Maximum Flow in Planar Networks Alon Itai and Yossi Shiloach 1979, - PowerPoint PPT Presentation

Maximum Flow in Planar Networks Alon Itai and Yossi Shiloach 1979, Society for Industrial and Applied Mathematics October 21, 2019

Planar Graph Problem Definition and Results Terminologies Berge’s Algorithm Modified Capacity Method Finding uppermost paths A validity proof of Algor Schedule Planar Graph 1 Problem Definition and Results 2 Terminologies 3 Berge’s Algorithm 4 Modified Capacity Method 5 Finding uppermost paths 6 A validity proof of Algorithm Uppermost 7 Efficient implementation 8 Analysis of Modified Capacity Algorithm 9 Itai,A.and Shiloach, Y. SIAM J. Comput. Maximum Flow in Planar Networks October 21, 2019 2 / 48

Planar Graph Problem Definition and Results Terminologies Berge’s Algorithm Modified Capacity Method Finding uppermost paths A validity proof of Algor Planar Graph Itai,A.and Shiloach, Y. SIAM J. Comput. Maximum Flow in Planar Networks October 21, 2019 3 / 48

Planar Graph Problem Definition and Results Terminologies Berge’s Algorithm Modified Capacity Method Finding uppermost paths A validity proof of Algor Planar Graph Definition A graph that can be drawn in the plane without any of its edges intersecting is called a planar graph Itai,A.and Shiloach, Y. SIAM J. Comput. Maximum Flow in Planar Networks October 21, 2019 4 / 48

Planar Graph Problem Definition and Results Terminologies Berge’s Algorithm Modified Capacity Method Finding uppermost paths A validity proof of Algor Planar Graph Definition A graph that can be drawn in the plane without any of its edges intersecting is called a planar graph Planar Embedding A particular drawing in the plane so that no two edges intersect. Itai,A.and Shiloach, Y. SIAM J. Comput. Maximum Flow in Planar Networks October 21, 2019 4 / 48

Planar Graph Problem Definition and Results Terminologies Berge’s Algorithm Modified Capacity Method Finding uppermost paths A validity proof of Algor Planar Graph Definition A graph that can be drawn in the plane without any of its edges intersecting is called a planar graph Planar Embedding A particular drawing in the plane so that no two edges intersect. Property 1 A given planar graph can give rise to several different planar embedding. Itai,A.and Shiloach, Y. SIAM J. Comput. Maximum Flow in Planar Networks October 21, 2019 4 / 48

Planar Graph Problem Definition and Results Terminologies Berge’s Algorithm Modified Capacity Method Finding uppermost paths A validity proof of Algor Planar Graph Face The edges and vertices of the graph divide the plane into regions. F 1 F 5 F 2 F 3 F 4 Itai,A.and Shiloach, Y. SIAM J. Comput. Maximum Flow in Planar Networks October 21, 2019 5 / 48

Planar Graph Problem Definition and Results Terminologies Berge’s Algorithm Modified Capacity Method Finding uppermost paths A validity proof of Algor Planar Graph Face The edges and vertices of the graph divide the plane into regions. F 1 F 5 F 2 F 3 F 4 Property 2 For every internal face F of a given planar embedding of a planar graph G , there is another planar embedding of G for which F is exterior. Itai,A.and Shiloach, Y. SIAM J. Comput. Maximum Flow in Planar Networks October 21, 2019 5 / 48

Planar Graph Problem Definition and Results Terminologies Berge’s Algorithm Modified Capacity Method Finding uppermost paths A validity proof of Algor Planar Graph Face The edges and vertices of the graph divide the plane into regions. F 1 F 5 F 2 F 3 F 4 Property 2 For every internal face F of a given planar embedding of a planar graph G , there is another planar embedding of G for which F is exterior. Property 3 If G be a connected planar simple graph with n ≥ 3 vertices and m edges. Then m ≤ 3 n − 6. Itai,A.and Shiloach, Y. SIAM J. Comput. Maximum Flow in Planar Networks October 21, 2019 5 / 48

Planar Graph Problem Definition and Results Terminologies Berge’s Algorithm Modified Capacity Method Finding uppermost paths A validity proof of Algor Problem Definition and Results Itai,A.and Shiloach, Y. SIAM J. Comput. Maximum Flow in Planar Networks October 21, 2019 6 / 48

Planar Graph Problem Definition and Results Terminologies Berge’s Algorithm Modified Capacity Method Finding uppermost paths A validity proof of Algor Max-Flow in Planar Network Problem Given a directed flow network N = ( G , s , t , c ) where G = ( V , E ) is a planar embedding of a directed planar graph , s and t are two distinct vertices of the graph known as source and terminal c : E → R + is a capacity function. The problem is to determine the maximum amount of flow that can be sent from the source node s to the sink node t in such a way that flow satisfies capacity rule and conservation rule . Itai,A.and Shiloach, Y. SIAM J. Comput. Maximum Flow in Planar Networks October 21, 2019 7 / 48

