Dynamic Programming(DP) is a technique to solve problems by breaking them down into overlapping sub-problems which follows the optimal substructure. There are various problems using DP like subset sum, knapsack, coin change etc. DP can also be applied on trees to solve some specific problems.
Given a tree with N nodes and N-1 edges, calculate the maximum sum of the node values from root to any of the leaves without re-visiting any node.
Given above is a diagram of a tree with N=14 nodes and N-1=13 edges. The values at node being 3, 2, 1, 10, 1, 3, 9, 1, 5, 3, 4, 5, 9 and 8 respectively for nodes 1, 2, 3, 4….14.
The diagram below shows all the paths from root to leaves :
All the paths are marked by different colors :
Path 1(red, 3-2-1-4) : sum of all node values = 10
Path 2(orange, 3-2-1-5) : sum of all node values = 11
Path 3(yellow, 3-2-3) : sum of all node values = 8
Path 4(green, 3-1-9-9) : sum of all node values = 22
Path 5(violet, 3-1-9-8) : sum of all node values = 21
Path 6(pink, 3-10-1) : sum of all node values = 14
Path 7(blue, 3-10-5) : sum of all node values = 18
Path 8(brown, 3-10-3) : sum of all node values = 16
The answer is 22, as Path 4 has the maximum sum of values of nodes in its path from a root to leaves.
The greedy approach fails in this case. Starting from the root and take 3 from the first level, 10 from the next level and 5 from the third level greedily. Result is path-7 if after following the greedy approach, hence do not apply greedy approach over here.
The problem can be solved using Dynamic Programming on trees. Start memoizing from the leaves and add the maximum of leaves to the root of every sub-tree. At the last step, there will be root and the sub-tree under it, adding the value at node and maximum of sub-tree will give us the maximum sum of the node values from root to any of the leaves.
The diagram above shows how to start from the leaves and add the maximum of leaves of a sub-tree to its root. Move upward and repeat the same procedure of storing the maximum of every sub-tree leaves and adding it to its root. In this example, the maximum of node 11 and 12 is taken to count and then added to node 5 (In this sub-tree, 5 is the root and 11, 12 are its leaves). Similarly, the maximum of node 13 and 14 is taken to count and then added to node 7. Repeat the steps for every sub-tree till we reach the node.
Let DPi be the maximum summation of node values in the path between i and any of its leaves moving downwards. Traverse the tree using DFS traversal. Store the maximum of all the leaves of the sub-tree, and add it to the root of the sub-tree. At the end, DP1 will have the maximum sum of the node values from root to any of the leaves without re-visiting any node.
Below is the implementation of the above idea :
Time Complexity: O(N), where N is the number of nodes.
Attention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the DSA Self Paced Course at a student-friendly price and become industry ready.
- Dynamic Programming on Trees | Set 2
- Dynamic Segment Trees : Online Queries for Range Sum with Point Updates
- Total number of possible Binary Search Trees and Binary Trees with n keys
- Generic Trees(N-array Trees)
- Overlapping Subproblems Property in Dynamic Programming | DP-1
- Optimal Substructure Property in Dynamic Programming | DP-2
- Travelling Salesman Problem | Set 1 (Naive and Dynamic Programming)
- Vertex Cover Problem | Set 2 (Dynamic Programming Solution for Tree)
- Bitmasking and Dynamic Programming | Set 1 (Count ways to assign unique cap to every person)
- Compute nCr % p | Set 1 (Introduction and Dynamic Programming Solution)
- Dynamic Programming | High-effort vs. Low-effort Tasks Problem
- Top 20 Dynamic Programming Interview Questions
- Dynamic Programming | Building Bridges
- Bitmasking and Dynamic Programming | Set-2 (TSP)
- Sum over Subsets | Dynamic Programming
- Number of Unique BST with a given key | Dynamic Programming
- Dynamic Programming vs Divide-and-Conquer
- Distinct palindromic sub-strings of the given string using Dynamic Programming
- Double Knapsack | Dynamic Programming
- Greedy approach vs Dynamic programming
If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to email@example.com. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.