Push-Relabel Algorithm for Maximum Flow HK Journalists Gehua Qin, - PowerPoint PPT Presentation

Introduction Proof of Correctness Analysis of Running time Push-Relabel Algorithm for Maximum Flow HK Journalists Gehua Qin, Chenlin Liu, Yihui Qian Algorithms run as fast as HK Journalists! November, 29, 2016 HK Journalists Gehua Qin, Chenlin Liu, Yihui Qian Push-Relabel Algorithm

Introduction Proof of Correctness Analysis of Running time Outline Introduction 1 Basic Features Basic Idea Proof of Correctness 2 Properties of Height Proof of Correctness Analysis of Running time 3 Analysis of the push-relabel method HK Journalists Gehua Qin, Chenlin Liu, Yihui Qian Push-Relabel Algorithm

Introduction Basic Features Proof of Correctness Basic Idea Analysis of Running time Outline Introduction 1 Basic Features Basic Idea Proof of Correctness 2 Properties of Height Proof of Correctness Analysis of Running time 3 Analysis of the push-relabel method HK Journalists Gehua Qin, Chenlin Liu, Yihui Qian Push-Relabel Algorithm

Introduction Basic Features Proof of Correctness Basic Idea Analysis of Running time Features a fairly di ff erent approach to finding maximum flows. Invented by Andrew Goldberg and Bob Tarjan in 1980s. Running time is O ( V 2 E ) , compared with O ( VE 2 ) for Edmonds-Karp’s Algorithm Do not maintain the flow-conservation — preflow HK Journalists Gehua Qin, Chenlin Liu, Yihui Qian Push-Relabel Algorithm

Introduction Basic Features Proof of Correctness Basic Idea Analysis of Running time Preflow Flow-conservation: X X u ∈ V − { s } , f ( v , u ) = f ( u , v ) v ∈ V v ∈ V HK Journalists Gehua Qin, Chenlin Liu, Yihui Qian Push-Relabel Algorithm

Introduction Basic Features Proof of Correctness Basic Idea Analysis of Running time Preflow In push-relabel algorithm: Flow-conservation: X X u ∈ V − { s } , f ( v , u ) ≥ f ( u , v ) v ∈ V v ∈ V HK Journalists Gehua Qin, Chenlin Liu, Yihui Qian Push-Relabel Algorithm

Introduction Basic Features Proof of Correctness Basic Idea Analysis of Running time Preflow Definition Excess flow: e ( u ) = P v ∈ V f ( v , u ) − P v ∈ V f ( u , v ) Definition If u ∈ V − { s } , e ( u ) > 0, then we call u is overflowing HK Journalists Gehua Qin, Chenlin Liu, Yihui Qian Push-Relabel Algorithm

Introduction Basic Features Proof of Correctness Basic Idea Analysis of Running time Outline Introduction 1 Basic Features Basic Idea Proof of Correctness 2 Properties of Height Proof of Correctness Analysis of Running time 3 Analysis of the push-relabel method HK Journalists Gehua Qin, Chenlin Liu, Yihui Qian Push-Relabel Algorithm

Introduction Basic Features Proof of Correctness Basic Idea Analysis of Running time Example HK Journalists Gehua Qin, Chenlin Liu, Yihui Qian Push-Relabel Algorithm

Introduction Basic Features Proof of Correctness Basic Idea Analysis of Running time Example HK Journalists Gehua Qin, Chenlin Liu, Yihui Qian Push-Relabel Algorithm

Introduction Basic Features Proof of Correctness Basic Idea Analysis of Running time Example HK Journalists Gehua Qin, Chenlin Liu, Yihui Qian Push-Relabel Algorithm

Introduction Basic Features Proof of Correctness Basic Idea Analysis of Running time Example HK Journalists Gehua Qin, Chenlin Liu, Yihui Qian Push-Relabel Algorithm

Introduction Basic Features Proof of Correctness Basic Idea Analysis of Running time Process Definition ∀ u ∈ V , u . h = the height of u . HK Journalists Gehua Qin, Chenlin Liu, Yihui Qian Push-Relabel Algorithm

Introduction Basic Features Proof of Correctness Basic Idea Analysis of Running time Initialization Initialization During the whole process, s . h ≡ | V | , t . h ≡ 0 In the beginning, ∀ u ∈ V − { s , t } , u . h = 0 HK Journalists Gehua Qin, Chenlin Liu, Yihui Qian Push-Relabel Algorithm

Introduction Basic Features Proof of Correctness Basic Idea Analysis of Running time Initialization HK Journalists Gehua Qin, Chenlin Liu, Yihui Qian Push-Relabel Algorithm

Introduction Basic Features Proof of Correctness Basic Idea Analysis of Running time Initialization Initialization During the whole process, s . h ≡ | V | , t . h ≡ 0 In the beginning, ∀ u ∈ V − { s , t } , u . h = 0 Initialize the preflow f as: ( c(u,v) if u = s ( u , v ) . f = 0 other HK Journalists Gehua Qin, Chenlin Liu, Yihui Qian Push-Relabel Algorithm

Introduction Basic Features Proof of Correctness Basic Idea Analysis of Running time Initialization HK Journalists Gehua Qin, Chenlin Liu, Yihui Qian Push-Relabel Algorithm

