Lecture 4: Linear Programming Computational Geometry Prof. Dr. Th. Ottmann 1 Linear Programming Overview Formulation of the problem and example Incremental, deterministic algorithm Randomized algorithm Unbounded linear programs Linear programming in higher dimensions

Lecture 4: Linear Programming Computational Geometry Prof. Dr. Th. Ottmann 2 Algorithm 2D-LP Input:A 2-Dimensional Linear Program (H, c ) Output: Either one optimal vertex or  or a ray along which (H, c ) is unbounded. if UnboundedLP(H, c ) reports (H, c ) is infeasible then return UnboundedLP(H, c ) elseh 1 := h; h 2 := h´ ; v 2 := l 1  l 2 h 3,...,h n := remaining half-planes of H for i:= 3 to n do if v i-1  h i then v i :=v i-1 else S i-1 := H i-1  * l i v i := 1-dim-LP(S i-1, c ) if v i not exists then return  return v n Running time: O(n²)

Lecture 4: Linear Programming Computational Geometry Prof. Dr. Th. Ottmann 3 New problem Find the point x on l i that maximizes cx, subject to the constraints x  h j, for 1  j  i –1 Observation: l i  h j is a ray Let S i-1 := { h 1  l i,..., h i-1  l i } 1.1-dim-LP{S i-1, c } 2.p 1 = s 1 3.for j := 2 to i-1 do 4. p j = p j-1  s j 5.if p i-1   then 6. return the optimal vertex of p i-1 else 7. return  Time: O(i)

Lecture 4: Linear Programming Computational Geometry Prof. Dr. Th. Ottmann 4 Sequences vivi GoodGood BadBad

Lecture 4: Linear Programming Computational Geometry Prof. Dr. Th. Ottmann 5 Optimal Vertex h1h1 h2h2 h3h3 h4h4 v 3 = v 4 c h1 h1 h2 h2 h3 h3 h4 h4 h5 h5 c v4 v4 v5v5

Lecture 4: Linear Programming Computational Geometry Prof. Dr. Th. Ottmann 6 Algorithm 2D-LP Input:A 2-Dimensional Linear Program (H, C ) Output: Either one optimal vertex or  or a ray along which (H, C ) is unbounded. if UnboundedLP(H, C )  {h, h´} then return UnboundedLP(H, C ) h 1 := h; h 2 := h´ ; v 2 := l 1  l 2 h 3,...,h n := remaining half-planes of H for i:= 3 to n do if v i-1  h i then v i := v i-1 else S i-1 := H i-1  * l i v i := 1-dim-LP(S i-1, C ) if v i does not exist then return  return v n Running time: O(n²)

Lecture 4: Linear Programming Computational Geometry Prof. Dr. Th. Ottmann 7 Sequences vivi GoodGood BadBad

Lecture 4: Linear Programming Computational Geometry Prof. Dr. Th. Ottmann 8 Algorithm 2D-LP Input:A 2-Dimensional Linear Program (H, C ) Output: Either one optimal vertex or  or a ray along which (H, C ) is unbounded. if UnboundedLP(H, C )  {h, h´} then return UnboundedLP(H, C ) h 1 := h; h 2 := h´ ; v 2 := l 1  l 2 h 3,...,h n := remaining half-planes of H compute a random permutation h 3,..., h n for i:= 3 to n do if v i-1  h i then v i :=v i-1 else S i-1 := H i-1  * l i v i := 1-dim-LP(S i-1, C ) if v i does not exist then return  return v n Running time: O(n²)

Lecture 4: Linear Programming Computational Geometry Prof. Dr. Th. Ottmann 9 Randomization Theorem: The 2-dimensional linear programming problem with n constraints can be solved in O(n) randomized expected time using worst-case linear storage.

Lecture 4: Linear Programming Computational Geometry Prof. Dr. Th. Ottmann 10 Random Variable x i X i = E[x i ] is the probability that v i-1  h i