Dinic Max Flow Algorithm Slides by Dominik Scheder Part I Dinic - - PowerPoint PPT Presentation

Dinic’ Max Flow Algorithm

Slides by Dominik Scheder

Part I

Dinic’ Algorithm in General Flow Networks

4 2 2 1 1 1 3 3 3 3 3 3 4 4 2 3 3 1

s t

2 3 2

4 2 2 1 1 1 3 3 3 3 3 3 4 4 2 3 3 1

s t Find out which edges lie on a shortest s-t-path,

2 3 2

4 2 2 1 1 1 3 3 3 3 3 3 4 4 2 3 3 1

s t Find out which edges lie on a shortest s-t-path,

2 3 2

4 2 2 1 1 1 3 3 3 3 3 3 4 4 2 3 3 1

s t Find out which edges lie on a shortest s-t-path, and forget the rest for the time being.

2 3 2

4 2 2 1 1 1 3 3 3 3 3 3 2 3 3

s t Find out which edges lie on a shortest s-t-path, and forget the rest for the time being.

4 2 2 1 1 1 3 3 3 3 3 3 2 3 3

s t

Now lets greedily route as much flow as possible in this network!

4 2 2 1 1 1 3 3 3 3 3 3 2 3 3

s t

4 2 2 1 1 1 3 3 3 3 3 3 2 3 3

s t Find an s-t-path with depth-first search.

4 2 2 1 1 1 3 3 3 3 3 3 2 3 3

s t Find an s-t-path with depth-first search.

4 2 2 1 1 1 3 3 3 3 3 3 2 3 3

s t Find an s-t-path with depth-first search.

4 2 2 1 1 1 3 3 3 3 3 3 2 3 3

s t Find an s-t-path with depth-first search.

4 2 2 1 1 1 3 3 3 3 3 3 2 3 3

s t Find an s-t-path with depth-first search.

4 2 2 1 1 1 3 3 3 3 3 3 2 3 3

s t Find an s-t-path with depth-first search. Route as much flow through it as possible: 1 unit.

4 2 2 1 1 1 3 3 3 3 3 3 2 3 3

s t Find an s-t-path with depth-first search. Route as much flow through it as possible: 1 unit.

(1) (1) (1) (1)

4 2 2 1 1 1 3 3 3 3 3 3 2 3 3

s t Find an s-t-path with depth-first search. Route as much flow through it as possible: 1 unit.

(1) (1) (1) (1)

Update capacities.

3 1 2 1 1 3 3 3 3 2 3 2 3 3

s t Find an s-t-path with depth-first search. Route as much flow through it as possible: 1 unit. Update capacities.

3 1 2 1 1 3 3 3 3 2 3 2 3 3

s t Find an s-t-path with depth-first search. Route as much flow through it as possible: 1 unit. Update capacities. Repeat!

3 1 2 1 1 3 3 3 3 2 3 2 3 3

s t Find an s-t-path with depth-first search.

3 1 2 1 1 3 3 3 3 2 3 2 3 3

s t Find an s-t-path with depth-first search.

3 1 2 1 1 3 3 3 3 2 3 2 3 3

s t Find an s-t-path with depth-first search.

3 1 2 1 1 3 3 3 3 2 3 2 3 3

s t Find an s-t-path with depth-first search.

3 1 2 1 1 3 3 3 3 2 3 2 3 3

s t Find an s-t-path with depth-first search.

3 1 2 1 1 3 3 3 3 2 3 2 3 3

s t Find an s-t-path with depth-first search.

3 1 2 1 1 3 3 3 3 2 3 2 3 3

s t Find an s-t-path with depth-first search.

3 1 2 1 1 3 3 3 3 2 3 2 3 3

s t Find an s-t-path with depth-first search. Delete dead ends.

3 2 1 1 3 3 3 3 2 3 2 3 3

s t Find an s-t-path with depth-first search. Delete dead ends.

3 2 1 1 3 3 3 3 2 3 2 3 3

s t Find an s-t-path with depth-first search. Delete dead ends. Route flow.

