22:03:18 5435
en18 English harshiscoding 2021-09-30 13:26:59 2377
en17 English harshiscoding 2021-09-30 01:11:18 910
en16 English harshiscoding 2021-09-30 00:51:45 2037
en15 English harshiscoding 2021-09-29 23:26:56 2409
en14 English harshiscoding 2021-09-29 22:37:35 1295
en13 English harshiscoding 2021-09-29 22:18:17 2063
en12 English harshiscoding 2021-09-29 21:42:26 112
en11 English harshiscoding 2021-09-29 21:40:43 802
en10 English harshiscoding 2021-09-29 21:29:57 2367
en9 English harshiscoding 2021-09-29 20:39:11 2367 Tiny change: '\n==================\n' -> '\n========\n'
en8 English harshiscoding 2021-09-27 19:09:00 231
en7 English harshiscoding 2021-09-27 18:46:00 1502
en6 English harshiscoding 2021-09-27 17:14:14 138
en5 English harshiscoding 2021-09-27 17:10:22 257
en4 English harshiscoding 2021-09-27 17:08:58 228
en3 English harshiscoding 2021-09-27 17:06:13 703
en2 English harshiscoding 2021-09-27 16:42:44 15 Tiny change: 'ologies:\n\nbhbm\n==================' -> 'ologies:\n==================\n\nhvh'
en1 English harshiscoding 2021-09-27 16:41:52 90 Initial revision (saved to drafts)