Todays topics More network flow reductions CSE 421 Airplane - PDF document

Today’s topics • More network flow reductions CSE 421 – Airplane scheduling – Image segmentation Algorithms – Baseball elimination Richard Anderson Lecture 27 Network Flow Applications Airplane Scheduling Compatible segments • Given an airline schedule, and starting locations • Segments S 1 and S 2 are compatible if the for the planes, is it possible to use a fixed set of same plane can be used on S 1 and S 2 planes to satisfy the schedule. – End of S 1 equals start of S 2 , and enough time for turn around between arrival and departure • Schedule times – [segments] Departure, arrival pairs (cities and times) – End of S 1 is different from S 2 , but there is • Approach enough time to fly between cities – Construct a circulation problem where paths of flow give segments flown by each plane Graph representation Setting up a flow problem • Each segment, S i , is represented as a pair 1,1 d i a i of vertices (d i , a i , for departure and arrival), with an edge between them. 0,1 a i d j d i a i • Add an edge between a i and d j if S i is 1 -1 P’ i compatible with S j . P i a i d j 1

Result Image Segmentation • Separate foreground • The planes can satisfy the schedule iff from background there is a feasible circulation Image analysis • a i : value of assigning pixel i to the foreground • b i : value of assigning pixel i to the background • p ij : penalty for assigning i to the foreground, j to the background or vice versa • A: foreground, B: background • Q(A,B) = Σ {i in A} a i + Σ {j in B} b j + Σ {(i,j) in E, i in A, j in B} p ij Pixel graph to flow graph Mincut Construction s s a v p vu u v p uv b v t t 2

Baseball elimination Baseball elimination • Can the Dung Beetles W L • Can the Fruit Flies W L win the league? win the league? Ants 4 2 Ants 17 12 • Remaining games: • Remaining games: Bees 4 2 Bees 16 7 – AB, AC, AD, AD, AD, – AC, AD, AD, AD, AF, Cockroaches 3 3 Cockroaches 16 7 BC, BC, BC, BD, CD BC, BC, BC, BC, BC, Dung Beetles 1 5 BD, BE, BE, BE, BE, Dung Beetles 14 13 BF, CE, CE, CE, CF, Earthworms 14 10 CF, DE, DF, EF, EF Fruit Flies 12 15 Assume Fruit Flies win remaining Remaining games games AC, AD, AD, AD, BC, BC, BC, BC, BC, BD, BE, BE, BE, BE, CE, CE, CE, DE • Fruit Flies are tied for first W L place if no team wins Ants 17 13 more than 19 games s • Allowable wins Bees 16 8 – Ants (2) Cockroaches 16 9 – Bees (3) AC AD BC BD BE CE DE – Cockroaches (3) Dung Beetles 14 14 – Dung Beetles (5) Earthworms 14 12 – Earthworms (5) E A B C D • 18 games to play Fruit Flies 19 15 T Network flow applications summary • Bipartite Matching • Disjoint Paths • Airline Scheduling • Survey Design • Baseball Elimination • Project Selection • Image Segmentation 3