Max Flows 2:
The Excess Scaling Algorithm
and Variants.
James Orlin
Algorithmic ideas from
the previous lecture
•
Augmenting path algorithm
• an improvement algorithm that is guaranteed to be
optimal
•
Max-Flow Min-Cut:
• using a variation of LP duality to establish optimality
and distance from optimality.
•
Scaling
• Solving a sequence of problems that converges to the
original problem.
• Efficient when reoptimizing can be done quickly.
•
Simple data structures
• keeping track of distance labels and current arcs sped
up the algorithm
2
Algorithmic Ideas in this lecture
•
Preflows:
• A relaxation of flows that leads to improved
algorithms.
•
A new scaling algorithm
•
Potential function analysis
• a method of establishing a faster running time
by using “global” information about the
algorithm.
3
Running times for max flow algorithms
O(nm2)
O(n2m)
O(n3)
O(n2 m½ )
O(nm log2 n)
O(nm log n)
O(nm log n2/m)
Edmonds and Karp (1972)
Dinic (1970)
Karzanov (1974)
Malhotra, Kumar & Maheshwari (1977)
Cherkasky (1977)
Galil and Naaman (1980)
Sleator and Tarjan (1983)
Goldberg and Tarjan (1988)
4
More Improvements in running time
O(nm+ n2+ε )
O(nm logα n )
King Rao and Tarjan (1992)
King Rao and Tarjan (1994)
α = m/(n log n)
O(n3 / log n )
O(nm)
O(nm / log n)
Cheriyan, Hagerup, and Mehlhorn (1996)
O. (2011)
O. (2011)
if m = O(n) and under some other conditions.
Remark 1: The max flow problem is solvable in
O(nm / log n) steps when m = Ω(n2) and when m = O(n).
Remark 2: The improvements by O. (2011) are very
involved.
5
Weakly polynomial running times
O(nm log U)
O(nm + n2 log U)
O(nm + n2 log½ U)
O(m3/2 log n log U)
O(n2/3 m log(n2/m) log U)
O(n2/3 m log U)
Gabow (1985)
Ahuja and Orlin (1989)
Ahuja, Orlin, and Tarjan (1989)
Goldberg and Rao (1998)
Goldberg and Rao (1998)
O. (2011)
Further improvements if log Umax/Umin = O(n1/3)
Remark: The improvement by O. (2011) is very simple
6
Preflows
At each intermediate stages we permit more flow
arriving at nodes than leaving (except for s)
A preflow is a function x: A  R s.t. 0  x  u and
such that
e(i) = ∑j∈N xji - ∑j∈N xij ≥ 0,
for all i  N – {s, t}.
i.e., e(i) = excess at i = net excess flow into node i.
The excess is required to be nonnegative.
7
A Feasible Preflow
2
2
3
0
2
5
2
1
3
s
2
4
0
3
2
3
1
The excess e(j) at
each node j ≠ s, t is
the flow in minus the
flow out.
t
Note: total excess =
flow out of s minus
flow into t.
In preflow push algorithms, we typically send
flow in only one arc at a time.
8
Converting a preflow into a flow
1
21
2
23
0
2
5
2
1
23
s
32
Any preflow can be
converted into a flow that
preserves the flow into t.
2
4
0
2
3
t
Run flow decomposition
and then keep only the
paths from s to t.
1
Path
Flow
Path
Flow
s-3-t
2
s-2
1
s-3
1
s-2-5-t
2
s-4-t
2
s-4-2
1
9
Distances
We will maintain distance labels.
Next: how to maintain distance labels efficiently.
10
Exact Distance Labels
Distance labels d( ) are exact for G(x) if for all j with a
path to t, d(j) is the length of the shortest path
from j to t.
d( )
4
3
2
1
i
0
t
P = the shortest path from i to t in G(x)
An arc is (i, j) is admissible if d(i) = d(j) + 1.
Lemma. Suppose all distance labels are exact.
Then, an arc (i, j) is admissible if and only if it is on
a shortest path from i to t.
11
Maintaining Exact Distance Labels
At each iteration, CurrentArc(i) is the first
admissible arc in A(i). The current arcs
form a tree (or forest), whose arcs are
directed into node t.
7
6
5
6
4
4
3
5
7
2
2
3
1
8
0
t
1
We will update CurrentArc(i) if it is
inadmissible. We will update d(i) if
there is no admissible arc out of i.
Suppose CurrentArc(i) = (i, j).
Arc (i, j) may become
inadmissible if
1.
