FAST ALGORITHMS FOR SURFACE EMBEDDED GRAPHS VIA HOMOLOGY The Final - - PowerPoint PPT Presentation

FAST ALGORITHMS FOR SURFACE EMBEDDED GRAPHS VIA HOMOLOGY

The Final Exam of

Kyle J. Fox

Doctoral committee: Jeff Erickson, Chair Chandra Chekuri Derek Hoiem David Eppstein, University of California, Irvine

Graph Algorithms

• Major target for algorithms research
• Provide natural abstractions for practical problems

2

Not Efficient

• Many problems are NP-hard
• Many polynomial time algorithms are quadratic or

worse

3

Planarity

• A natural assumption for many problem domains
• Speeds up computation
• Polynomial time → near-linear
• NP-hard → polynomial time

4

Beyond Planarity

• Many planar graph algorithms generalize
• Bounded genus surface graphs
• Minor free graphs
• Generalizations are sparse and have small well-

balanced separators

• But that’s often not enough!

5

Surface Graphs

• Drawn on 2-manifolds with boundary
• Vertices map to points, edges to internally disjoint

curves

6

Flows and Cuts

• Network flow algorithms for planar graphs do not

generalize easily

• First results for max flows/min cuts on surfaces in

2009 [Chambers, Erickson, and Nayyeri STOC/SoCG ’09]

• Algorithms either have high dependency on g or

require very complicated subroutines

7

Thesis: Homology Helps

• Many algorithms for surface embedded graphs can

be improved by using homology

• Homology was the key tool in the Chambers et al.

flow/cut papers

• Homology characterizes curves by whether their

difference bounds a weighted sum of faces

• Intuitively, how certain cycles or flows travel around

features of the surface

8

The Dissertation

• Algorithms for combinatorial optimization problems

related to maximum flows and minimum cuts

• Assume graphs with n vertices and O(n) edges on
• rientable surfaces of genus g = O(n)
• All algorithms use results in homology theory

9

Results

• 1. A gO(g)n log log n time algorithm for global

minimum cut [Erickson, F, and Nayyeri SODA ’12]

• 2. Algorithms for shortest non-trivial cycles tailored

for undirected and directed graphs [F SODA ’13]

• 3. A 2O(g)n² time algorithm for counting minimum


s,t-cuts [Chambers, F, and Nayyeri SoCG ’13]

• 4. Preliminary work on computing maximum s,t-flows

[Erickson and F ‘13]

10

Results

• 1. A gO(g)n log log n time algorithm for global

minimum cut [Erickson, F, and Nayyeri SODA ’12]

• 2. Algorithms for shortest non-trivial cycles tailored

for undirected and directed graphs [F SODA ’13]

• 3. A 2O(g)n² time algorithm for counting minimum


s,t-cuts [Chambers, F, and Nayyeri SoCG ’13]

• 4. Preliminary work on computing maximum s,t-flows

[Erickson and F ‘13]

11

Global Minimum Cut

• Remove smallest weight set of edges to separate

graph into non-empty components

12

• gO(g)n log log n time algorithm matches best result

for planar graphs when g = O(1) [Łącki and Sankowski ’11]

Dual Graph

u* v*

f* g*

f g

u v

• vertices ⇄ faces
• edges ⇄ edges
• faces ⇄ vertices

13

Separating Subgraphs

• Find a minimum weight set of cycles in the dual

graph that separate the faces

14

• Reduce problem to two cases
• 1. some minimum weight subgraph bounds a disk
• 2. all minimum weight subgraphs decomposable

into topologically non-trivial cycles

Results

• 1. A gO(g)n log log n time algorithm for global

minimum cut [Erickson, F, and Nayyeri SODA ’12]

• 2. Algorithms for shortest non-trivial cycles tailored

for undirected and directed graphs [F SODA ’13]

• 3. A 2O(g)n² time algorithm for counting minimum


s,t-cuts [Chambers, F, and Nayyeri SoCG ’13]

• 4. Preliminary work on computing maximum s,t-flows

[Erickson and F ‘13]

15

Non-trivial Cycles

• Want short cycles that are topologically non-trivial
• Particularly non-contractible or non-separating

16

Contractible & Separating Non-contractible & Separating Non-contractible & Non-separating

Non-trivial Running Times

When Directed? Running Time Citation Before Undirected gO(g)n log log n

[Italiano et al. ’11]