3 2 1 1 3 3 3 3 2 3 2 3 3

s t Find an s-t-path with depth-first search. Delete dead ends. Route flow.

(1) (1) (1) (1)

3 2 1 1 3 3 3 3 2 3 2 3 3

s t Find an s-t-path with depth-first search. Delete dead ends. Route flow.

(1) (1) (1) (1)

Update capacities.

2 1 1 3 3 3 3 1 3 2 3 3

s t Find an s-t-path with depth-first search. Delete dead ends. Route flow. Update capacities.

2 1 1 3 3 3 3 1 3 2 3 3

s t Find an s-t-path with depth-first search. Delete dead ends. Route flow. Update capacities. Repeat!

2 1 1 3 3 3 3 1 3 2 3 3

s t Find an s-t-path with depth-first search.

2 1 1 3 3 3 3 1 3 2 3 3

s t Find an s-t-path with depth-first search.

2 1 1 3 3 3 3 1 3 2 3 3

s t Find an s-t-path with depth-first search.

2 1 1 3 3 3 3 1 3 2 3 3

s t Find an s-t-path with depth-first search.

2 1 1 3 3 3 3 1 3 2 3 3

s t Find an s-t-path with depth-first search.

2 1 1 3 3 3 3 1 3 2 3 3

s t Find an s-t-path with depth-first search.

2 1 1 3 3 3 3 1 3 2 3 3

s t Find an s-t-path with depth-first search.

2 1 1 3 3 3 3 1 3 2 3 3

s t Find an s-t-path with depth-first search.

2 1 1 3 3 3 3 1 3 2 3 3

s t Find an s-t-path with depth-first search.

2 1 1 3 3 3 3 1 3 2 3 3

s t Find an s-t-path with depth-first search.

2 1 1 3 3 3 3 1 3 2 3 3

s t Find an s-t-path with depth-first search.

2 1 1 3 3 3 3 1 3 2 3 3

s t Find an s-t-path with depth-first search. Delete dead ends.

1 3 3 3 3 1 3 2 3

s t Find an s-t-path with depth-first search. Delete dead ends.

1 3 3 3 3 1 3 2 3

s t Find an s-t-path with depth-first search. Delete dead ends. Route as much flow as possible.

1 3 3 3 3 1 3 2 3

s t Find an s-t-path with depth-first search. Delete dead ends. Route as much flow as possible.

(1) (1) (1) (1)

1 3 3 3 3 1 3 2 3

s t Find an s-t-path with depth-first search. Delete dead ends. Route as much flow as possible.

(1) (1) (1) (1)

Update capacities.

2 3 3 3 2 2 3

s t Find an s-t-path with depth-first search. Delete dead ends. Route as much flow as possible. Update capacities.

2 3 3 3 2 2 3

s t Find an s-t-path with depth-first search. Delete dead ends. Route as much flow as possible. Update capacities. Repeat!

2 3 3 3 2 2 3

s t Find an s-t-path with depth-first search.

2 3 3 3 2 2 3

s t Find an s-t-path with depth-first search.

2 3 3 3 2 2 3

s t Find an s-t-path with depth-first search.

2 3 3 3 2 2 3

s t Find an s-t-path with depth-first search.

2 3 3 3 2 2 3

s t Find an s-t-path with depth-first search.

2 3 3 3 2 2 3

s t Find an s-t-path with depth-first search.

2 3 3 3 2 2 3

s t Find an s-t-path with depth-first search.

2 3 3 3 2 2 3

s t Find an s-t-path with depth-first search.

2 3 3 3 2 2 3

s t Find an s-t-path with depth-first search.

2 3 3 3 2 2 3

s t Find an s-t-path with depth-first search. Delete dead ends.

2 3 3 3

s t Find an s-t-path with depth-first search. Delete dead ends.

2 3 3 3

s t Find an s-t-path with depth-first search. Delete dead ends. Route as much flow as possible.

2 3 3 3

s t Find an s-t-path with depth-first search. Delete dead ends. Route as much flow as possible.

