This talk improves the known asymptotic complexity of online lowest common ancestor search from O(h) to O(log h), opening the door to new uses in distributed computing and version control.

1. Purely Functional Data Structures
for On-line LCA
Edward Kmett

2. Overview
The Lowest Common Ancestor (LCA) Problem
Tarjan’s Off-line LCA
Off-line Tree-Like LCA
Off-line Range-Min LCA
Naïve On-line LCA
Data Structures from Number Systems
Skew-Binary Random Access Lists
Skew-Binary On-line LCA

3. The Lowest Common Ancestor Problem
Given a tree, and two nodes in the tree, find the lowest
entry in the tree that is an ancestor to both.
A
B E
C D F G I
H J

5. The Lowest Common Ancestor Problem
Given a tree and two nodes in the tree, find the lowest
entry in the tree that is an ancestor to both.
Applications:
Computing Dominators in Flow Graphs
Three-Way Merge Algorithms in Revision Control
Common Word Roots/Suffixes
Range-Min Query (RMQ) problems
Computing Distance in a Tree
…

6. The Lowest Common Ancestor Problem
Given a tree and two nodes in the tree, find the lowest
entry in the tree that is an ancestor to both.
First formalized by Aho, Hopcraft, and Ullman in 1973.
They provided ephemeral on-line and off-line versions of
the problem in terms of two operations, with their off-line
version of the algorithm requiring O(n log*(n)) and their
online version requiring O(n log n) steps.
Research has largely focused on the off-line versions of
this problem where you are given the entire tree a priori.

7. cons, link, or grow?
The original formulation of LCA was in terms of two
operations link x y which grafts an unattached tree x
on as a child of y, and lca x y which computes the
lowest common ancestor of x and y.
Alternately, we can work with lca x y and cons a
y, which returns a new extended version of the path y
grown downward with the globally unique node ID a, and
We can replace cons a y with a monadic grow y, which
tracks the variable supply internally. By using a concurrent
variable supply like the one supplied by the concurrent-
supply package enables you to grow the tree in parallel.

8. Tarjan’s Off-line LCA
In 1979, Robert Tarjan found a way to compute a
predetermined set of distinct LCA queries at the same
time given the complete tree by creatively using disjoint-
set forests in O(nα(n)). (This is stronger condition than the usual offline problem
statement.) TarjanOLCA(u)
function
MakeSet(u);
u.ancestor := u;
for each v in u.children do
TarjanOLCA(v);
Union(u,v);
Find(u).ancestor := u;
u.colour := black;
for each v such that {u,v} in P do
if v.colour == black
print "The LCA of “+u+" and “+v+" is " + Find(v).ancestor;

9. Tarjan’s Off-line LCA
In 1979, Robert Tarjan found a way to compute a
predetermined set of distinct LCA queries at the same
time given the complete tree by creatively using disjoint-
set forests in O(nα(n)).
In 1983, Harold Gabow and Robert Tarjan improved the
asymptotics of the preceding algorithm to O(n) by noting
special-case opportunities not available in general
purpose disjoint-set forest problems.

10. Tree-Like Off-line LCA
In 1984, Dov Harel and Robert Tarjan provided the first
asymptotically optimal off-line solution, which converts the
tree in O(n) into a structure that can be queried in O(1).
In 1988, Baruch Scheiber and Uzi Vishkin simplified that
structure, by building arbitrary-fanout trees out of paths
and binary trees, and providing fast indexing into each
case.

11. Range-Min Off-line LCA
In 1993, Omer Berkman and Uzi Vishkin found another
conversion with the same O(n) preprocessing using an
Euler tour to convert the tree structure into a Range-Min
structure, that can be queried in O(1) time.
This was improved in 2000 by Michael Bender and Martin
Farach-Colton.
Alstrup, Gavoille, Kaplan and Rauhe focused on
distributing this algorithm.
Fischer and Heun reduced the memory requirements, but
also show logarithmically slower RMQ algorithms are
often faster the common problem sizes of today!

12. Backup Plans

13. Naïve On-line LCA
Build paths as lists of node IDs, using cons as you go.
x = [5,4,3,2,1] :# 5
y = [6,3,2,1] :# 4
To compute lca x y, first cut both lists to have the same
length.
x’ = [4,3,2,1], y’ = [6,3,2,1], len = 4
Then keep dropping elements from both until the IDs
match.
lca x y = [3,2,1] :# 3

14. Naïve On-line LCA
No preprocessing step.
O(h) LCA query time where h is the length of the path.
O(1) to extend a path.
No need to store the entire tree, just the paths you are
currently using. This helps with distribution and
parallelization.
As an on-line algorithm, the tree can grow without
requiring costly recalculations.