Introduction Basic Features Proof of Correctness Basic Idea Analysis of Running time NOTE!!! The whole process is within the residual network . HK Journalists Gehua Qin, Chenlin Liu, Yihui Qian Push-Relabel Algorithm

Introduction Basic Features Proof of Correctness Basic Idea Analysis of Running time NOTE!!! HK Journalists Gehua Qin, Chenlin Liu, Yihui Qian Push-Relabel Algorithm

Introduction Basic Features Proof of Correctness Basic Idea Analysis of Running time Process Relabel(u) Applies when: u is overflowing and ∀ v ∈ V , such that ( u , v ) ∈ E f , we have u . h ≤ v . h . Action: Increase the height of u . u . h = 1 + min { v . h : ( u , v ) ∈ E f } HK Journalists Gehua Qin, Chenlin Liu, Yihui Qian Push-Relabel Algorithm

Introduction Basic Features Proof of Correctness Basic Idea Analysis of Running time Relabel HK Journalists Gehua Qin, Chenlin Liu, Yihui Qian Push-Relabel Algorithm

Introduction Basic Features Proof of Correctness Basic Idea Analysis of Running time Relabel HK Journalists Gehua Qin, Chenlin Liu, Yihui Qian Push-Relabel Algorithm

Introduction Basic Features Proof of Correctness Basic Idea Analysis of Running time Relabel HK Journalists Gehua Qin, Chenlin Liu, Yihui Qian Push-Relabel Algorithm

Introduction Basic Features Proof of Correctness Basic Idea Analysis of Running time Relabel HK Journalists Gehua Qin, Chenlin Liu, Yihui Qian Push-Relabel Algorithm

Introduction Basic Features Proof of Correctness Basic Idea Analysis of Running time Relabel HK Journalists Gehua Qin, Chenlin Liu, Yihui Qian Push-Relabel Algorithm

Introduction Basic Features Proof of Correctness Basic Idea Analysis of Running time Relabel HK Journalists Gehua Qin, Chenlin Liu, Yihui Qian Push-Relabel Algorithm

Introduction Basic Features Proof of Correctness Basic Idea Analysis of Running time Process HK Journalists Gehua Qin, Chenlin Liu, Yihui Qian Push-Relabel Algorithm

Introduction Basic Features Proof of Correctness Basic Idea Analysis of Running time Process HK Journalists Gehua Qin, Chenlin Liu, Yihui Qian Push-Relabel Algorithm

Introduction Basic Features Proof of Correctness Basic Idea Analysis of Running time Process Push(u,v) Applies when: u is overflowing, c f ( u , v ) > 0, and u . h = v . h + 1. Action: Push ∆ f ( u , v ) = min ( u . e , c f ( u , v )) units of flow from u to v . ∆ f ( u , v ) = min ( u . e , c f ( u , v )) if ( ( u , v ) ∈ E ( u , v ) . f = ( u , v ) . f + ∆ f ( u , v ) else ( u , v ) . f = ( u , v ) . f − ∆ f ( u , v ) u . e = u . e − ∆ f ( u , v ) v . e = v . e − ∆ f ( u , v ) HK Journalists Gehua Qin, Chenlin Liu, Yihui Qian Push-Relabel Algorithm

Introduction Basic Features Proof of Correctness Basic Idea Analysis of Running time When we cannot push... HK Journalists Gehua Qin, Chenlin Liu, Yihui Qian Push-Relabel Algorithm

Introduction Basic Features Proof of Correctness Basic Idea Analysis of Running time When we cannot push... HK Journalists Gehua Qin, Chenlin Liu, Yihui Qian Push-Relabel Algorithm

Introduction Basic Features Proof of Correctness Basic Idea Analysis of Running time When we can push!!! HK Journalists Gehua Qin, Chenlin Liu, Yihui Qian Push-Relabel Algorithm

Introduction Basic Features Proof of Correctness Basic Idea Analysis of Running time When we can push!!! HK Journalists Gehua Qin, Chenlin Liu, Yihui Qian Push-Relabel Algorithm

Introduction Basic Features Proof of Correctness Basic Idea Analysis of Running time Process HK Journalists Gehua Qin, Chenlin Liu, Yihui Qian Push-Relabel Algorithm

Introduction Basic Features Proof of Correctness Basic Idea Analysis of Running time Process HK Journalists Gehua Qin, Chenlin Liu, Yihui Qian Push-Relabel Algorithm

Introduction Basic Features Proof of Correctness Basic Idea Analysis of Running time Process HK Journalists Gehua Qin, Chenlin Liu, Yihui Qian Push-Relabel Algorithm

Introduction Basic Features Proof of Correctness Basic Idea Analysis of Running time Process HK Journalists Gehua Qin, Chenlin Liu, Yihui Qian Push-Relabel Algorithm

Introduction Basic Features Proof of Correctness Basic Idea Analysis of Running time Process HK Journalists Gehua Qin, Chenlin Liu, Yihui Qian Push-Relabel Algorithm

Introduction Basic Features Proof of Correctness Basic Idea Analysis of Running time Process HK Journalists Gehua Qin, Chenlin Liu, Yihui Qian Push-Relabel Algorithm

Introduction Basic Features Proof of Correctness Basic Idea Analysis of Running time Process HK Journalists Gehua Qin, Chenlin Liu, Yihui Qian Push-Relabel Algorithm