(i, j) is saturated or
2.
the distance label of j is
increased.
12
An update of CurrentArc
Example 1. Suppose that flow is sent in
(7, 2) and the arc is saturated.
7
Then (7, 3) becomes admissible.
6
5
4
6
1
4
6
Tail
Head
7
7
7
7
7
2
3
4
6
8
4
3
5
7
2
2
3
1
8
0
t
7
2
3
Res.
Cap
0
1
4
2
0
8
13
Example 2: there is a cascading effect
Example 2. Suppose now that (3, 8) is
saturated.
Node 3 increases its distance label to 4.
(3, 5) is admissible. (7, 3) is inadmissible
and node 7 is updated next.
7
6
5
6
1
Tail
Head
Res. Cap
4
4
3
5
7
3
5
6
2
2
3
3
7
2
1
8
3
8
0
0
t
14
Example 3: a gap is created
Example 3. Suppose now that (5, 2) is
saturated.
Node 5 increases its distance label to 5.
But now there is no node with distance
label 3.
7
6
5
6
4
4
3
5
5
2
0
2
2
5
3
3
1
8
5
4
6
0
t
1
7
3
Tail
Head
Res. Cap
15
Example 3: continued
∞7
1
3
4
5
6
7
6
5
When a gap is created, all
nodes with distance level
above the gap have their
distance level set to ∞.
4
3
2
2
1
8
0
t
16
Time for updating distance labels
Lemma.
1.
The arc list of each node i is scanned at most
once between updates of d(i).
2.
The arc list is scanned once if d(i) needs to be
updated.
3.
d(i) is updated at most n-1 times for each i.
Conclusion. Each arc is scanned O(n) times. The
total running time for updating CurrentArc and d( )
is O(nm).
17
On the order in which things are done
Typically, the generic preflow push algorithm is
presented before the excess scaling algorithm.
But not here.
18
Scaling Algorithms
1.
Define a concept called Δoptimal, where Δ is some
positive integer, and where a
1-optimal solution is optimal
for the original problem.
2.
Develop a subroutine that
efficiently determines Δ0optimum solution where
Δ0 is some (possibly large)
power of 2
3.
Develop a subroutine
Improve-Approx that
transforms a Δ-optimal
solution into a Δ/2-optimal
solution.
Generic Scaling
Algorithm
Δ := 2K for some selected
value K
determine a Δ-optimal
solution x
while Δ > 1 do
y := ImproveApprox(x, Δ)
x := y
Δ := Δ/2
19
Initialize
1.
Determine exact distance labels for each node.
2.
For each arc (s, j), send usj units of flow from s to j.
This results in an excess e(j) = usj.
3.
d(s) := ∞;
4.
Choose Δ0 to be the minimum power of 2 that is
greater than max (usj : j ∈ N}.
5
e(5) = us5
7
e(7) = us7
s
d(s) := ∞
20
Excess- Scaling Preliminaries
A preflow x is called Δ-optimal if
1. 0 ≤ e(i) < Δ for all i with 1 ≤ d(i) < ∞.
2. the distance labels are exact
3. d(s) = ∞; that is, there is no path from s to t in G(x).
Subroutine ImproveApprox(x,Δ): takes a flow that is
-maximum and outputs a flow that is /2-maximum.
ImproveApprox terminates when
e(i) < Δ/2 for all i with 1 ≤ d(i) < ∞.
After initialization, the flow is Δ0-optimal.
21
Push-Relabel(i)
INPUT: a node i with e(i) ≥ Δ/2.
Push-Relabel(i)
begin
while e(i) > 0 do
if there is an admissible arc (i, j) then
send min {e(i), Δ - e(j) – 1, rij) units
of flow in (i, j)
else replace d(i) by 1 + min{d(j) : rij > 0}
update r, e, CurrentArc, and d( );
end
22
Examples of Push-Relabel
Δ = 64
level
k+2
6
k+1
4
k
Case 1. e(7) = 62
r73 = 40
e(3) = 10
Case 2. e(7) = 62
r73 = 40
e(3) = 30
7
62
40
k-1
3
Case 3. e(7) = 62
r73 = 0
10
22
29
62
0
40
7
0
52
63
30
23
Excess Scaling Algorithm
Procedure Improve-Approx(r, Δ)
begin
while there is a Δ/2-active node do
begin
let i = argmin (d(i) : i is Δ/2-active}
Push-Relabel(i)
end
end;
24
Pushing in the 64-scaling phase
s
9
j
32 ≤ e(j) < 64
“large excess”
8
i
7
e(i) < 32
6
62
7
5
4
6
3
9
3
2
2
s
7
1
4
55
0
t
25
2
8
4
40
5
0
Choose a large
excess node i at
minimum distance
level.
