Data Structures & Algorithms Network Flow

Data Structures & Algorithms Network Flow Richard Newman based on book by R. Sedgewick

Network Flow Problems • Weighted, digraph G, or network • May have cost per unit flow for edges • May have maximum flow per edge • May have max production rates • May have required consumption rate per sink

Distribution Problems • Merchandise distribution • Sources with production rates • Sinks with consumption rates • Distribution centers • Input rate = output rate • Channels with maximum rate and unit cost for distribution

Merchandise distribution Factories 0 0 1 1 2 2 2 7 3 3 3 4 4 4 Distribution Centers 5 5 5 6 6 6 Retail Locations 7 7 7 8 8 8 9 9 9 Get product to retail locations cheaply

Transportation Problems • Communications • Max total data rate between a source and sink • Cheapest way to move a given amount of data from s to t • Traffic flow • Minimize evacuation time • Minimize total cost

Transportation Problem 0 0 1 1 2 2 2 3 3 3 4 4 4 Supply 7 Channels Demand 5 5 5 6 6 6 7 7 7 8 8 8 9 9 9 No channel capacity restrictions Get product to retail locations cheaply

Matching Problems • Job placement • Interviews + job offers • Maximize number of placements • Min-distance point matching • Two sets A and B of N points each • Find set of N segments matching an element from A with one from B that has lowest cost

Matching Problem 0 0 1 1 2 2 2 3 3 3 4 4 4 Employer 7 Offers Applicant 5 5 5 6 6 6 7 7 7 8 8 8 9 9 9 Maximize placements (matching)

Cut Problems • Network reliability • What is minimum number of lines that must be cut to disconnect two switches? • Supply line cutting • What is the minimum supply line destruction required to ensure no troops get supplies?

Cut Problem Supplies 0 0 1 1 2 2 2 7 3 3 3 4 4 4 Delivery paths 5 5 5 6 6 6 Troops 7 7 7 8 8 8 9 9 9 How few edges must be cut to disrupt delivery May have edge weights also

Network Flow Problems • Generic problems • Maxflow • What is maximum flow between s and t? • Mincost-flow • What is cheapest cost way to achieve a particular flow?

Network Flow • Flow Networks • Maxflow Algorithms • Maxflow Reductions • Mincost Flows • Network Simplex Algorithm

Flow Networks • Defn 22.1: A network with a single source and a single sink is an s-t network

Flow Networks • Defn 22.2: A flow network is an s-t network with positive edge weights, called capacities. • A flow in a flow network is a set of non-negative edge weights called edge flows satisfying: • No edge flow exceeds capacity • In flow = out flow for interior nodes

Flow Network 3 1 3 1 2 2 1 0 5 3 1 1 Oil Field 3 1 3 Refinery 1 1 2 4 Pipelines and Valves Maximize flow subject to capacity and conservation of flow Is this OK?

Flow Network 1 3 1 3 1 2 2 1 1 0 5 2 1 1 Oil Field 3 1 3 Refinery 1 1 2 4 Pipelines and Valves Can we do better?

Flow Network 1 3 1 3 2 2 2 1 1 1 0 5 2 1 1 Oil Field 3 1 3 Refinery 1 1 2 4 Pipelines and Valves Are we done? Is this OK?

Flow Network 1 3 1 3 2 2 2 1 2 1 0 5 2 1 1 Oil Field 3 2 3 Refinery 1 1 2 4 Pipelines and Valves Now are we done? Yep

Flow Network • Sum of flows into a node is called inflow • Sum of flows out of a node is called outflow • Conservation of flow: except for source and sink, inflow = outflow • Feasible flow = obeys constraints (max flow and conservation of flow)

Flow Network • Set outflow from sink to zero • Set inflow to source to zero • Outflow of source = inflow of sink • This is called network's value

Maximum Flow • Given an s-t network, find a flow such that no other flow from s to t has a larger value. • A flow like this is called a maxflow. • Problem of finding one is called the maxflow problem.

Augmenting Path Maximum Flow • First algorithm like this due to Ford and Fulkerson • Iteratively: • Find a feasible path from s to t • Find the max residual capacity on it • Saturate the path by adding flow along it equal to the minimum residual capacity

Ford-Fulkerson 2 3 1 3 2 2 2 1 2 s 0 5 t 1 New flow 3 3 1 2 4 Find path with capacity left: 0-1-3-5 Find minimum residual capacity: Add flow to saturate: 2 2

Ford-Fulkerson 2 3 Saturated edge, flow 1 3 2 2 2 2 1 s 0 5 t 1 1 Unsaturated flow 3 1 3 1 1 2 4 Find path with capacity left: 0-2-4-5 Find minimum residual capacity: Add flow to saturate: 1 1

Ford-Fulkerson 2 3 1 3 2 2 2 2 1 s 0 5 t 1 1 3 1 3 “Reverse edge” 1 1 2 4 Find path with capacity left: Wait a doggone second... Consider flow in one direction as capacity in reverse direction! There are none!

Ford-Fulkerson 2 3 1 3 2 2 2 2 2 1 s 0 5 t 1 1 3 1 3 “Reverse edge” 1 1 2 4 1 2 (the flow) What is “residual capacity” of 3-1? Find path with capacity left: What is the minimum RC on this path? Add flow to saturate... 0-2-3-1-4-5 1

Ford-Fulkerson 1 3 1 3 2 1 2 1 2 2 1 1 s 0 5 t 2 1 1 3 2 3 1 1 1 2 4 1 There is none. Find path with capacity left: We are done.