(2) (2) (2) (2)

2 3 3 3

s t Find an s-t-path with depth-first search. Delete dead ends. Route as much flow as possible.

(2) (2) (2) (2)

Update capacities.

1 1 1

s t Find an s-t-path with depth-first search. Delete dead ends. Route as much flow as possible. Update capacities.

1 1 1

s t Find an s-t-path with depth-first search. Delete dead ends. Route as much flow as possible. Update capacities. Repeat.

1 1 1

s t Find an s-t-path with depth-first search.

1 1 1

s t Find an s-t-path with depth-first search.

1 1 1

s t Find an s-t-path with depth-first search.

1 1 1

s t Find an s-t-path with depth-first search.

1 1 1

s t Find an s-t-path with depth-first search.

1 1 1

s t Find an s-t-path with depth-first search.

1 1 1

s t Find an s-t-path with depth-first search.

1 1 1

s t Find an s-t-path with depth-first search. Delete dead ends.

t Find an s-t-path with depth-first search. Delete dead ends. The source s has been deleted. The phase ends.

4 2 2 1 1 1 3 3 3 3 3 3 4 4 2 3 3 1

s t

2 3 2

4 2 2 1 1 1 3 3 3 3 3 3 4 4 2 3 3 1

s t

2 3 2

Collect the flow of this phase.

4 2 2 1 1 1 3 3 3 3 3 3 4 4 2 3 3 1

s t

2 3 2

Collect the flow of this phase.

4 2 2 1 1 1 3 3 3 3 3 3 4 4 2 3 3 1

s t

2 3 2

Collect the flow of this phase.

(1) (1) (1) (1)

4 2 2 1 1 1 3 3 3 3 3 3 4 4 2 3 3 1

s t

2 3 2

Collect the flow of this phase.

(1) (1) (1) (1)

4 2 2 1 1 1 3 3 3 3 3 3 4 4 2 3 3 1

s t

2 3 2

Collect the flow of this phase.

(1) (1) (1) (1)

4 2 2 1 1 1 3 3 3 3 3 3 4 4 2 3 3 1

s t

2 3 2

Collect the flow of this phase.

(1) (1) (2) (2) (1) (1)

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s t

2 3 2

Collect the flow of this phase.

(1) (1) (2) (2) (1) (1)

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s t

2 3 2

Collect the flow of this phase.

(1) (1) (2) (2) (1) (1)

4 2 2 1 1 1 3 3 3 3 3 3 4 4 2 3 3 1

s t

2 3 2

Collect the flow of this phase.

(1) (1) (2) (3) (1) (1) (1) (1) (1)

4 2 2 1 1 1 3 3 3 3 3 3 4 4 2 3 3 1

s t

2 3 2

Collect the flow of this phase.

(1) (1) (2) (3) (1) (1) (1) (1) (1)

4 2 2 1 1 1 3 3 3 3 3 3 4 4 2 3 3 1

s t

2 3 2

Collect the flow of this phase.

(1) (1) (2) (3) (1) (1) (1) (1) (1)

4 2 2 1 1 1 3 3 3 3 3 3 4 4 2 3 3 1

s t

2 3 2

Collect the flow of this phase.

(1) (1) (2) (3) (1) (1) (1) (1) (2) (1) (1) (1)

4 2 2 1 1 1 3 3 3 3 3 3 4 4 2 3 3 1

s t

2 3 2

Collect the flow of this phase.

(1) (1) (2) (3) (1) (1) (1) (1) (2) (1) (1) (1)

4 2 2 1 1 1 3 3 3 3 3 3 4 4 2 3 3 1

s t

2 3 2

Collect the flow of this phase.

(1) (1) (2) (3) (1) (1) (1) (1) (2) (1) (1) (1)

Observation: Only edges on a shortest s-t-path carry flow.

4 2 2 1 1 1 3 3 3 3 3 3 4 4 2 3 3 1

s t

2 3 2

Collect the flow of this phase.