25
For each admissible arc (i,j), j is
not large excess.
Pushing in the 64-scaling phase.
62
7
2
j
5
0
7
e(i) < 32
28
Rule: push as before, except that
no node excess can exceed Δ-1.
25
2
4
“large excess”
40
i
4
32 ≤ e(j) < 64
5
35
2
5
63
Push 2 units in (7,4).
Then push 35 units in (7,5)
Send min{e(i), rij, Δ-e(j)-1} units of flow in (i,j)
26Any
non-saturating push has at least Δ/2 units of flow
Bounding the number of pushes
Each arc (i, j) has at most one saturating push in
between relabels of i. It may have several nonsaturating pushes.
We will show that the number of non-saturating
pushes per scaling phase is O(n2).
We will use a potential function argument.
We will define a potential function Φ.
• Each non-saturating push will decrease Φ by .5.
We will bound this number by bounding the total
increase in Φ over all iterations.
27
Elevators movements
10
9
8
7
6
5
4
3
2
1
0
An elevator is in a building that
is n stories high. It can go up
multiple floors at a time, but it
can only go down one floor at a
time.
In m movements of the elevator,
how many floors can it travel
upwards?
Note: it can travel n floors in a
single move.
28
UP ≤ DOWN + n.
10
9
It can go up at most n floors
more than it goes down.
8
7
6
5
4
3
2
It can go down at most m floors
in m moves.
In m moves, it can travel at most
2m + n floors.
If m > n, it travels at most 3m
floors, or fewer than 3 floors per
move on average.
1
0
29
Finding a min-distance Δ/2-active node.
10
9
8
7
Δ/2-active
nodes at
level k
6
{j, j’}
5
4
∅
∅
3
∅
2
{i}
1
∅
0
∅
For each distance level store a
doubly linked list of Δ/2-active
nodes.
After pushing flow from a node
i, the next push is from a node
with d(j) ≥ d(i) – 1.
Start with distance level d(i) – 1
and seek the first distance level
with a Δ/2-active node.
Number of steps in m pushes is
at most 2m + n.
30
Next: Bounding the number of pushes
We will create a potential function Φ.
We will use the up and down argument to bound the
amount of decrease in Φ over all iterations.
This will bound the number of pushes.
31
A potential function for excess-scaling
s
9
Φ = jN e(j) d(j) / Δ
8
7
6
5
4
6
3
9
3
2
2
s
7
1
4
55
0
t
32
node
2
3
4
5
6
7
8
9
e(j)
4
0
1
0
5
35
55
40
e(j)d(j)
8
0
1
0
20
70
165 120
8
Φ = 384 / 64 = 6
The potential is the “gravitational
potential” measured in units of Δ
Increases in the potential function
s
9
Φ = jN e(j) d(j) / Δ
8
Relabels:
7
Φ increases by at most k if
d(j) increases by k.
6
Φ increases by O(n2) over all
relabels.
5
4
6
3
9
3
2
2
s
7
1
4
55
0
t
33
8
Start of scaling phase.
Φ doubles when Δ is replaced by
Δ/2.
Φ increases by O(n2 log U) over
all scaling phases.
Decreases in the potential function
s
9
Φ = jN e(j) d(j) / Δ
8
Push:
7
Φ decreases by δ/Δ if δ units of
flow are sent in (i, j)
6
Φ decreases by at least ½ in any
large push
5
4
6
3
9
3
2
2
s
7
1
4
55
0
t
34
8
Conclusion: number of large
pushes is at most 2n2 plus the
total increase in Φ. This is
O(n2 log U) large pushes.
Summary of Excess Scaling Algorithm
•
Converts Δ-optimal flows into Δ/2-optimal flows
•
O(nm) running time (excluding time for large
pushes)
•
O(n2) time for large pushes per scaling phase.
•
O(nm + n2 log U) time overall.
35
On the analysis of the potential function
Note: the potential function Φ increased O(n2)
because of relabels, and increased O(n2 log U)
when Δ was replaced by Δ/2.
Speeding up the algorithm:
•
If we can reduce the total flow at the end of the
algorithm to O(nΔ/K) for some K, then the
potential increase would be O( (n2 log U)/K).
•
We will show how to do this for K = log½ U.