15. Naïve On-line LCA
To go faster we’d need to extract a common suffix in
sublinear time. Very Well…

16. Data Structures from
Number Systems
We are already familiar with at least one data structure
derived from a number system.
data Nat = Zero | Succ Nat
data List a = Nil | Cons a (List a)
O(1) succ grants us O(1) cons

17. Binary Random-Access
Lists
We could construct a data structure from binary numbers
as well, where you have a linked list of “flags” with 2n
elements in them.
However, adding 1 to a binary number can affect all log n
digits in the number, yielding O(log n) cons.

18. Skew-Binary Numbers 15 7 3 1
0
1
2
1 0
The nth digit has value2n+1-1, and each
1 1
digit has a value of 0,1, or 2. 1 2
2 0
We only allow a single 2 in the
1 0 0
number, which must be the first non-zero
1 0 1
digit. 1 0 2
1 1 0
Every natural number can be uniquely
1 1 1
represented by this scheme.
1 1 2
1 2 0
succ is an O(1) operation.
2 0 0
There are 2n+1-1 nodes in a complete tree 1 0 0 0
of height n.

19. Skew-Binary Random Access
Lists
We store a linked list of complete trees, where we are
allowed to have two trees of the same size at the front of
the list, but after that all trees are of strictly increasing
height.
data Tree a = Tip a | Bin a (Tree a) (Tree a)
data Path a = Nil | Cons !Int !Int (Tree a) (Path a)
length :: Path a -> Int
length Nil = 0
length (Cons n _ _ _) = n
I call these random-access lists a Path here, because of our use case.

20. Skew-Binary On-line LCA
Naïve On-line LCA:
Build paths as lists of node IDs, using cons as you go.
To compute lca x y, first cut both lists to have the same length.
Then keep dropping elements until the IDs match.

21. Skew-Binary On-line LCA
Naïve On-line LCA:
Build paths as lists of node IDs, using cons as you go.
To compute lca x y, first cut both lists to have the same length.
Then keep dropping elements until the IDs match.

22. Skew-Binary On-line LCA
Naïve On-line LCA:
Build paths as lists of node IDs, using cons as you go.
To compute lca x y, first cut both lists to have the same length.
Then keep dropping elements until the IDs match.
1

23. Skew-Binary On-line LCA
Naïve On-line LCA:
Build paths as lists of node IDs, using cons as you go.
To compute lca x y, first cut both lists to have the same length.
Then keep dropping elements until the IDs match.
2 1

24. Skew-Binary On-line LCA
Naïve On-line LCA:
Build paths as lists of node IDs, using cons as you go.
To compute lca x y, first cut both lists to have the same length.
Then keep dropping elements until the IDs match.
3
2 1

25. Skew-Binary On-line LCA
Naïve On-line LCA:
Build paths as lists of node IDs, using cons as you go.
To compute lca x y, first cut both lists to have the same length.
Then keep dropping elements until the IDs match.
4 3
2 1

26. Skew-Binary On-line LCA
Naïve On-line LCA:
Build paths as lists of node IDs, using cons as you go.
To compute lca x y, first cut both lists to have the same length.
Then keep dropping elements until the IDs match.
5 4 3
2 1

27. Skew-Binary On-line LCA
Naïve On-line LCA:
Build paths as lists of node IDs, using cons as you go.
To compute lca x y, first cut both lists to have the same length.
Then keep dropping elements until the IDs match.
6 3
5 4 2 1

28. Skew-Binary On-line LCA
Naïve On-line LCA:
Build paths as lists of node IDs, using cons as you go.
To compute lca x y, first cut both lists to have the same length.
Then keep dropping elements until the IDs match.
7
6 3
5 4 2 1

29. Skew-Binary On-line LCA
Naïve On-line LCA:
Build paths as lists of node IDs, using cons as you go.
To compute lca x y, first cut both lists to have the same length.
Then keep dropping elements until the IDs match.
8 7
6 3
5 4 2 1

30. Skew-Binary On-line LCA
Naïve On-line LCA:
Build paths as lists of node IDs, using cons as you go.
To compute lca x y, first cut both lists to have the same length.
Then keep dropping elements until the IDs match.
-- O(1)
cons :: a -> Path a -> Path a
cons a (Cons n w t (Cons _ w' t2 ts))
| w == w' = Cons (n + 1) (2 * w + 1) (Bin a t t2) ts
cons a ts = Cons (length ts + 1) 1 (Tip a) ts

31. Skew-Binary On-line LCA
Naïve On-line LCA:
Build paths as lists of node IDs, using cons as you go.
To compute lca x y, first cut both lists to have the same length.
Then keep dropping elements until the IDs match.
lca :: Eq a => Path a -> Path a -> Path a
lca xs ys = case compare nxs nys of
LT -> lca' xs (keep nxs ys)
EQ -> lca' xs ys
GT -> lca' (keep nys xs) ys
where
nxs = length xs
nys = length ys

32. Skew-Binary Keep
O(log (h - k)) to keep the top k elements of path of height
h
keep 2 (fromList [6,5,4,3,2,1])
6 3
5 4 2 1