(1) (1) (2) (3) (1) (1) (1) (1) (2) (1) (1) (1)

Observation: Only edges on a shortest s-t-path carry flow. Build residual network (and introduce back edges)!

2 1 1 1 2 2 2 3 2 4 4 2 3 3 1

s t

2 3 2

Collect the flow of this phase.

1 1 2 1 1 1 1 2 1 1 1

Observation: Only edges on a shortest s-t-path carry flow. Build residual network (and introduce back edges)!

4 2 2 1 1 1 3 3 3 3 3 3 4 4 2 3 3 1

s t

2 3 2

4 2 2 1 1 1 3 3 3 3 3 3 4 4 2 3 3 1

s t

2 3 2

dist(s, t) = 4 in the original network.

2 1 1 1 2 2 2 3 2 4 4 2 3 3 1

s t

2 3 2 1 1 2 1 1 1 1 2 1 1 1

dist(s, t) = 4 in the original network.

2 1 1 1 2 2 2 3 2 4 4 2 3 3 1

s t

2 3 2 1 1 2 1 1 1 1 2 1 1 1

dist(s, t) = 4 in the original network. Back edges go backwards. Not part of any length-4-path.

2 1 1 1 2 2 2 3 2 4 4 2 3 3 1

s t

2 3 2 1 1 2 1 1 1 1 2 1 1 1

dist(s, t) = 4 in the original network. Back edges go backwards. Not part of any length-4-path. dist(s, t) > 4 in the residual network.

Runtime Analysis

dist(s, t) increases in every phase.

dist(s, t) increases in every phase. At most n − 2 phases.

dist(s, t) increases in every phase. At most n − 2 phases. Every phase performs a couple of depth-first searches.

dist(s, t) increases in every phase. At most n − 2 phases. Every phase performs a couple of depth-first searches. How many?

dist(s, t) increases in every phase. At most n − 2 phases. Every phase performs a couple of depth-first searches. How many? How much time does each take?

• Claim. Each phase performs at most m depth-first searches.

• Claim. Each phase performs at most m depth-first searches.

Proof.

• Claim. Each phase performs at most m depth-first searches.

Proof. There are at most m edges in the network.

• Claim. Each phase performs at most m depth-first searches.

Proof. There are at most m edges in the network. After every depth-first search we delete at least

• ne edge.

• Claim. Each phase performs at most m depth-first searches.

Proof. There are at most m edges in the network. After every depth-first search we delete at least

• ne edge.

• Observation. The ith depth-first search takes time O(n + di),

where di is the number of edges deleted as dead ends.

• Observation. The ith depth-first search takes time O(n + di),

where di is the number of edges deleted as dead ends. Thus, the total running time of all (at most m) depth-first searches is:

• Observation. The ith depth-first search takes time O(n + di),

where di is the number of edges deleted as dead ends. Thus, the total running time of all (at most m) depth-first searches is: O (m

i=1(n + di))

• Observation. The ith depth-first search takes time O(n + di),

where di is the number of edges deleted as dead ends. Thus, the total running time of all (at most m) depth-first searches is: O (m

i=1(n + di)) = O (nm + m i=1 di)

• Observation. The ith depth-first search takes time O(n + di),

where di is the number of edges deleted as dead ends. Thus, the total running time of all (at most m) depth-first searches is: O (m

i=1(n + di)) = O (nm + m i=1 di)

= O (nm + m)

• Observation. The ith depth-first search takes time O(n + di),

where di is the number of edges deleted as dead ends. Thus, the total running time of all (at most m) depth-first searches is: O (m

i=1(n + di)) = O (nm + m i=1 di)

= O (nm + m) = O (nm)

• Observation. The ith depth-first search takes time O(n + di),

where di is the number of edges deleted as dead ends. Thus, the total running time of all (at most m) depth-first searches is: O (m

i=1(n + di)) = O (nm + m i=1 di)

= O (nm + m) = O (nm) Thus, each phase takes O(nm) steps.