•
Another bottleneck appears; so one cannot make
K any larger.
36
General Algorithmic Rule
Single bottleneck rule
If the bottleneck in an algorithm
is caused by one “thing”, then
try to speed up the “thing.”
37
Excess Scaling with Waves
Carry out a sequence of “waves” during a scaling phase.
Rules for the wave.
1. Select the lowest level Δ/2-active node, if one exists.
Else, select the highest level node i with 1 < e(i) < Δ/2.
Push flow from the selected node.
2. If a node has no admissible arcs, then delete it from
G(x) for the rest of the wave (and add it back after the
wave).
3. The wave ends when there are no nodes with positive
excess remaining.
4. After the wave, update the distance labels and the
CurrentArc( ).
38
Excess Scaling with Waves
Procedure Improve-Approx(r, Δ)
begin
while the total excess is greater than nΔ / log½ U do
perform a wave
update distance labels and current arcs
end;
39
Examples of Pushes During a Wave
There are no Δ/2-active nodes.
Node 7 is the highest level node with excess.
Arcs (7, 3) and (7, 4) are admissible.
Δ = 64
level
k
30
15
20
k-1
20
2
7
30Example
7
3
4
5
Example 1
Case 1. Send 20
10 7
units in arc (7, 3).
Node 3 becomes
15
0
Δ/2-active.
40
3
4
15
20
5
8
3
4
3
7
30 Example
6
5
Case 2. Send 20
0 7
units in arc (7, 3).
Send 10 units
5 in
0
arc (7, 4). Delete
node 7 from LIST.
28 3
4 15
10 3
4
4
5
Case 3. Send 6
20 7
units in arc (7, 3).
Send 4 units
0 in
0
arc (7, 4). Delete
node 7 from LIST.
16 3
4 9
40
Waves during the Δ-scaling Phase
1.
No node is ever Δ-active during the Δ-scaling
phase.
2.
At the end of the wave, each active node j
(that is, e(j) > 0) will increase its distance label.
(There is no path from j to t of admissible arcs).
4
6
3
7
3
2
2
1
An arc may become inadmissible
when it is saturated. This can
cause a “cascading effect of
deletions.
1
4
55
e.g., suppose (1, 5) is saturated.
0
t
suppose (2, 4) is saturated.
41
Waves during the Δ-scaling Phase
3.
There is at most
one non-saturating
push from a node
with e(i) < Δ/2.
4
6
3
7
3
2
2
1
1
4
55
0
t
4.
If no nodes are Δ/2
active, then the
highest level active
node will never
receive any more
flow during the wave.
If e(6) := 0, then e(6) = 0 for the
rest of the wave.
If the Δ-scaling phase
does not end after a
wave, there were at
least n/log½ U distance
increases.
At the end of the
wave, each node with
excess will be
relabeled.
If the Δ-scaling phase
didn’t end, then the
total excess is at
least nΔ/log½ U.
42
Putting it all together
1.
2.
There are
O(log U + n log½ U )
waves. Each wave
leads to O(n) small
non-saturating
pushes.
3.
Each non-saturating
push from a Δ/2-active
node decreases Φ by at
least .5
4.
Φ increases by
O(n2 / log½ U ) following
a scaling phase. It
increases by
O(n2 log½ U ) in total.
5.
Conclusion: Number of
non-saturating pushes
is O(n2 log½ U ).
Total run time
because of the waves
is O(n2 log½ U ),
if log U < n2
43
summary of techniques
•
Preflows: A relaxation of flows that leads to improved
algorithms. It is quite common to solve problems by
solving relaxations.
•
Potential function analysis
• a method of establishing a faster running time by using
“global” information about the algorithm.
• surprisingly useful in analyzing data structures and
algorithms
•
Excess scaling
•
Excess scaling with waves. (If there is a single bottleneck
operation or a single cause of the bottleneck, sometimes
the running time can be improved by combining two
approaches.)
44
Research notes on preflow push
Pushing from the active node with the largest distance
label leads to O(n2 m.5) nonsaturating pushes.
A very efficient data structure called dynamic trees
reduces the running time to O(nm log n2/m). GoldbergTarjan (1986)
The “excess scaling technique” of Ahuja and Orlin (1989)
reduced the running time to O(nm + n2 log U).
Ahuja, Orlin, and Tarjan (1989): further very small
improvements.
Goldberg and Rao (1998). An even more efficient
algorithm for max flows.
Orlin [2011] max flows in O(nm) time and sometimes less.
45