33. Skew-Binary Keep
O(log (h - k)) to keep the top k elements of path of height
h
keep 2 (fromList [6,5,4,3,2,1])
=
keep 2 (fromList [3,2,1])
6 3
5 4 2 1

34. Skew-Binary Keep
O(log (h - k)) to keep the top k elements of path of height
h
keep 2 (fromList [6,5,4,3,2,1])
6 3
5 4 2 1

35. Skew-Binary Keep
O(log (h - k)) to keep the top k elements of path of height
h
keep :: Int -> Path a -> Path a
keep _ Nil = Nil
keep k xs@(Cons n w t ts)
| k >= n = xs
| otherwise = case compare k (n - w) of
GT -> keepT (k - n + w) w t ts
EQ -> ts
LT -> keep k ts
consT :: Int -> Tree a -> Path a -> Path a
consT w t ts = Cons (w + length ts) w t ts
keepT :: Int -> Int -> Tree a -> Path a -> Path a
keepT n w (Bin _ l r) ts = case compare n w2 of
LT -> keepT n w2 r ts
EQ -> consT w2 r ts
GT | n == w - 1 -> consT w2 l (consT w2 r ts)
| otherwise -> keepT (n - w2) w2 l (consT w2 r ts)
where w2 = div w 2
keepT _ _ _ ts = ts

36. Skew-Binary On-line LCA
Naïve On-line LCA:
Build paths as lists of node IDs, using cons as you go.
To compute lca x y, first cut both lists to have the same length.
Then keep dropping elements until the IDs match.
lca :: Eq a => Path a -> Path a -> Path a
lca xs ys = case compare nxs nys of
LT -> lca' xs (keep nxs ys)
EQ -> lca' xs ys
GT -> lca' (keep nys xs) ys
where
nxs = length xs
nys = length ys

37. Comparing Node IDs
We can check to see if two paths have the same head or
are both empty in O(1).
infix 4 ~=
(~=) :: Eq a => Path a -> Path a -> Bool
Nil ~= Nil = True
Cons _ _ s _ ~= Cons _ _ t _ = sameT s t
_ ~= _ = False
sameT :: Eq a => Tree a -> Tree a -> Bool
sameT xs ys = root xs == root ys
root :: Tree a -> a
root (Tip a) = a
root (Bin a _ _) = a

38. Monotonicity
We can modify the algorithm
for keep into an algorithm that
takes any monotone predicate
that only transitions from False
to True once during the walk
up the path and yields a result
in O(log h)
We have exactly one shape for a given number of elements,
so we can walk the spine of the two random access lists at
the same time in lock-step. This lets us, modify this algorithm
to work with a pair of paths, because the shapes agree.
(~=) is monotone given using globally unique IDs.

39. Finding the Match
lca’ requires the invariant that both paths have the same
length. This is provided by the fact that lca, shown earlier,
trims the lists first.
lca' :: Eq a => Path a -> Path a -> Path a
lca' h@(Cons _ w x xs) (Cons _ _ y ys)
| sameT x y = h
| xs ~= ys = lcaT w x y xs
| otherwise = lca' xs ys
lca' _ _ = Nil
lcaT :: Eq a => Int -> Tree a -> Tree a -> Path a -> Path a
lcaT w (Bin _ la ra) (Bin _ lb rb) ts
| sameT la lb = consT w2 la (consT w2 ra ts)
| sameT ra rb = lcaT w2 la lb (consT w ra ts)
| otherwise = lcaT w2 ra rb ts
where w2 = div w 2
lcaT _ _ _ ts = ts

40. Skew-Binary On-line LCA
Naïve On-line LCA:
Build paths as lists of node IDs, using cons as you go. O(1)
To compute lca x y, first cut both lists to have the same length. O(h)
Then keep dropping elements until the IDs match. O(h)
Skew-Binary On-line LCA:
Build paths as lists of node IDs, using cons as you go. O(1)
To compute lca x y, first cut both lists to have the same length. O(log
h)
Then keep dropping elements until the IDs match. O(log h)

41. Skew-Binary On-line LCA
No preprocessing step.
O(log h) LCA query time where h is the length of the path.
O(1) to extend a path.
No need to store the entire tree, just the paths you are currently
using. This helps with distribution and parallelization when
working on large trees.
As an on-line algorithm, the tree can grow without requiring
costly recalculations.
Preserves all of the benefits of the naïve algorithm, while
drastically reducing the costs.

42. Now What?
We found that skew-binary random access lists can be used to
accelerate the naïve online LCA algorithm while retaining the
desirable properties.
You can install a working version of this algorithm from hackage
cabal install lca
Next time I’ll talk about the applications of this algorithm to a
“revision control” monad which can be used for parallel and
incremental computation in Haskell.
I am working with Daniel Peebles on a proof of correctness and
asymptotic performance in Agda.

43. Any Questions?