• Observation. The ith depth-first search takes time O(n + di),

where di is the number of edges deleted as dead ends. Thus, the total running time of all (at most m) depth-first searches is: O (m

i=1(n + di)) = O (nm + m i=1 di)

= O (nm + m) = O (nm) Thus, each phase takes O(nm) steps. Dinic’ algorithm performs at most n − 2 phases.

• Observation. The ith depth-first search takes time O(n + di),

where di is the number of edges deleted as dead ends. Thus, the total running time of all (at most m) depth-first searches is: O (m

i=1(n + di)) = O (nm + m i=1 di)

= O (nm + m) = O (nm) Thus, each phase takes O(nm) steps. Dinic’ algorithm performs at most n − 2 phases. Dinic’ algorithm performs O(n2m) steps.

• Theorem. Dinic’ algorithm finds a maximum flow in

O(n2m) steps.

Part II

Dinic’ Algorithm in Unit Capacity Networks

s t All capacities are 1

s t All capacities are 1 k := dist(s, t). Here, k = 4.

s t All capacities are 1 Ignore edges not on a shortest s-t-path. k := dist(s, t). Here, k = 4.

s t All capacities are 1 Ignore edges not on a shortest s-t-path. k := dist(s, t). Here, k = 4.

s t All capacities are 1 Ignore edges not on a shortest s-t-path. k := dist(s, t). Here, k = 4.

s t Find s-t-path by depth-first search.

s t Find s-t-path by depth-first search.

s t Find s-t-path by depth-first search.

s t Find s-t-path by depth-first search.

s t Find s-t-path by depth-first search.

s t Find s-t-path by depth-first search. Route 1 unit of flow.

s t Find s-t-path by depth-first search. Route 1 unit of flow. Update capacities.

s t Find s-t-path by depth-first search. Route 1 unit of flow. Update capacities.

s t Find s-t-path by depth-first search. Route 1 unit of flow. Update capacities. This deletes k edges!

• Observation. Let k = dist(s, t). Every capacity update

removes k edges.

• Observation. Let k = dist(s, t). Every capacity update

removes k edges. Thus, each phase performs at most m/k depth first searches.

• Observation. Let k = dist(s, t). Every capacity update

removes k edges. Thus, each phase performs at most m/k depth first searches. Every depth-first search takes time O(k + di), where di is the number of edges deleted as dead ends.

• Observation. Let k = dist(s, t). Every capacity update

removes k edges. Thus, each phase performs at most m/k depth first searches. Every depth-first search takes time O(k + di), where di is the number of edges deleted as dead ends. Number of steps performed in phase stage is: O m/k

i=1 (k + di)

• Observation. Let k = dist(s, t). Every capacity update

removes k edges. Thus, each phase performs at most m/k depth first searches. Every depth-first search takes time O(k + di), where di is the number of edges deleted as dead ends. Number of steps performed in phase stage is: O m/k

i=1 (k + di)

• = O
• m + m/k

i=1 di

• Observation. Let k = dist(s, t). Every capacity update

removes k edges. Thus, each phase performs at most m/k depth first searches. Every depth-first search takes time O(k + di), where di is the number of edges deleted as dead ends. Number of steps performed in phase stage is: O m/k

i=1 (k + di)

• = O
• m + m/k

i=1 di

• = O(m).

• Observation. Let k = dist(s, t). Every capacity update

removes k edges. Thus, each phase performs at most m/k depth first searches. Every depth-first search takes time O(k + di), where di is the number of edges deleted as dead ends. Number of steps performed in phase stage is: O m/k

i=1 (k + di)

• = O
• m + m/k

i=1 di

• = O(m).

There are at most n − 2 phases.

• Theorem. On a network in which all capacities are 1,

Dinic’ algorithm finds a maximum flow in O(nm) steps.

• Theorem. On a network in which all capacities are 1,

Dinic’ algorithm finds a maximum flow in O(nm) steps. even better: Lemma 1. On a network with unit capacities, Dinic’ algorithm performs at most √m phases, and thus O(m1.5) steps.