17

Non-trivial Running Times

When Directed? Running Time Citation Before Undirected gO(g)n log log n

[Italiano et al. ’11]

Now Undirected 2O(g)n log log n

[F ’13]

17

Non-trivial Running Times

When Directed? Running Time Citation Before Undirected gO(g)n log log n

[Italiano et al. ’11]

Now Undirected 2O(g)n log log n

[F ’13]

Before Directed Non-separating: O(g² n log n) Non-contractible: O(gO(g) n log n) [Erickson ’11]

17

Non-trivial Running Times

When Directed? Running Time Citation Before Undirected gO(g)n log log n

[Italiano et al. ’11]

Now Undirected 2O(g)n log log n

[F ’13]

Before Directed Non-separating: O(g² n log n) Non-contractible: O(gO(g) n log n) [Erickson ’11] Now Directed Non-contractible: O(g³ n log n) [F ’13]

17

Covering Spaces

• Spaces that map down to the input
• Non-trivial cycles gain structure in the right cover

18

… …

Results

• 1. A gO(g)n log log n time algorithm for global

minimum cut [Erickson, F, and Nayyeri SODA ’12]

• 2. Algorithms for shortest non-trivial cycles tailored

for undirected and directed graphs [F SODA ’13]

• 3. A 2O(g)n² time algorithm for counting minimum


s,t-cuts [Chambers, F, and Nayyeri SoCG ’13]

• 4. Preliminary work on computing maximum s,t-flows

[Erickson and F ‘13]

19

Counting Cuts

• Input gives vertices s and t
• An s,t-cut is a subset of vertices containing s but

not t

• Want to count s,t-cuts that minimize weight of

edges leaving cut

20

Counting Cuts Example

s t

21

Counting Cuts Example

s t s t

21

Counting Cuts Example

s t s t s t

21

Counting Cuts Example

s t s t s t s t

21

Counting Cuts Results

• 2O(g)n² time algorithm for counting minimum s,t-cuts

matches the best results for planar graphs when
 g = O(1) [Bezáková and Friedlander ’12]

• Count forward t,s-cuts in a directed acyclic graph
• Edges leaving forward t,s-cuts are dual to directed

cycles in a particular cohomology class

• Afterward, sample minimum cuts in O(n log n) time

per sample

22

Results

• 1. A gO(g)n log log n time algorithm for global

minimum cut [Erickson, F, and Nayyeri SODA ’12]

• 2. Algorithms for shortest non-trivial cycles tailored

for undirected and directed graphs [F SODA ’13]

• 3. A 2O(g)n² time algorithm for counting minimum


s,t-cuts [Chambers, F, and Nayyeri SoCG ’13]

• 4. Preliminary work on computing maximum s,t-flows

[Erickson and F ‘13]

23

Maximum Flow

• Classic non-trivial polynomial time algorithm
• Planar case considered since the seminal maximum

flow/minimum cut paper [Ford and Fulkerson ’56]

24

[Harris and Ross ’55]

Flow in the Plane

• General sparse graphs: O(n² / log n) time


[Orlin ’13]

• For undirected graphs: O(n log log n) time


[Italiano et al. ’11]

• For directed graphs: O(n log n) time 


[Borradaile and Klein ’09]

25

Flow in Surfaces

• For integer capacities summing to C:


O(g8n log²n log²C) [Chambers, Erickson, and Nayyeri ’09]

• For arbitrary real capacities:


gO(g)n3/2 [Chambers, Erickson, and Nayyeri ’09]

• Both algorithms are very complicated and probably
• utperformed by more general quadratic time

algorithms

• Is there a clean O(gO(1) n log n) time algorithm?

26

A New Algorithm

• A true generalization of Borradaile and Klein’s

algorithm

• Uses duality between minimum cost flow problems

in the primal and dual graphs

• Provably runs in strongly polynomial time
• May actually run in O(gO(1) n log n) time

27

Chains and Circulations

• Each (oriented) edge uv has two darts u→v and

v→u

• A [0,1,2]-chain assigns real values to the vertices,
• riented edges, or faces.
• Flow is a synonym for 1-chain
• A flow f is a circulation if for each vertex v,


∑uv f (uv) - ∑vu f (vw) = 0

28

Capacities and Feasibility

• For flow f and edge uv, we say


f (u→v) = f (uv) and f (v→u) = -f (uv)

