Back Tracking

约 173 个字 17 行代码 预计阅读时间 1 分钟

• Game Tree

  Each path from the root to a leaf defines an element of the solution space.

• Template

  bool Backtracking ( int i )
  {   Found = false;
      if ( i > N )
          return true; /* solved with (x1, …, xN) */
      for ( each xi in Si ) { 
          /* check if satisfies the restriction R */
          OK = Check((x1, …, xi) , R ); /* pruning */
          if ( OK ) {
              Count xi in;
              Found = Backtracking( i+1 );
              if ( !Found )
                  Undo( i ); /* recover to (x1, …, xi-1) */
          }
          if ( Found ) break; 
      }
      return Found;
  }
  

• Efficiency

  回溯的效率跟S的规模、约束函数的复杂性、满足约束条件的结点数相关。

  约束函数决定了剪枝的效率，但是如果函数本身太复杂也未必合算。

  满足约束条件的结点数最难估计，使得复杂度分析很难完成。

• \(\alpha-\beta\) pruning

  Limits search to \(O(\sqrt N)\) nodes, where \(N\) is the total nodes in a game tree.

Examples

• The Turnpike Reconstruction Problem

  Given \(N ( N – 1 ) / 2\) distances. Reconstruct a point set from the distances.

  Solution: Reconstruct from the largest distance. Two options (\(X_{l}, X_r\))

  Answers are symetric.

• Tic-tac-toe

  \(f(P) = W_{Computer} - W_{human}\)

  \(W\) is the number of potential wins at P.