• Theorem. On a network in which all capacities are 1,

Dinic’ algorithm finds a maximum flow in O(nm) steps. even better: Lemma 1. On a network with unit capacities, Dinic’ algorithm performs at most √m phases, and thus O(m1.5) steps. Lemma 2. On a network with unit capacities, Dinic’ algorithm performs at most O

• n2/3

phases, and thus O(n2/3m) steps.

Part III

Number of Phases in Unit Capacity Networks

Lemma 1. On a network with unit capacities, Dinic’ algorithm performs at most 2√m phases, and thus O(m1.5) steps.

Lemma 1. On a network with unit capacities, Dinic’ algorithm performs at most 2√m phases, and thus O(m1.5) steps.

• Proof. Let t ∈ N, to be determined later.

Lemma 1. On a network with unit capacities, Dinic’ algorithm performs at most 2√m phases, and thus O(m1.5) steps.

• Proof. Let t ∈ N, to be determined later.

Consider Gt, the residual network after t phases.

Lemma 1. On a network with unit capacities, Dinic’ algorithm performs at most 2√m phases, and thus O(m1.5) steps.

• Proof. Let t ∈ N, to be determined later.

Consider Gt, the residual network after t phases. k := dist(s, t) ≥ t in Gt.

Lemma 1. On a network with unit capacities, Dinic’ algorithm performs at most 2√m phases, and thus O(m1.5) steps.

• Proof. Let t ∈ N, to be determined later.

Consider Gt, the residual network after t phases. k := dist(s, t) ≥ t in Gt. Vi := {v | dist(s, v) = i in Gt}.

Lemma 1. On a network with unit capacities, Dinic’ algorithm performs at most 2√m phases, and thus O(m1.5) steps.

• Proof. Let t ∈ N, to be determined later.

Consider Gt, the residual network after t phases. k := dist(s, t) ≥ t in Gt. Vi := {v | dist(s, v) = i in Gt}. s t . . . . . . . . . . . . V0 V1 Vi Vi+1 Vk+1 Vk

Vi := {v | dist(s, v) = i in Gt}. s t . . . . . . . . . . . . V0 V1 Vi Vi+1 Vk+1 Vk

Vi := {v | dist(s, v) = i in Gt}. s t . . . . . . . . . . . . V0 V1 Vi Vi+1 Vk+1 Vk Ei := {(u, v) ∈ E | u ∈ Vi, v ∈ Vi+1}

Vi := {v | dist(s, v) = i in Gt}. s t . . . . . . . . . . . . V0 V1 Vi Vi+1 Vk+1 Vk Ei := {(u, v) ∈ E | u ∈ Vi, v ∈ Vi+1}

Vi := {v | dist(s, v) = i in Gt}. s t . . . . . . . . . . . . V0 V1 Vi Vi+1 Vk+1 Vk Ei := {(u, v) ∈ E | u ∈ Vi, v ∈ Vi+1} |E0| + |E1| + . . . + |Ek−1| ≤ m.

Vi := {v | dist(s, v) = i in Gt}. s t . . . . . . . . . . . . V0 V1 Vi Vi+1 Vk+1 Vk Ei := {(u, v) ∈ E | u ∈ Vi, v ∈ Vi+1} |E0| + |E1| + . . . + |Ek−1| ≤ m. |Ei| ≤ m

k ≤ m t for some i.

Vi := {v | dist(s, v) = i in Gt}. s t . . . . . . . . . . . . V0 V1 Vi Vi+1 Vk+1 Vk Ei := {(u, v) ∈ E | u ∈ Vi, v ∈ Vi+1} |E0| + |E1| + . . . + |Ek−1| ≤ m. |Ei| ≤ m

k ≤ m t for some i.

S := V0 ∪ V1 ∪ . . . ∪ Vi. is an s-t-cut!

Vi := {v | dist(s, v) = i in Gt}. s t . . . . . . . . . . . . V0 V1 Vi Vi+1 Vk+1 Vk Ei := {(u, v) ∈ E | u ∈ Vi, v ∈ Vi+1} |E0| + |E1| + . . . + |Ek−1| ≤ m. |Ei| ≤ m

k ≤ m t for some i.