Planar Graph Problem Definition and Results Terminologies Berge’s Algorithm Modified Capacity Method Finding uppermost paths A validity proof of Algor Max-Flow in Planar Network Problem Given a directed flow network N = ( G , s , t , c ) where G = ( V , E ) is a planar embedding of a directed planar graph , s and t are two distinct vertices of the graph known as source and terminal c : E → R + is a capacity function. The problem is to determine the maximum amount of flow that can be sent from the source node s to the sink node t in such a way that flow satisfies capacity rule and conservation rule . 10 1 4 v 1 v 2 3 v 3 1 3 3 2 2 s v 4 t 5 2 2 v 5 Itai,A.and Shiloach, Y. SIAM J. Comput. Maximum Flow in Planar Networks October 21, 2019 7 / 48

Planar Graph Problem Definition and Results Terminologies Berge’s Algorithm Modified Capacity Method Finding uppermost paths A validity proof of Algor Results Past Result Ford and Fulkerson proved Max Flow-Min Cut theorem and established the technique of augmenting paths for finding a maximum flow. All algorithms for finding maximum flow in general graph runs in O ( n 3 ) time for planar graphs. Berge proposed an algorithm to find max-flow in a planar network where s and t are on the same face, whose straight forward implementation requires O ( n 2 ) time . Itai,A.and Shiloach, Y. SIAM J. Comput. Maximum Flow in Planar Networks October 21, 2019 8 / 48

Planar Graph Problem Definition and Results Terminologies Berge’s Algorithm Modified Capacity Method Finding uppermost paths A validity proof of Algor Results Past Result Ford and Fulkerson proved Max Flow-Min Cut theorem and established the technique of augmenting paths for finding a maximum flow. All algorithms for finding maximum flow in general graph runs in O ( n 3 ) time for planar graphs. Berge proposed an algorithm to find max-flow in a planar network where s and t are on the same face, whose straight forward implementation requires O ( n 2 ) time . Overall Idea of the Paper O ( n log n ) implementation of Berge’s algorithm. O ( n log n ) is a lower bound to any implementation of Berge algorithm. Find a flow of value D ≥ 0 in a directed planar network if such a flow exists, otherwise this fact is indicated in O ( n 2 log n ) time. O ( n 2 log n ) algorithm for finding a minimum ( s , t ) cut in an undirected planar network and so is the maximum flow. Itai,A.and Shiloach, Y. SIAM J. Comput. Maximum Flow in Planar Networks October 21, 2019 8 / 48

Planar Graph Problem Definition and Results Terminologies Berge’s Algorithm Modified Capacity Method Finding uppermost paths A validity proof of Algor Terminologies Itai,A.and Shiloach, Y. SIAM J. Comput. Maximum Flow in Planar Networks October 21, 2019 9 / 48

Planar Graph Problem Definition and Results Terminologies Berge’s Algorithm Modified Capacity Method Finding uppermost paths A validity proof of Algor Teminology Data Structure Graph is represented by incidence lists. Each vertex v has a list E v of all the edges to which v is an end point. The set E v is represented by a circular list corresponding to the circular clockwise ordering of the edges around v . Each edge e ∈ E v has a unique successor edge succ v ( e ) in E v . Assumption Graph G has a fixed planar representation. s and t belong to the same face and WLOG, t → s ∈ E . t → s is incident with the exterior face. Itai,A.and Shiloach, Y. SIAM J. Comput. Maximum Flow in Planar Networks October 21, 2019 10 / 48

Planar Graph Problem Definition and Results Terminologies Berge’s Algorithm Modified Capacity Method Finding uppermost paths A validity proof of Algor Uppermost Path s t Itai,A.and Shiloach, Y. SIAM J. Comput. Maximum Flow in Planar Networks October 21, 2019 11 / 48

Planar Graph Problem Definition and Results Terminologies Berge’s Algorithm Modified Capacity Method Finding uppermost paths A validity proof of Algor Uppermost Path s t s t Itai,A.and Shiloach, Y. SIAM J. Comput. Maximum Flow in Planar Networks October 21, 2019 11 / 48

Planar Graph Problem Definition and Results Terminologies Berge’s Algorithm Modified Capacity Method Finding uppermost paths A validity proof of Algor Uppermost Path s t s t s t Itai,A.and Shiloach, Y. SIAM J. Comput. Maximum Flow in Planar Networks October 21, 2019 11 / 48

Planar Graph Problem Definition and Results Terminologies Berge’s Algorithm Modified Capacity Method Finding uppermost paths A validity proof of Algor Berge’s Algorithm Itai,A.and Shiloach, Y. SIAM J. Comput. Maximum Flow in Planar Networks October 21, 2019 12 / 48