Graph Algorithms Minimum Spanning Tree (Chap 23) Shortest Path (Chap. 24, 25) Maximum Flow (Chap. 26, optional)

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

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, ,….

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

Shortest paths among all pairs— Floyd-Warshall algorithm Given G=(V, E, w) suppose V={1,2,…,n} are vertices and w is weight matrix on edges Consider a subset of vertices {1,2,…, k} for some k, for any pair of i and j, consider all the paths from i to j including only vertices from 1,2,…,to k and let p be one of the minimum such paths (also a simple path).

Floyd-Warshall algorithm (cont.) There are two cases: K is not on path p. In this case, path p will be a minimum path including only vertices of 1,2,…, k-1. k is on path p. In this case, the portion of i to k on path p will be a minimum path from i to k including only vertices 1,2, …, k-1, similarly for the pair of k and j.

Floyd-Warshall algorithm (cont.) Let D(i,j,k) denote the (length of) minimum path between i and j including only vertices 1,2,…, k. D(i,j,n) will be the answer for any pair i and j. Recursive definition: Algorithm: Initialize for k=0 For (k=1 to n) For (i=1 to n) and for (j=1 to n) D(i,j,k)=min{D(i,j,k-1), D(i,k,k-1)+D(k,j,k-1)}

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 : VVR that satisfies: Capacity constraints: for all u,vV, f(u,v) c(u,v). Skew symmetry: for all u,vV, f(u,v)= -f(v,u). Flow conservation: for all uV-{s,t}, vV 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|=vVf(s,v), i.e., the total flow out of the source s. |f|=uVf(u,t), i.e., the total flow into the sink t.

Example of max-flow problem

Ford-Fulkerson method Contains several algorithms: Residue networks Augmenting paths

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)VV: cf(u,v)=c(u,v)-f(u,v), and cf(u,v)>0} a network with left capacity >0, also a flow network.

Residual network and augmenting path

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'|.

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.}

Augmenting path Lemma 26.3 (page 654): Corollary 26.4 (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: VVR 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|.

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|.

Execution of Ford-Fulkerson

An example of loop |f*| times Note: if finding an augmenting path uses breadth-first search, i.e., each augmenting path 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.

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)=

Maximum bipartite matching Matching in a undirected graph G=(V,E) A subset of edges ME, such that for all vertices vV, at most one edge of M is incident on v. Maximum matching M For any matching M′, |M|| M′|. Bipartite: V=LR 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 =LR: V′=V{s,t}, introducing a source and a sink E′={(s,u): uL}  E {(v,t): vR} 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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