S := V0 ∪ V1 ∪ . . . ∪ Vi. cap(S, V \ S) = |Ei| ≤ m

t .

is an s-t-cut!

Vi := {v | dist(s, v) = i in Gt}. s t . . . . . . . . . . . . V0 V1 Vi Vi+1 Vk+1 Vk Ei := {(u, v) ∈ E | u ∈ Vi, v ∈ Vi+1} |E0| + |E1| + . . . + |Ek−1| ≤ m. |Ei| ≤ m

k ≤ m t for some i.

S := V0 ∪ V1 ∪ . . . ∪ Vi. cap(S, V \ S) = |Ei| ≤ m

t .

maxflow(Gt) ≤ m

t .

is an s-t-cut!

Gt := the residual network after t phases.

Gt := the residual network after t phases. maxflow(Gt) ≤ m

t .

Gt := the residual network after t phases. maxflow(Gt) ≤ m

t .

Every phase increases the flow by at least 1.

Gt := the residual network after t phases. maxflow(Gt) ≤ m

t .

Every phase increases the flow by at least 1. At most t + m

t phases.

Gt := the residual network after t phases. maxflow(Gt) ≤ m

t .

Every phase increases the flow by at least 1. At most t + m

t phases.

At most 2√m phases (setting t = √m).

Gt := the residual network after t phases. maxflow(Gt) ≤ m

t .

Every phase increases the flow by at least 1. At most t + m

t phases.

At most 2√m phases (setting t = √m). Lemma 1. On a network with unit capacities, Dinic’ algorithm performs at most 2√m phases, and thus O(m1.5) steps.

Gt := the residual network after t phases. maxflow(Gt) ≤ m

t .

Every phase increases the flow by at least 1. At most t + m

t phases.

At most 2√m phases (setting t = √m). Lemma 1. On a network with unit capacities, Dinic’ algorithm performs at most 2√m phases, and thus O(m1.5) steps.

Lemma 2. On a network with unit capacities, Dinic’ algorithm performs at most O

• n2/3

phases, and thus O(n2/3m) steps.

Lemma 2. On a network with unit capacities, Dinic’ algorithm performs at most O

• n2/3

phases, and thus O(n2/3m) steps. Proof.

Lemma 2. On a network with unit capacities, Dinic’ algorithm performs at most O

• n2/3

phases, and thus O(n2/3m) steps.

• Proof. Vi := {v | dist(s, v) = i in Gt}.

s t . . . . . . . . . . . . V0 V1 Vi Vi+1 Vk+1 Vk

Lemma 2. On a network with unit capacities, Dinic’ algorithm performs at most O

• n2/3

phases, and thus O(n2/3m) steps.

• Proof. Vi := {v | dist(s, v) = i in Gt}.

s t . . . . . . . . . . . . V0 V1 Vi Vi+1 Vk+1 Vk Vi is big if |Vi| ≥ 2n/k.

Lemma 2. On a network with unit capacities, Dinic’ algorithm performs at most O

• n2/3

phases, and thus O(n2/3m) steps.

• Proof. Vi := {v | dist(s, v) = i in Gt}.

s t . . . . . . . . . . . . V0 V1 Vi Vi+1 Vk+1 Vk Vi is big if |Vi| ≥ 2n/k. At most

n 2n/k = k 2 sets Vi can be big.

s t . . . . . . . . . . . . V0 V1 Vi Vi+1 Vk+1 Vk Vi is big if |Vi| ≥ 2n/k. At most

n 2n/k = k 2 sets Vi can be big.

s t . . . . . . . . . . . . V0 V1 Vi Vi+1 Vk+1 Vk Vi is big if |Vi| ≥ 2n/k. At most

n 2n/k = k 2 sets Vi can be big.

