DP on Trees for Competitive Programming

Dynamic Programming (DP) on trees is a powerful algorithmic technique commonly used in competitive programming. It involves solving various tree-related problems by efficiently calculating and storing intermediate results to optimize time complexity. By using the tree structure, DP on trees allows programmers to find solutions for a wide range of problems, making it an essential skill in competitive programming.

Dynamic Programming (DP) on trees is a technique used to efficiently solve problems involving tree structures by breaking them down into smaller subproblems and storing intermediate results to avoid redundant calculations.

• Clearly define the problem you want to solve in the context of a tree structure. Identify what information or value you need to compute or optimize within the tree.
• Choose a suitable node as the root of the tree. This choice can significantly impact the efficiency of the DP approach. In many cases, the choice of the root node is arbitrary, but it should make the problem-solving process more straightforward.
• Decide on a traversal order, such as post-order, pre-order, or bottom-up, based on the problem's nature and requirements. The traversal order dictates how you move through the tree and calculate solutions.
• Define the state for each node in the tree. The state represents the information needed to solve the problem for that node. Formulate recurrence relations that describe how to compute the solution for a node based on the solutions of its child nodes. This is where the DP magic happens, as you express how the solution at a given node depends on the solutions of its subproblems.
• Initialize a DP table or array to store intermediate results. Use the chosen traversal order to propagate values through the tree, updating the DP table as you traverse. At each node, calculate and store the result for that node based on the recurrence relations.
• Optimize the calculation of subproblem solutions by avoiding redundant calculations. This often involves memoization, where you store and reuse previously computed results for nodes you've already processed.
• Once you've traversed the entire tree and filled in the DP table, the final result is often found in the root node's entry in the DP table. Depending on the problem, you may need to process and analyze the DP table further to extract the answer.

Lets, understand the Dynamic Programming (DP) on by a problem:

Given a tree consisting of N Nodes and N-1 edges. The task is to select the maximum set of edges such that each vertex is part of at most one of the selected edges (no two edges share a common end point) i.e., if we select an edge connecting vertex u and v, then we cannot select any other edge connected by vertex u or vertex v.

The problem can be solved using DP on Tree in the following way:

Root the tree at any of node(say 1).

For each node we have 2 options, select a edge from the current from the current vertex to any of his child or do not select any edge passing through the current vertex.

dp[v][0]= Maximum set of edges selected in the subtree rooted at vertex v such that no two edges have a common end point, if we do not select any edge passing through vertex v.

dp[v][1]= Maximum set of edges selected in the subtree rooted at vertex v such that no two edges have a common end point, if we select a edge passing through vertex v.

Calculating dp[v][0]: Since dp[v][0] defines the state where we do not take any edge passing through vertex v (i.,e, v->u , where u is child of vertex). So for the all child nodes of vertex v we can either take an edge passing through it or do not take, and all the children of vertex v will be independent of each other in this case.

, where u is child of vertex v.

Calculating dp[v][1]: dp[v][1] defines the state where we do take a edge passing through vertex v (i.,e, v->u , where u is child of vertex).

Suppose we select a edge (v->u), then we cannot choose any other edge passing through vertex u. So will need dp[u][0] in this case, and for all other children of vertex v, we can either choose an edge passing through it or do not choose it as done in above case and we can use that state (dp[v][0]) for calculating this state.

, for all child u of vertex v.

Since we have chosen vertex as the 1 root. Our answer will be max(dp[1][0], dp[1][1]).

Below is the implementation of above approach:

2

Time Complexity: O(N)
Auxilary Space: O(N)

The Rerooting technique is a method used in tree algorithms to efficiently compute values associated with nodes or subtrees of a rooted tree when the tree's root can be moved or "rerooted" to different nodes in the tree. This technique is particularly useful when you want to calculate values for various nodes or subtrees in a tree, but you don't want to traverse the entire tree for each query.