Ford-Fulkerson 1 3 1 3 2 1 2 1 2 2 1 1 s 0 5 t 2 1 1 3 2 3 1 1 1 2 4 1 Removing saturated edges partitions the network

Ford-Fulkerson 3 1 3 s 0 5 t 2 3 2 3 2 4 Removing saturated edges partitions the network Source s is in one part, t is in another Note: cut is subset of saturated edges

s-t Cut • Defn. 22.3: An s-t cut is a cut that places node s in one of its sets and node t in the other. Its capacity is the sum of its edge weights. • The flow across an s-t cut is the sum of the flows across its s-t edges, less the sum of the flows across its t-s edges.

MinCut Problem • Given an s-t network, find an s-t cut such that the capacity of no other cut is smaller. We call this a mincut. • Maxflow-mincut theorem – min cut capacity and maxflow value are equal

Residual Network • Given flow network G and a flow F, the residual network R for the flow has the same nodes, and one or two edges in R for each edge in the original: for edge (v,w) in G, let f be the flow and c be the capacity. If f<c, include (v,w) with capacity c-f; if>0 include (w,v) with capacity f.

Residual Network • If flow is 0, then the original edge with original capacity is used • For positive flows, decrease capacity by the flow (unless the residual capacity is zero – the edge is saturated – remove the edge) • For positive flows, add a reverse directed edge with the flow as its capacity (can reduce flow that much).

Flow Network with Flow 2 3 1 3 2 2 2 2 1 2 2 2 Reverse edges s 0 5 t 1 3 3 Flow 1 2 4 Make reverse edges with capacity = flow Decrease capacity by flow on forward edges

Residual Network 1 Saturated edge 1 3 2 0 0 1 2 2 Reverse edges s 0 5 t 1 3 3 Residual capacity 1 2 4 Make reverse edges with capacity = flow Decrease capacity by flow on forward edges Remove edges with zero residual capacity (saturated edges)

Residual Network 1 1 3 2 1 2 2 Reverse edges s 0 5 t 1 3 3 Residual capacity 1 2 4 Residual network remains Note that reverse edge to s or from t don’t help, so can be omitted in practice

Variants on Ford-Fulkerson • Shortest Augmenting Path • Measured by number of edges • Build “layer graph” – like BFS • Which nodes can be reached by 1 edge, then by 2 edges, etc. • Stop when t is reached and add flow to saturate edge on path to t

Variants on Ford-Fulkerson • Maximum Flow Augmenting Path • Measured by max flow along path • Pick edges that give max flow to next layer • Take max of (min flow along path to predecessor u, capacity (u,v)) to find max flow to v through u at that layer

Ford-Fulkerson Complexity • Prop. 22.6: Let M be the maximum edge capacity in G. The number of augmenting paths needed by any implementation of F-F is at most VM. • Every AP adds at least one unit of flow to every cut; any cut has at most V edges; hence the algorithm must terminate after VM passes since any cut must be saturated by then.

Ford-Fulkerson Complexity • Cor: The time required to find a maxflow is O(VEM), which is O(V2M) for sparse networks. • Linear (in edges) graph search per pass. • Need extra lg V factor if using priority queue fringe implementation • Actual performance is quite good

Ford-Fulkerson Complexity • Prop. 22.7: The number of augmenting paths needed in the shortest augmenting path F-F algo is at most VE/2.

Ford-Fulkerson Complexity • The length of the APs monotone non-decreasing. • Every AP has a critical edge that is saturated in its pass. • Each time edge e is the critical edge, the AP must be at least 2 hops longer. • The longest path has < V edges

Ford-Fulkerson Complexity • Prop. 22.7 (again): The number of augmenting paths needed in the shortest augmenting path F-F algo is at most VE/2. • Cor: The time required to find a maxflow in a sparse network is O(V3) • O(E) time per pass, VE/2 passes, and if G is sparse, E is O(V).

Ford-Fulkerson Complexity • Prop. 22.8: The number of augmenting paths needed in the maximal augmenting path F-F algo is at most 2E lg M. • Cor: The time required to find a maxflow in a sparse network is O(V2 lg M lg V)

Network Flow • Flow Networks • Maxflow Algorithms • Maxflow Reductions • Mincost Flows • Network Simplex Algorithm

Maxflow in General Networks • Maximize the total outflow from sources in a network. (Zero by convention if no sources or sinks) • Multiple sources • Multiple sinks • Still need feasible flows!

Maxflow in General Networks • Prop. 22.14: The maxflow in general networks is equivalent to the maxflow problem for general networks. • The general case subsumes the special case of s-t networks. • Add dummy source connected to all sources and dummy sink to all sinks by high-capacity edges

Vertex-capacity Constraints • Given a flow network, find a maxflow satisfying additional constraints that the flow through each node v must not exceed the capacity of that node.

Vertex-capacity Constraints • Prop. 22.15: The maxflow with node capacity problem is equivalent to maxflow problem. • Setting node capacity high subsumes the special case of s-t networks. • Node capacity can be set to max of in-capacity and out-capacity (sum of in- and out-edge capacities, resp.)

Vertex-capacity Constraints • Prop. 22.15: The maxflow with node capacity problem is equivalent to maxflow problem. • Split each node into in-part and out-part, with all in-edge to in-part and all out-edge from out-part • Make edge from in-part to out-part with edge capacity = node capacity in original problem

Data Structures & Algorithms Network Flow