Thus, V0, . . . , Vk contains two consecutive “small” sets Vi, Vi+1. . . . . . . Vi Vi+1

s t . . . . . . . . . . . . V0 V1 Vi Vi+1 Vk+1 Vk Vi is big if |Vi| ≥ 2n/k. At most

n 2n/k = k 2 sets Vi can be big.

Thus, V0, . . . , Vk contains two consecutive “small” sets Vi, Vi+1. |E(Vi, Vi+1)| ≤ |Vi| · |Vi+1| ≤ 4n2

k2 .

. . . . . . Vi Vi+1

s t . . . . . . . . . . . . V0 V1 Vi Vi+1 Vk+1 Vk Vi is big if |Vi| ≥ 2n/k. At most

n 2n/k = k 2 sets Vi can be big.

Thus, V0, . . . , Vk contains two consecutive “small” sets Vi, Vi+1. |E(Vi, Vi+1)| ≤ |Vi| · |Vi+1| ≤ 4n2

k2 .

. . . . . . Vi Vi+1 S := V0 ∪ V1 ∪ . . . Vi is an s-t-cut.

s t . . . . . . . . . . . . V0 V1 Vi Vi+1 Vk+1 Vk Vi is big if |Vi| ≥ 2n/k. At most

n 2n/k = k 2 sets Vi can be big.

Thus, V0, . . . , Vk contains two consecutive “small” sets Vi, Vi+1. |E(Vi, Vi+1)| ≤ |Vi| · |Vi+1| ≤ 4n2

k2 .

. . . . . . Vi Vi+1 S := V0 ∪ V1 ∪ . . . Vi is an s-t-cut. cap(S, V \ S) ≤ 4n2

k2 ≤ 4n2 t2 .

s t . . . . . . . . . . . . V0 V1 Vi Vi+1 Vk+1 Vk Vi is big if |Vi| ≥ 2n/k. At most

n 2n/k = k 2 sets Vi can be big.

Thus, V0, . . . , Vk contains two consecutive “small” sets Vi, Vi+1. |E(Vi, Vi+1)| ≤ |Vi| · |Vi+1| ≤ 4n2

k2 .

. . . . . . Vi Vi+1 S := V0 ∪ V1 ∪ . . . Vi is an s-t-cut. cap(S, V \ S) ≤ 4n2

k2 ≤ 4n2 t2 .

maxflow(Gt) ≤

Gt := the residual network after t phases.

Gt := the residual network after t phases. maxflow(Gt) ≤ 4n2

t2 .

Gt := the residual network after t phases. maxflow(Gt) ≤ 4n2

t2 .

Every phase increases the flow by at least 1.

Gt := the residual network after t phases. maxflow(Gt) ≤ 4n2

t2 .

Every phase increases the flow by at least 1. At most t + 4n2

t2 phases.

Gt := the residual network after t phases. maxflow(Gt) ≤ 4n2

t2 .

Every phase increases the flow by at least 1. At most t + 4n2

t2 phases.

At most 5n2/3 phases (setting t = n2/3).

Gt := the residual network after t phases. maxflow(Gt) ≤ 4n2

t2 .

Every phase increases the flow by at least 1. At most t + 4n2

t2 phases.

At most 5n2/3 phases (setting t = n2/3). Lemma 2. On a network with unit capacities, Dinic’ algorithm performs at most O

• n2/3

phases, and thus O(n2/3m) steps.

Gt := the residual network after t phases. maxflow(Gt) ≤ 4n2

t2 .

Every phase increases the flow by at least 1. At most t + 4n2

t2 phases.

At most 5n2/3 phases (setting t = n2/3). Lemma 2. On a network with unit capacities, Dinic’ algorithm performs at most O

• n2/3

phases, and thus O(n2/3m) steps.

• Theorem. Dinic’ algorithm finds a maximum flow in

O(n2m) steps.

• Theorem. On a network with unit capacities, Dinic’

algorithm finds a maximum flow in O(m1.5) steps.

• Theorem. On a network with unit capacities, Dinic’

algorithm finds a maximum flow in O(n2/3m) steps.