Concepts32
State Space Reduction
State space reduction shrinks the number of dynamic programming or search states by keeping only the information that truly affects future decisions.
Bellman Equations
Bellman equations express how the value of a state or action equals immediate reward plus discounted value of what follows.
Reinforcement Learning Theory
Reinforcement Learning (RL) studies how an agent learns to act in an environment to maximize long-term cumulative reward.
DP on Broken Profile - Plug DP
Plug DP (DP on broken profile with plugs) sweeps a grid cell by cell while remembering how partial path segments cross the frontier as labeled “plugs.”
Partition Function
The partition function p(n) counts the number of ways to write n as a sum of positive integers where order does not matter.
Derangements
A derangement is a permutation with no element left in its original position, often written as !n or D(n).
Stars and Bars
Stars and Bars counts the ways to distribute n identical items into k distinct bins using combinations.
Permutations and Combinations
Permutations count ordered selections, while combinations count unordered selections.
Matrix Exponentiation
Matrix exponentiation turns repeated linear transitions into a single fast power of a matrix using exponentiation by squaring.
Broken Profile DP
Broken Profile DP is a dynamic programming technique that sweeps a grid one cell or one column at a time while encoding the boundary between processed and unprocessed cells as a compact state.
Exchange Arguments in DP
An exchange argument proves that any optimal solution can be reordered to satisfy a simple sorting rule by showing that swapping adjacent out-of-order elements never helps.
Knuth Optimization
Knuth Optimization speeds up a class of interval dynamic programming (DP) from O(n^3) to O(n^2) by exploiting the monotonicity of optimal split points.