Leftist tree
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A leftist tree or leftist heap is a priority queue implemented with a variant of a binary heap. Every node has an s-value which is the distance to the nearest leaf. In contrast to a binary heap, a leftist tree attempts to be very unbalanced. In addition to the heap property, leftist trees are maintained so the right descendant of each node has the lower s-value.
The leftist tree was invented by Clark Allan Crane. The name comes from the fact that the left subtree is usually taller than the right subtree.
When inserting a new node into a tree, a new one-node tree is created and merged into the existing tree. To delete a minimum item, we remove the root and the left and right sub-trees are then melded. Both these operations take O(log n) time. For insertions, this is slower than binary heaps which support insertion in amortized constant time, O(1) and O(log n) worst-case.
Leftist trees are advantageous because of their ability to merge quickly, compared to binary heaps which take Θ(n). In almost all cases, skew heaps have better performance.
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[edit] Bias
The usual leftist tree is a height-biased leftist tree. However, other biases can exist, such as in the weight-biased leftist tree.
[edit] S-value
The s-value of a node is the distance from that node to the nearest leaf. In the diagram, the s-values are shown as calculated by including both the starting node and the leaf in the s-value. Thus the s-value of the top node (with value 4) is 2 because there are 2 nodes on the way to the leaf (the starting node, and the leaf itself). For the other three nodes, the s-value is 1 because those nodes are leaves themselves.
[edit] Merging height biased leftist trees
Merging two nodes together depends on whether the tree is a min or max height biased leftist tree. For a min height biased leftist tree, set the higher valued node as its right child of the lower valued node. If the lower valued node already has a right child, then merge the higher valued node with the sub-tree rooted by the right child of the lower valued node.
After merging, the s-value of the lower valued node must be updated (see above section, s-value). Now check if the lower valued node has a left child. If it does not, then move the right child to the left. If it does have a left child, then the child with the highest s-value should go on the left.
[edit] Java code for merging a min height biased leftist tree
public Node merge(Node x, Node y) { if(x == null) return y; if(y == null) return x; // if this was a max height biased leftist tree, then the // next line would be: if(x.element < y.element) if(x.element.compareTo(y.element) > 0) { // x.element > y.element Node temp = x; x = y; y = temp; } x.rightChild = merge(x.rightChild, y); if(x.leftChild == null) { // left child doesn't exist, so move right child to the left side x.leftChild = x.rightChild; x.rightChild = null; x.s = 1; } else { // left child does exist, so compare s-values if(x.leftChild.s < x.rightChild.s) { Node temp = x.leftChild; x.leftChild = x.rightChild; x.rightChild = temp; } // since we know the right child has the lower s-value, we can just // add one to its s-value x.s = x.rightChild.s + 1; } return x; }
[edit] Initializing a height biased leftist tree
Initializing a height biased leftist tree is primarily done in one of two ways. The first is to merge each node one at a time into one HBLT. This process is inefficient and takes O(nlogn) time. The other approach is to use a queue to store each node and resulting tree. The first two items in the queue are removed, merged, and placed back into the queue. This can initialize a HBLT in O(n) time. This approach is detailed in the three diagrams supplied. A min height biased leftist tree is shown.
To initialize a min HBLT, place each element to be added to the tree into a queue. In the example (see Part 1 to the left), the set of numbers [4, 8, 10, 9, 1, 3, 5, 6, 11] are initialized. Each line of the diagram represents another cycle of the algorithm, depicting the contents of the queue. The first five steps are easy to follow. Notice that the freshly created HBLT is added to the end of the queue. In the fifth step, the first occurrence of an s-value greater than 1 occurs. The sixth step shows two trees merged with each other, with predictable results.
In part 2 a slightly more complex merge happens. The tree with the lower value (tree x) has a right child, so merge must be called again on the subtree rooted by tree x's right child and the other tree. After the merge with the subtree, the resulting tree is put back into tree x. The s-value of the right child (s=2) is now greater than the s-value of the left child (s=1), so they must be swapped. The s-value of the root node 4 is also now 2.
Part 3 is the most complex. Here, we recursively call merge twice (each time with the right child 's subtree that is not grayed out). This uses the same process described for part 2.
[edit] External links
- Leftist Trees at Dr. Sartaj Sahni's website (Department Chair in CISE at University of Florida)
- Java implementation of leftist tree
