# Root Switching DP

2 min readApr 7, 2024

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# What is root switching DP?

Based on GPT

Root Switching DP is a dynamic programming technique used to solve problems on tree data structures where it’s necessary to compute answers with every node considered as the root of the tree. This approach is particularly useful in situations where the optimal solution or certain properties need to be recalculated as the root of the tree changes, allowing for efficient computation without redoing the entire process from scratch for each potential root.

The core idea behind Root Switching DP involves two main steps:

1. **Initial Root Calculation**: First, a root is arbitrarily chosen, and dynamic programming (DP) values are calculated typically in a bottom-up manner. This step involves computing some property or value for every node, considering the chosen root as the base of the tree. These computations often involve aggregating information from child nodes to their parents until reaching the initial root.

2. **Root Switching and Recalculation**: After computing the DP values with the initial root, the algorithm then iteratively “switches” the root to each of the other nodes in the tree, recalculating the necessary DP values to reflect this change. The trick here lies in efficiently updating these values based on the previously computed results rather than recomputing everything from scratch. This is often achieved by maintaining and updating certain global and local information as the root changes.

The technique leverages the parent-child relationships within the tree to perform updates. When the root switches from a parent to one of its children, the information that was previously propagated upwards needs to be adjusted to reflect the change in perspective (now considering the parent as a descendant of the child in terms of DP calculations).

**Applications**: Root Switching DP is widely used in various tree problems, such as those requiring calculations of paths, subtrees optimizations, or even counting problems where the structure or property of the tree affects the outcome. It is an advanced technique that can significantly reduce the computational complexity of tree-based problems by avoiding redundant recalculations, making it a powerful tool in competitive programming and algorithm design.

In summary, Root Switching DP is about efficiently recalculating dynamic programming solutions for tree-based problems by intelligently updating DP values as the root of the tree changes, thus saving on the computational cost of starting the calculations anew for each possible root.

# An example

E — Minimize Sum of Distances

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