Figure 1. Finding the shortest path using optimal substructure. Numbers represent the length of the path; straight lines indicate single edges, wavy lines indicate shortest paths, i.e., there might be other vertices that are not shown here.
In computer science, a problem is said to have optimal substructure if an optimal solution can be constructed from optimal solutions of its subproblems. This property is used to determine the usefulness of greedy algorithms for a problem.[1]
Typically, a greedy algorithm is used to solve a problem with optimal substructure if it can be proven by induction that this is optimal at each step.[1] Otherwise, provided the problem exhibits overlapping subproblems as well, divide-and-conquer methods or dynamic programming may be used. If there are no appropriate greedy algorithms and the problem fails to exhibit overlapping subproblems, often a lengthy but straightforward search of the solution space is the best alternative.
In the application of dynamic programming to mathematical optimization, Richard Bellman's Principle of Optimality is based on the idea that in order to solve a dynamic optimization problem from some starting period t to some ending period T, one implicitly has to solve subproblems starting from later dates s, where t<s<T. This is an example of optimal substructure. The Principle of Optimality is used to derive the Bellman equation, which shows how the value of the problem starting from t is related to the value of the problem starting from s.
^ abCormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2009). Introduction to Algorithms (3rd ed.). MIT Press. ISBN 978-0-262-03384-8.
and 27 Related for: Optimal substructure information
computer science, a problem is said to have optimalsubstructure if an optimal solution can be constructed from optimal solutions of its subproblems. This property...
Hamilton–Jacobi–Bellman equation – An optimality condition in optimal control theory Markov decision process – Mathematical model Optimal control theory – Mathematical...
solved optimally by breaking it into sub-problems and then recursively finding the optimal solutions to the sub-problems, then it is said to have optimal substructure...
to the solution. Optimalsubstructure "A problem exhibits optimalsubstructure if an optimal solution to the problem contains optimal solutions to the...
In computer science, an optimal binary search tree (Optimal BST), sometimes called a weight-balanced binary tree, is a binary search tree which provides...
due to an exponential complexity. If the problem also shares an optimalsubstructure property, dynamic programming is a good way to work it out. Consider...
solved optimally by breaking it into sub-problems and then recursively finding the optimal solutions to the sub-problems, then it is said to have optimal substructure...
complexity must be at least exponential. The LCS problem has an optimalsubstructure: the problem can be broken down into smaller, simpler subproblems...
problems exhibiting the properties of overlapping subproblems and optimalsubstructure Ellipsoid method: is an algorithm for solving convex optimization...
programming When a problem shows optimalsubstructures—meaning the optimal solution to a problem can be constructed from optimal solutions to subproblems—and...
optimization — studies problems in which one problem is embedded in another Optimalsubstructure Dykstra's projection algorithm — finds a point in intersection of...
maximum subarray as well. Because of the way this algorithm uses optimalsubstructures (the maximum subarray ending at each position is calculated in a...
Hamilton–Jacobi–Bellman equation Markov decision process Optimal control theory Optimalsubstructure Recursive competitive equilibrium Bellman pseudospectral...
Straddling checkerboard Subsequence Longest common subsequence problem Optimal-substructure Subset sum problem Symmetric functions Szemerédi's theorem Thue–Morse...
positive probability ε (and assuming the optimal value of k is used) can be computed by substituting the optimal value of k in the probability expression...
FETI method (finite element tearing and interconnect) is an iterative substructuring method for solving systems of linear equations from the finite element...
graph theory studies how global properties of a graph influence local substructure. Results in extremal graph theory deal with quantitative connections...
especially in remote homologs. The optimal "threading" of a protein sequence onto a known structure and the production of an optimal multiple sequence alignment...
extensions. If substructures are obtained from the experimental data, the generation starts with these substructures. These substructures provide known...
Substructure Analysis that was created in 1973. Each fragment substructure make a continuous contribution an activity of specific type. Substructure is...
nearly 52 m (171 ft; 99 cu) tall pyramid to a mound of ruins, with a substructure that is dangerous to enter due to the risk of cave-ins. Adjoining the...
screen facade made of porcelain stoneware slabs (55×120×1.4 cm). The substructure is composed of aluminum uprights. Similarly, the walls separating the...
alkaline phosphatase production. The optimal pH for the activity of the E. coli enzyme is 8.0 while the bovine enzyme optimum pH is slightly higher at 8.5. Alkaline...
Cold Spring Harbor Laboratory formulated the optimal balance of the four variables and calculated the optimal ratio of axon plus dendrite volume (i.e. the...
organization of cells. This new area of research concerned itself with substructure, also known as the ultrastructure. Many scientists use ultrastructural...
allowing for the identification of substructures such as globules, crystals, membranes, fibrils and tubules. The substructures found in chromoplasts are not...
root. Like ordinary Fibonacci heaps, strict Fibonacci heaps possess substructures similar to binomial heaps. To identify these structures, we label every...
Global Information