Design & Analysis of Algorithms COMP 482 / ELEC 420

John Greiner

Flow networks

What if weights in a graph are maximum capacities of some flow of material?

• Pipe network to transport fluid/gas (e.g., water, oil, natural gas, CO2)– Edges – pipes– Vertices – junctions of pipes

• Data communication network – Edges – network connections of different capacity– Vertices – routers (do not produce or consume data just move it)

• Also used in resource planning, economics, ecosystem network analysis

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To do:[CLRS] 26#7

Flow Network Definitions

Directed graph G=(V,E)

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Each vertex is on some path from s to t.

One source and one sink.

Each edge has capacity c(u,v)

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Flow Definitions

How much is currently flowing – f : V V ®

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Must satisfy 3 properties:• Capacity constraint: u,v V: f(u,v) c(u,v)• Skew symmetry: u,v V: f(u,v) = –f(v,u)• Flow conservation: u V–{s,t}: f(u,V) = f(V,u) = 0 What goes in must go out.

Total value of flow f:|f| = f(s,V) = f(V,t)

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Example flows

Valid or invalid? Why?

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Maximize the flow

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Maximum flow: Algorithm idea• If we have some flow, …

• …and can find an augmenting path can add a constant amount of flow along path: a>0, (u,v)p, f(u,v) + a c(u,v)

• Then just do it, to get a better flow!

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Ford-Fulkerson method

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Ford-Fulkerson(G,s,t) 1 initialize flow f to 0 everywhere2 while there is an augmenting path p do3 augment flow f along p4 return f

• How do we find/choose an augmenting path?• How much additional flow can we send through that

path?• Does the algorithm always find the maximum flow?

Augmenting

Augmenting path – any path in the residual network:– Residual network: Gf=(V,Ef)

Ef = {(u,v) V V : cf(u,v) > 0}– Residual capacities: cf(u,v) = c(u,v) – f(u,v)

– Residual capacity of path p:

9 Observe – Edges in Ef are either edges in E or their reversals: |Ef| 2|E|.

Residual ? ?

vu,cminpc fpu,vf

Residual capacity of path (s,c,d,t)?

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Ford-Fulkerson method, with details

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Ford-Fulkerson(G,s,t) 1 for each edge (u,v)G.E do 2 f(u,v) = f(v,u) = 0 3 while path p from s to t in residual network Gf do4 cf = min{cf(u,v): (u,v)p} 5 for each edge (u,v) in p do6 f(u,v) = f(u,v) + cf

7 f(v,u) = -f(u,v)8 return f

Algorithms based on this method differ in how they choose p.

Does it always find a maximum flow?

Cut – a partition of V into S,T such that sS, tT

Minimum cut – a cut with minimum capacity

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TS,vu

vu,fTS,f

TS,vu

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|f| = f(S,T)

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Does it always find a maximum flow?

Max-flow min-cut theorem:

The following are equivalent statements:1. f is a maximum flow in G.2. The residual network Gf contains no augmenting paths.3. |f| = c(S,T), for some cut (S,T) of G.

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We will prove three parts: From this we have 2.®1., which means that the Ford-

Fulkerson method always correctly finds a maximum flow.

1.2.3.

What is the worst-case running time?• Augmentation = ?• How many augmentations?

– Let’s assume integer flows.– Each increases the value of the flow by some integer.– O(|f*|), where f* is the max-flow.

• Total worst-case = O(E|f*|).

– How an augmenting path is chosen is very important! 13

O(E)

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Edmonds-Karp Algorithm

Use shortest augmenting path (in #edges).

Run algorithm on our example:

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Edmonds-Karp algorithm analysis: 1

Augmentation = O(E) – Breadth-first search

Will prove: #augmentations = O(VE).Let d(v) be distance from s to v in residual network. Will prove: Every |E| iterations, d(t) increases by 1.d(t) can increase at most |V| times ® O(VE) iterations

Total = O(VE2)

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Edmonds-Karp algorithm analysis: 2Will prove: Every |E| iterations, d(t) increases by 1.

Consider the residual network in levels according to d(v):

As long as d(t) doesn't change, the paths found will only use forward edges.• Each iteration saturates & removes at least 1 forward edge, and adds only

backward edges (so no distance ever drops).• After removing |E| - d(t) +1 forward edges, t will be disconnected from s.

So, within |E| iterations, either• t is disconnected, & algorithm terminates, or• A non-forward edge used, & d(t) increased.

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Antiparallel edges

Allow:

Flows cancel.

Residual (multi)graph can have parallel edges.

Disallow:

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What if we have multiple sources or sinks?

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Maximum Flow: Another Algorithm Idea

Greedily fill outgoing edges to capacity. Later edges’ capacity constraints can lead us to reduce those flows.

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Push-Relabel Example – Goldberg & Tarjan (1986)

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Correctness Overview

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Is this a flow?Is it maximum? – What is the residual graph?

Correctness Depends on Height Invariant

Invariant: Residual edges go down in height by at most one.

But, height(s) – height(t) = |V|, longer than any s-t path.

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Running Time Overview

Max height = .

Overall algorithm is .– Requires amortized analysis.– See text & homework.

Choosing operation order wisely, can reduce to .

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Two Other Approaches

Blocking flows:– Each s-t path in blocking flow contains a saturated edge.– Dinitz/Dinic (1970)

O(VE log V)– Goldberg & Rao (1997)

O(min(V2/3,E1/2) E log(V2/E) log (max(u,v)E c(u,v)))

Combinatorial game:– Cheriyan, Hagerup (1989)

O(VE + V2 log2 V) expected– King, Rao, & Tarjan (1994)

O(VE logE/(V log V) V)

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An Application of Max-Flow

Maximum bipartite matching problem

Matching in a graph is a subset M of edges such that each vertex has at most one edge of M incident on it. It puts vertices in pairs.

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E.g.?

A maximumBipartite: V = L+R E L×R

E.g., dating agency

Maximum Bipartite Matching

• How can we reformulate as a max-flow problem?

• What is the running time, using Edmonds-Karp?

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V = L+RE L×R