• A capacity function c assigns real values to the

darts

• Flow f is feasible if for each dart u→v, we have


f (u→v) ≤ c(u→v)

• f induces residual capacity function cf, where


cf (u→v) = c(u→v) - f (u→v)

29

Maximum s,t-flow

• An s,t-flow has ∑uv f (uv) - ∑vu f (vu) = 0 for each

vertex v except s and t

• Value of an s,t-flow is ∑s→v f (s→v)
• Given capacity function c, want to find a feasible

s,t-flow of maximum value

• Value equal to minimum capacity of darts leaving

any s,t-cut [Ford and Fulkerson ’56]

30

Borradaile and Klein

• Algorithm of Borradaile and Klein removes residual

clockwise cycles, and then sends flow along leftmost residual s,t-paths

• But what does leftmost mean in surfaces with

genus?

• Erickson [’10] offers a different interpretation

31

Borradaile and Klein


(according to Erickson) A planar graph with capacity function c has a feasible s,t-flow matching the value of s,t-flow f if and

• nly if there are no negative length dual cycles wrt cf.

32

[Erickson ‘10]

s t

C

• s*

t*

• Negative length cycles are over-saturated s,t-cuts

[Venkatesan ‘83]

Parametric Shortest Paths

• Compute a dual shortest path tree from any dual

vertex o incident to t using c for the dart lengths

• Continuously increase flow from s to t along any

path, changing residual capacities

33

[Erickson ‘10]

Parametric Shortest Paths

• Shortest path tree pivots at critical moments…

34

[Erickson ‘10]

Parametric Shortest Paths

• Shortest path tree pivots at critical moments…

34

[Erickson ‘10]

Parametric Shortest Paths

• Shortest path tree pivots at critical moments…

34

[Erickson ‘10]

Parametric Shortest Paths

• Shortest path tree pivots at critical moments…

34

[Erickson ‘10]

Parametric Shortest Paths

• Shortest path tree pivots at critical moments…

34

[Erickson ‘10]

Parametric Shortest Paths

• Shortest path tree pivots at critical moments…

34

[Erickson ‘10]

Parametric Shortest Paths

• Shortest path tree pivots at critical moments…

34

[Erickson ‘10]

• …until a directed cycle is formed

Parametric Shortest Paths

• The directed cycle is dual to a minimum s,t-cut
• Each pivot can be performed in O(log n) time
• O(n) pivots (not obvious), so O(n log n) overall

35

[Erickson ‘10]

… on Surfaces

• The flow you send from s to t affects how the

shortest path tree changes

• A flow f may induce negative cycles in the dual

residual graph even if there are feasible flows of the same value [Chambers, Erickson, and Nayyeri ’09]

• And it may take Ω(n²) pivots to reach a negative

cycle [Erickson ’10]

36

Boundary Circulations

• The boundary of a 2-chain α is the flow ∂α where

∂α(uv) = α(right(uv)) - α(left(uv))

• Boundary circulations are the boundaries of 2-

chains

37

7 2 5

Homology

• Two flows f and f’ are homologous if f - f’ is a

boundary circulation

• Forms an equivalence relation between flows

38

Coflows and Capacities

• Flows, circulations, etc. are called coflows,

cocirculations, etc. in the dual

• Let Θ be a coflow and c be a capacity function.

Define c’ on the edges so that

• c’(uv) = c(u→v) if Θ(uv) ≥ 0
• c’(uv) = -c(v→u) otherwise
• The capacity of Θ wrt c is the dot product ⟨Θ, c’⟩

39

Homological
 Max Flow/Min Cut

• Extends a theorem of Chambers, Erickson, and

Nayyeri [’09]

40

Theorem: Let c be a capacity function and Θ be a

• coflow. The maximum cost ⟨Φ, Θ⟩ of any feasible

circulation Φ is equal to the minimum capacity of any coflow cohomologous to Θ.

