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Leftist Heap & Skew Heap

约 177 个字 20 行代码 预计阅读时间 1 分钟

Idea: faster merge

Leftist Heap

\(O(logN)\)

  • Keep the length right path low.(logN)

    Balanced tree can do this, but too more constraint.

    A more relaxed constraint: \(NPL\) (NULL path length)

    Shortest path to a node without 2 child.

    \(NPL(single node) = 0\)

    \(NPL(NULL) = -1\)

    \(NPL(x) = min_{c \in child(x)}\{NPL(c) + 1\}\)

  • Maintain NPL

    Swap childs.

  • Merge Operation

    Heap A, B

    1. Sort right path of A B, let it to be the new right path.

      image-20240318220214707

    2. Attach remained left sub tree.

    3. Maintain NPL.

  • Time complexity

    \(O(logN)\)

  • Codes

    Don't forget to maintain NPL.

    PriorityQueue  Merge ( PriorityQueue H1, PriorityQueue H2 )
{ 
  if ( H1 == NULL )   return H2;  
  if ( H2 == NULL )   return H1;  
  if ( H1->Element < H2->Element )  return Merge1( H1, H2 );
  else return Merge1( H2, H1 );
}
static PriorityQueue
Merge1( PriorityQueue H1, PriorityQueue H2 )
{ 
  if ( H1->Left == NULL )     /* single node */
      H1->Left = H2;          /* H1->Right is already NULL and H1->Npl is already 0 */
  else {
      H1->Right = Merge( H1->Right, H2 );     /* Recursive Merge */
      if ( H1->Left->Npl < H1->Right->Npl )   /* Maintain subtree */
          SwapChildren( H1 ); 
      H1->Npl = H1->Right->Npl + 1;           /* Update NPL value*/
  } /* end else */
  return H1;
}

Skew Heap

  • No maintaining of NPL anymore.

    Length of right path is ONLY guaranteed under amortized analysis.

  • Merge Operation

    1. Sort right path of A B, let it to be the new left path.
    2. Attach left-sub tree to the corresponding right child. (except the largest on new left path)

  • Amortized analysis

    \(\Phi = \) number of heavy nodes.

    After swap, heavy nodes MUST becomes light nodes.

    Light nodes MAY become heavy nodes.

    New heavy nodes \(\leq \) Old light nodes.

    image-20240318220754013