Incremental topological ordering
with Bernhard Haeupler, Sid Sen
• Two problems on digraphs:
Cycle detection
Topological ordering: a total order O such that
v  w  O(v) < O(w)
Incremental topological ordering
• A digraph is acyclic if it has at least one
topological ordering
3
5
1
2
4
• Static: O(n + m) time, n = # vertices, m = # arcs
Assume m ≥ n (simplicity)
Dynamic: on-line, incremental
• Start with vertex set, no arcs
• Add one arc at a time
Maintain a topological order
Report a cycle as soon as one is created
How to maintain ordering?
• 1-1 mapping: V  {1,2,…,n}
via 1 or 2 arrays (inverse?)
• Dynamic ordered list
query: v < w ?
update: delete v
insert v (before or after) w
a, b, c, d, e, f, g, h
move c after f
O(1) time per operation (Dietz, Sleator 1987; Bender, Cole,
Demaine, Farach-Colton, Zito 2002)
Adding an arc
• Given an existing order, how to handle a new
arc v  w?
If v < w: do nothing
If w < v: search for a cycle or a set of vertices to
reorder to restore topological order
• Affected region: all vertices x : w ≤ x ≤ v
Adding an arc
• Search forward from w, but not from vertices > v
w
v
x
• Vertex states during search:
U = unlabeled
L = labeled, initially {w}
S = scanned
Adding an arc
• scan(x):
for x  y
if y = v stop (cycle)
if y < v and y  U add y to L
• while L ≠ Ø:
delete some x from L, scan(x), add x to S
• reorder: move S after v (in order)
Adding an arc
• Running time?
Count related pairs: some paths contain more than one
type
vertex, vertex pairs –
arc, vertex pairs – nm
m
arc, arc pairs –  2 

n

 2

 

• One-way search: x  y traversed  v, x  y newly
related
 O(m) amortized time per arc addition; O(nm) total (vs. O(m2))
• Marchetti-Spaccamela, Nanni, Rohnert 1996
Two-way (bidirectional) search
• Forward from w, backward from v concurrently
v
w
?
w
v
?
When to stop searching?
When to pay for a search step?
Two-way (bidirectional) search
• Stop when x with:
no forward labeled vertices < x
no backward labeled vertices > x
• Reorder:
Move forward scanned vertices (≠ x) after x
Move backward scanned vertices (≠ x) before x
v
w
zU
x
v
w
x
z
How to pay?
• Traverse arcs in pairs: x  y forward, z  u backward
allowed if x < u
• Adding v  w relates x  y, z  u (unless cycle)
• Search time = O(1) per arc traversal + overhead
• k traversal steps  k2/4 new related arc pairs
m
(out of  2  )
k ≤ m1/2  O(m1/2) time per arc addition
k > m1/2  (km1/2)/4 new pairs
 k = O(m3/2)  O(m1/2) amortized per arc
addition
How to implement?
• Ordered search:
Scan smallest forward labeled vertex or largest
backward labeled vertex
Stop when next forward vertex > next backward
vertex
 All forward traversed arcs form pairs with all
backward traversed arcs
• But: need a heap (priority queue) for:
F = forward labeled vertices
R = backward labeled vertices
• Logarithmic (?) overhead
History of two-way search method
• Idea: Alpern, Hoover, Rosen, Sweeney, Zadeck
1990
Incremental bound (per arc addition)
• Katriel, Bodlaender 2005
O(min{m3/2 logn, m3/2 + n2 logn})
• Liu, Chao 2007
O(m3/2 + mn1/2 logn)
• Kavitha, Mathew 2007
O(m3/2 + nm1/2 logn)
• Improvable, but…
All bounds for m arc additions
Avoid heaps?
• Can we eliminate the heap altogether?
Get O(m1/2) per arc amortized time?
• Balanced safe search
F=A∪B
bypassed (temporarily)
R=C∪D
Choose x  A, u  C
If x < u, traverse x y, z  u
If x > u, bypass x (move to B) or u (move to D)
Split:
• Which? What if A or C is empty?
Soft threshold
• Maintain a tentative threshold t, initially w or v
If x > u, bypass x if x > t, u if u < t (if both, choose one)
if A is empty, replace A by B, choose new t uniformly
at random from A (or median)
Similarly if C is empty
 Amortized O(1) bypassed vertices per search
step (expected or worst-case)
 O(1) overhead per search step
 O(m1/2) amortized time per arc addition
forward vertex
unreached vertex
backward vertex
scanned vertex
(v,w)
a
w
l,sl
b
c
d
s
A = {w}, B = {}, D = {}, C = {v}
e
v
f
h
a
forward vertex
unreached vertex
backward vertex
scanned vertex
w
l
b
l,ssl
c
d
s
A = {}, B = {b,d}, D = {}, C = {c,e}
e
v
f
h
a
forward vertex
unreached vertex
backward vertex
scanned vertex
w
b
l
c
s
d
s
e
hh,s
A = {}, B = {d}, D = {c}, C = {}
v
f
h
a
forward vertex
unreached vertex
backward vertex
scanned vertex
w
b
l
c
s
ds
e
h
A = {}, B = {}, D = {c}, C = {}
v
f
a
w
forward vertex
unreached vertex
backward vertex
scanned vertex
b
X
c
s
e
v
Y
f
a
c
forward vertex
unreached vertex
backward vertex
scanned vertex
e
v
w
b
s
f
• Lower bounds
(n logn) vertex reorderings for any algorithm
Ramalingam and Reps 1994
(n2) vertex reorderings if
order = 1-1 mapping
reordering only within affected region
(nm) vertex reorderings if
reordering only within affected region
• Dense case
O(n2.75) Ajwani, Friedrich, Meyer 2006
Õ(n2.5) Liu, Chao 2007
O(n2.5) Kavitha, Mathew 2007