Min Capacity Coflows

• Fix a circulation Φ and capacity function c. If coflow

Θ is minimum capacity for its cohomology class wrt c then Θ is minimum capacity wrt cΦ

41

Back to the Plane

• Find a minimum cost coflow
• Continuously increase flow from s to t along any

path, changing residual capacities

42

1 1 1 1 1 1 2 2 3 3 4 5 7

Back to the Plane

• Minimum cost cocirculation pivots at critical

moments…

43

1 1 1 1 1 1 2 2 3 3 4 5 7 1 1 1 1 2 3 2 1 4 5 6 6 1

• …by pushing coflow around s,t-cuts

Optimal Circulations and Coflows

• Fix a coflow Θ and capacity function c. Circulation

Φ maximizes ⟨Φ, Θ⟩ and Θ is minimum capacity for its cohomology class wrt c if and only if
 Φ(u→v) = c(u→v) for every dart u→v such that Θ(u→v) > 0

• Φ saturates the darts carrying coflow

44

Generalizations

• Extend parametric minimum capacity coflows to

arbitrary surfaces and cohomology classes

• Minimum cost coflow is resilient to circulations; can

push flow from s to t along any path

• Maintain feasible flow saturating darts with positive

coflow

45

1 2 s t coflow

Augmentations

• D: edges carrying non-zero coflow
• Cannot send primal flow through edges in D if we

want to keep darts saturated

• Push flow through G \ D until there is an s,t-cut S

with saturated outgoing darts C in G \ D

46

1 2 s t 1 2 s t

• utgoing darts C in G \ D

coflow edges D

Pivots

• Some edges from C + D bound s,t-cut S in G
• Push coflow around coboundary of S until an edge

goes empty

47

1 2 s t 1 3 s t coboundary of s,t-cut S in G

Pivots

• Pushed coflow around a coboundary so the new

coflow is in the same cohomology class

• Every dart carrying positive coflow is saturated, so

coflow is still minimum cost

48

1 2 s t 1 3 s t

Termination

49

• Process ends when no edge can go empty
• We’ve saturated a minimum s,t-cut in G and

computed a maximum s,t-flow

3 s t 1 3 s t 1 saturated s,t-cut

Efficient Pivots

• Initial coflow runs along a shortest path tree from o
• o is the only “source” so edges with dual coflow

stay connected

• Both D and G \ D have a simple structure, so


O(g log n + g²) time per pivot

50

s t 1 1 1 1 3 3 7

Time Analysis

Theorem: Let N be the number of pivots. Our algorithm computes a maximum s,t-flow in
 O(n log n + gN(g + log n)) time.

51

• Ideally, N = O(gO(1)n), even for arbitrary

cohomology classes

• Right now, can prove N = O(n²)

Winding Numbers

• For s,t-path π, let f π be the flow that sends 1 unit on

each dart of π

• The winding number of coflow Θ wrt π is ⟨f π, Θ⟩

52

1 2 s t winding number = 2

More Winding

53

1 2 s t 1 3 s t

• The winding number of a fixed path π wrt each

minimum cost coflow Θ increases by at least 1 with each pivot

winding number = 2 winding number = 3

Means Less Winding

54

1 2 s t

• Each iteration’s augmenting paths avoid edges

used for the minimum cost coflow

• The winding number of each augmenting path π

wrt the original minimum cost coflow Θ₀ decreases by at least 1 with each pivot

winding number = 0 winding number = -1 1 2 s t

Bounding Pivots

55

s t 1 1 1 1 3 3 7

• In the shortest path coflow Θ₀, Θ₀(u→v) ≤ O(n)
• So lowest winding number possible for any

augmenting path is -O(n²)

• First augmenting paths have winding number 0 wrt

Θ₀

Bounding Pivots

56

• So after O(n²) pivots, there can never be another

augmenting path avoiding edges with coflow Theorem: Our algorithm computes a maximum s,t- flow in O(gn²(g + log n)) time.

Next Steps in Flows

• Prove O(gO(1)n) bound on number of pivots
• If it applies to arbitrary coflows, there are likely

applications to minimum cost flow, multiple-source multiple-sink maximum flow, …

57

Results

• 1. A gO(g)n log log n time algorithm for global

minimum cut [Erickson, F, and Nayyeri SODA ’12]

• 2. Algorithms for shortest non-trivial cycles tailored

for undirected and directed graphs [F SODA ’13]

• 3. A 2O(g)n² time algorithm for counting minimum


s,t-cuts [Chambers, F, and Nayyeri SoCG ’13]

• 4. Preliminary work on computing maximum s,t-flows

[Erickson and F ‘13]

58

What Comes Next?

• Postdoc at ICERM at Brown University participating

in the Spring 2014 program Network Science and Graph Algorithms

59

Thank you!