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Review of some graph algorithms

• Graph G(V,E) (Chapter 22)– Directed, undirected– Representation

• Adjacency-list, adjacency-matrix

• Breadth-first search (BFS), O(V+E)

• Depth-first search (DFS), O(V+E), recursive– Topological Sort – Strong connected components

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Review of some graph algorithms

• Minimum Spanning Tree (Chapter 23)– Greedy algorithm– Kruskal’s algorithm: O(ElgV)

• Using disjoint set algorithm, similar to connected components– Prim’s algorithm: O(E+VlgV)

• Similar to Dijkstra, using min-priority QUEUE• Single-source shortest paths (Chapter 24)

– Bellman-Ford algorithm: O(VE)• dj=min{dj,dk+w(j,k)} where k is j’s neighbor

– Dijkstra’s algorithm: O(E+VlgV)• Find the closest node n1 which is s’s neighbor, modify other nodes distance• Find the second closest node n2 which is the neighbor of s or n1, modify the

distance• Find the third closest node n3, which is the neighbor of s, n1, or n2, ,….

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Graph algorithms (cont.)

• All-pairs shortest paths (Chapter 25)– Floyd-Warshall algorithms: O(V3). – Dynamic programming, similar to Matrix Chain

Multiplications

– dij(0)=wij

– For k1 to n • for i1 to n

– for j1 to n

» dij(k)=min(dij

(k-1),dik(k-1)+dkj

(k-1))

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Maximum Flow (chap. 26)

• Max-flow problem:– A directed graph G=<V,E>, a capacity function on

each edge c(u,v) 0 and a source s and a sink t. A flow is a function f : VVR that satisfies:

• Capacity constraints: for all u,vV, f(u,v) c(u,v).• Skew symmetry: for all u,vV, f(u,v)= -f(v,u).• Flow conservation: for all uV-{s,t}, vV f(u,v)=0, or to say,

total flow out of a vertex other s or t is 0, or to say, how much comes in, also that much comes out.

– Find a maximum flow from s to t.– Denote the value of f as |f|=vVf(s,v), i.e., the total

flow out of the source s.• |f|=uVf(u,t), i.e., the total flow into the sink t.

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Example of max-flow problem

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Ford-Fulkerson method• Contains several algorithms:

– Residue networks– Augmenting paths

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Residual Networks

• Given a flow network G=<V,E> and a flow f, – the residual network of G induced by f is

Gf=<V,Ef> where Ef={(u,v)VV: cf(u,v)=c(u,v)-f(u,v), and cf(u,v)>0}

– a network with left capacity >0, also a flow network.

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Residual network and augmenting path

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Residual network and flow theorem

• Lemma 26.2 (page 653):– Let G=<V,E> be a flow network with source s and

sink t, and let f be a flow, – Let Gf be the residual network of G induced by f, and

let f' be a flow of Gf. – Define the flow sum: f+f' as: – (f+f')(u.v)=f(u.v)+f'(u.v), then – f+f' is a flow in G with value |f+f'|=|f|+|f'|.

• Proof:– Capacity constraint, skew symmetry, and flow

conservation and finally |f+f'|=|f|+|f'|.

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Augmenting paths• Let G=<V,E> be a flow network with source s and

sink t, and let f be a flow, • An augmenting path p in G is a simple path from s

to t in Gf, the residual network of G induced by f.

• Each edge (u,v) on an augmenting path admits some additional positive flow from u to v without violating the capacity constraint.

• Define residual capacity of p is the maximum amount we can increase the flow:– cf(p)=min{cf(u,v): (u,v) is on p.}

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Augmenting path

• Lemma 26.3 (page 654):– Let G=<V,E> be a flow network with source s and

sink t, let f be a flow, and let p be an augmenting path in Gf. Define fp: VVR by:

• fp(u,v)= cf(p) if (u,v) is on p.• -cf(p) if (v,u) is on p.• 0 otherwise

– Then fp is a flow in Gf with value |fp|=cf(p) >0.

• Corollary 26.4 (page 654):– Define f'=f+fp, then f' is a flow in G with value |f'|=|f|+|

fp|>|f|.

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Basic Ford-Fulkerson algorithm

Running time: if capacities are in irrational numbers, the algorithm may not terminate.Otherwise, O(|E||f*|) where f* is the maximum flow found by the algorithm: while loop runs f* times, increasing f* by one each loop, finding an augmenting path using depth-first search or breadth-first search costs |E|.

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Execution of Ford-Fulkerson

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An example of loop |f*| times

Note: if finding an augmenting path uses breadth-first search, i.e., each augmentingpath is a shortest path from s to t in the residue network, while loop runs at most O(|V||E|) times (in fact, each edge can become critical at most |V|/2-1 times), so the total cost is O(|V||E|2). Called Edmonds-Karp algorithm.

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Network flows with multiple sources and sinks

• Some problems can be reduced to maximum flow problem. Here give two examples.

• Reduce to network flow with single source and single sink

• Introduce a supersource s which is directly connected to each of the original sources si with a capacity c(s,si)=

• Introduce a supersink t which is directly connected from each of the original sinks ti with a capacity c(si,s)=

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Maximum bipartite matching

• Matching in a undirected graph G=(V,E) – A subset of edges ME, such that for all vertices

vV, at most one edge of M is incident on v.• Maximum matching M

– For any matching M′, |M|| M′|.• Bipartite: V=LR where L and R are distinct and

all the edges go between L and R. • Practical application of bipartite matching:

– Matching a set L of machines with a set R of tasks to be executed simultaneously.

– The edge means that a machine can execute a task.

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Finding a maximum bipartite matching

• Construct a flow network G′=(V′,E′,C) from G=(V,E) as follows where =LR: – V′=V{s,t}, introducing a source and a sink– E′={(s,u): uL} E {(v,t): vR}– For each edge, its capacity is unit 1.

• As a result, the maximum flow in G′ is a maximum matching in G.

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