• Start with the original root of the tree, where the tree is rooted. This is often an arbitrary choice or based on the problem's context.
• Use a depth-first search (DFS) to process the tree starting from the original root. During the DFS, calculate and store values for each node and its subtree rooted at that node. These values can be specific to the problem you're solving. For example, you might calculate the sum of values in the subtree or the maximum value in the subtree.
• After processing the entire tree with the original root, you can reroot the tree at any node. This means you can choose a different node in the tree to be the new root.
• When you reroot the tree, you don't need to perform a full DFS traversal again. Instead, you can update the values of the nodes and subtrees based on the values already calculated in the previous DFS traversal. This updating process is efficient and can be done in a way that avoids redundant computations.
• Once you have updated the values for the new root, you can efficiently answer queries related to the values of nodes or subtrees in the tree. By rerooting the tree as needed, you can answer queries at different nodes without recomputing the values from scratch.

The rerooting technique is commonly used in problems that require dynamic programming on trees, such as finding optimal paths, maximum values, or minimum values in subtrees. It allows you to reposition the root and update the values in a way that minimizes redundant work, resulting in more efficient algorithms for various tree-related problems.

Lets, understand the Rerooting Techique by a problem:

Given a tree consisting of N Nodes and N-1 edges. The task is to determine for each node the sum of distance to all other nodes.
The problem can be solved using Rerooting technique in the following way:

Root the tree at node 1.

We will need 2 dp arrays for this problem.

dp1[v]: The sum of distance of node v from all the other nodes in subtree rooted at vertex v.

dp2[v]: The sum of distance of node v from all the other other nodes in the tree.

For the root, dp1[1] will be equal to dp2[1] since, all the nodes of the tree lie in the subtree rooted at vertex 1.

dp1[v] can be calculated by:-

, where u is the child of vertex v and sub[u] is the number of nodes in subtree rooted at vertex u.

Since dp2[1]=dp1[1], answer for the root have been determined.

Now suppose we know the final answer for vertex v, dp2[v] (sum of distance of node v from all other nodes). Now we want to compute the answer for node u, which is a child of vertex v.

dp2[u], (sum of distance from node u to all other nodes in tree) will be the sum of following two parts:

1. 
   We already computed dp1[u], which is sum of distance from node u to all other nodes in the subtree rooted at node u.
   
2. 
   This will be calculated with the help of dp2[v](sum of distance from node v to all other nodes in tree)

Sum of distance from node v to all other nodes excluding the subtree rooted at vertex v will be

From this we can calculate sum of distance from node u to all other nodes excluding the subtree rooted at vertex u by adding (N-sub[u]) to above equation, since N- sub[u] gives the total number of nodes excluding the nodes in the subtree rooted at vertex u.

Summing the above two parts we get:

This can be further simplified to:

Below is the implementation of above approach:

6 9 5 8 8 

Time Complexity: O(N)
Auxiliary Space: O(N)

Dynamic Programming (DP) on trees is a powerful technique for solving a variety of problems involving tree structures. Here is a comprehensive overview of its applications and considerations:

• DP on trees is commonly used to find the maximum or minimum sum of a path from one node to another. This can be applied to problems like finding the longest or shortest path in a tree.
• DP helps determine the diameter of a tree, which is the longest path between any two nodes. This involves finding the maximum distance in the tree.
• DP can be used to find the maximum independent set of nodes in a tree. This is helpful in problems where you want to select nodes with certain constraints, such as no two adjacent nodes in the set.
• DP is often used in binary trees to solve problems like finding the maximum sum of values along a path or identifying the least common ancestor (LCA) of two nodes.
• For binary search trees, DP can help find the minimum or maximum element in the tree, or check if a given tree is a valid BST.
• DP on trees can be used to analyze games or decision-making processes represented as trees, helping to determine optimal strategies.
• When you need to find the best solution within a subtree, DP can be applied to determine the maximum sum, product, or other properties of a subtree.
• DP is applicable to dynamic tree problems, where the structure of the tree changes over time with nodes being added or removed.
• Certain graph problems can be transformed into tree problems, making DP on trees a valuable technique for various graph algorithms.

Related Article:

• Introduction to Tree – Data Structure and Algorithm Tutorials
• Dynamic Programming (DP) Tutorial with Problems