Combinatorial search

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A* heuristic


  • A systematic way to run through all the possible configurations of a search space.
    • Configurations
      • permutations:= all possible arrangements of objects
      • subsets:= all possible ways of building a collection of them
      • enumerating all spanning trees of a graph, all paths between two vertices, or all possible ways to partition vertices into colour classes
  • We must generate each possible configuration exactly once.
  • At each step in the backtracking algorithm, we try to extend a given partial solution $a = (a_1, a_2, …, a_k)$ by adding another element at the end. After this extension, we must test whether what we now have is a complete solution: if so, we should print it or count it. If not, we must check whether the partial solution is still potentially extendable to some complete solution.


Constructing all paths in a graph

Constructing all permutations

Constructing all subsets

Search pruning