Purely Functional Data Structures ex3.3

1. Leftist Heap
Definition
A Leftist (min)Heap is a binary tree that satisfies the follow-ing conditions.
If X is a node and L and R are its left and right children, then:
– X.value ≤ L.value
– X.value ≤ R.value
– null path length of L ≥ null path length of R
where the null path length of a node is the shortest distance between
from that node to a descendant with 0 or 1 child.
If a node is null, its null path length is −1.

2. Example: Null Path Length
•node 4 violates leftist heap property
4  fix by swapping children
8 0 19 1
12 1 27 0 20 0
15 0 25 0 43 0
node 4 8 19 12 15 25 27 20 43
npl 1 0 1 1 0 0 0 0 0

3. Example: Null Path Length
4
19 1 8 0
27 0 20 0 12 1
43 0 15 0 25 0
node 4 8 19 12 15 25 27 20 43
npl 1 0 1 1 0 0 0 0 0

4. FromListの実装
(言語ゲーム 2009-11-23 Leftist Heap (http://d.hatena.ne.jp/propella/20091123/p1) のサンプルプログラムに追加して使って下さい)
-- Copied from CS231 Algorithm Prof. Lyn Turbak Wellesley College
-- make a leftist heap with one node.
singleton x = T 1 x E E
-- make a leftist heap from list
fromList :: Ord a => [a] -> Heap a
fromList [] = E
fromList xs = mergeLoop (map singleton xs)
where
mergeLoop [] = error "shouldnt: mergeLoop []"
mergeLoop [t] = t
mergeLoop ts = mergeLoop(mergePairs ts)
mergePairs [] = []
mergePairs [t] = [t]
mergePairs (t1:t2:ts) = (merge t1 t2):(mergePairs ts)
prop_Heap2 :: [Int] -> Bool
prop_Heap2 xs = isSorted (heapToList (fromList xs))
test2 = verboseCheck prop_Heap2

5. 動作
3段階目
2段階目
1段階目

6. fromListの計算量がO(n)であること
n n n 2 n n
+ 2 log 2 + 3 log 2 + ⋯ + (log n+1) log 2
2 2 2 2
2 n
n n log 2 log 2 log 2
= + ( + 2 + ⋯+ n )
2 2 2 2 2
2 n
1 log 2 log 2 log 2
（ ( + 2
+⋯+ n
) < 1 よ り）
2 2 2 2
n 3n
< +n=
2 2

7. 参考文献
・Clemson University CpSc 212, Section 001 (Roy P. Pargas)
http://www.cs.clemson.edu/~pargas/courses/cs212/common/notes/ppt/1
4LeftistSkewHeaps.ppt
p.3, p.6-7
・Wellesley College CS 231 2001 (Lyn Turbak)
http://cs.wellesley.edu/~cs231/fall01/leftist.pdf
p.2, p.5
・言語ゲーム 2009-11-23 Leftist Heap
http://d.hatena.ne.jp/propella/20091123/p1