Max-Independent-Set

Calculating Max Independent Set with greedy algorithm

Tech 🛠️ Languages and Tools :

   

For Tree : anytree Lib

For Graph : networkX Lib

Max Independent Set for Tree

Time complexity : O(n)

Always return correct answer (Always return Max Independent Set)

Prove :

Step 1 : Initialization The initial set unCutNodesSet is constructed by adding the root and all descendants. Using Root.descendants from the anytree library, this operation takes O(n), where n is the number of nodes in the tree.

Step 2 : Independent Set Iteration

• Checking if a node has no children (leaf check) is O(1).
• Appending to IndependentSet and removing from unCutNodesSet are both O(1).
• For each non-leaf node, the function calls itself recursively on each child.

Step 3 : Main Loop The main loop iterates as long as unCutNodesSet is not empty. Within each iteration:

• Choosing a node in unCutNodesSet: The line next(iter(unCutNodesSet)) takes O(1) time.
• Calling IndependentSetIteration: The core of the complexity lies in this recursive function.

The loop runs O(n) times since it processes each node once O(1), with each iteration involving O(1) operations and at most one call to IndependentSetIteration.

Total Complexity : Each step costs O(n) then total complexity is O(n) + O(n) + O(n) = O(n)

Max Independent Set for Graph

Time complexity : O(V^2+VE)

Sometimes return correct answer (because its greedy algorithm)

Correct answer :

Wrong answer :

Conclusion :

Greedy algorithm works correct for trees not graphs

Also greedy algorithm for tree has better complexity