1. Linear Programming Computational Geometry, WS 2007/08 Lecture 7, Part II Prof. Dr. Thomas Ottmann Algorithmen & Datenstrukturen, Institut für Informatik Fakultät für Angewandte Wissenschaften Albert-Ludwigs-Universität Freiburg

2. Overview • Problem formulation and example. • Incremental, deterministic algorithm. • Randomized algorithm. • Unbounded linear programs. • Linear programming in higher dimensions. • Half-plane intersection. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

3. Side-Track: Half-Plane Intersection Problem Given a set of n half-planes H = {h1, h2, …, hn}, compute their intersection. • A half-plane hi is a convex set, and hence the intersection of any number of half-planes is • In the worst case, how many sides can bound the intersection of n half-planes? • [Open discussion] How fast can we compute the intersection of the half-spaces? Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

4. Recall: Problem Description Maximize c1x1 + c2x2 + ... + cdxd Subject to the conditions: a1,1x1 + ... a1,dxd b1 a2,1x1 + ... a2,dxd b2 : : : an,1x1 + ... an,dxd bn Linear program of dimension d: c = (c1,c2,...,cd) hi = {(x1,...,xd) ; ai,1x1 + ... + ai,dxd bi} li = hyperplane that bounds hi (straight lines, ifd=2) H = {h1, ... , hn} Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

5. Recall: Geometric Interpretation • Each constraint hi represents a half-space in Rd • The intersection of these half-spaces forms the feasible region • The feasible region is a convex polyhedron in Rd • A convex polyhedron is not necessarily bounded Feasible region Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

6. c Recall: Geometric Interpretation • Given c = (c1, c2, …, cd) • We want to find the optimal vertex vopt of the feasible region, such that c is the outer normal at vopt, if one exists. • Without loss of generality, we assume that the vector c is pointing straight down, and the problem is just finding the lowest point of the feasible region. Feasible region vopt Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

7. c Recall: Degenerate Case • Degenerate case: An LP may have an infinite number of solution. Feasible region f(x,x) = opt Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

8. c Recall: Infeasible • The feasible region can be empty; in this case there is no solution to the LP, and the program is said to be infeasible. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

9. Feasible region c Recall: Unbounded • The feasible region may be unbounded in the direction of c. • Our task is then to determine if the given set H = {h1, h2, …, hn} is unbounded in the direction of the optimization function c. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

10. Recall: Randomized Algorithm 2D-LP Input: A 2-dimensional Linear Program (H, c) Output: Either (i) one optimal vertex, or, (ii) , or (iii) a ray along which (H, c) is unbounded IfUnbounded_LP(H, c) does not report 2 half-planes Then Return the ray along which (H, c) is unbounded Else h1 := h; h2 := h’; v2 := l1 l2; h3, …, hn := the remaining half-planes in H Compute the random permutation of h3, …, hn Fori := 3 to nDo Ifvi-1  hiThenvi := vi-1 ElseSi-1 := Hi-1 * Ii vi := 1_Dim_LP(Si-1, c) Ifvi does not exist ThenReturn  Loop End If Returnvn Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

11. hj hk hi* c Unbounded Linear Programs • Begin by finding li* from the given set H • The ray li* makes the smallest angle i, amongst the half-planes in H, between its outward normal i and c. • The ray li* = hi*, where hi* is the most restrictive half-plane among all the half-planes with the same smallest angle i. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

12. i i hi c Unbounded Linear Programs i : The outward normal of hi i := The smaller angle that i makes with c imin, an index with i min = min j,1  j  n Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

13. c Finding the Test Plane li* -j hj Hmin = {hj H | j = min} Hpar = {hj H | j = -min} hk hi* i i Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

14. c Lemma Consider the set of half-planes H = {h1, h2, …, hn} Assuming that  (Hmin  Hpar) is not empty. • Case 1 If li*  hj is unbounded in the direction c, for  hj  H \ (Hmin  Hpar), then (H, c) is unbounded along a ray contained in li*. li* i* i* Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

15. c Lemma (continued) • Case 2 If li*  hj* is bounded in the direction c, for  hj*  H \ (Hmin  Hpar), then the LP ({hi*, hj*}, c) is bounded. hj* li* i* j* i* Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

16. Algorithm: Unbounded_LP Input: A 2-dimensional Linear Program (H, c) Output: Either (i) a pair of half-planes bounding the LP, or, (ii) , or (iii) a ray along which (H, c) is unbounded For Each half-plane hi H, compute i Let hmin be the half-plane with min; where min = Min(j), for 1  j  n Set Hmin := {hj H | j = min}; Set Hpar := {hj H | j = -min} If Hmin Hpar is empty ThenReturn  Let hi*  Hmin be the half-plane whose line li* bounds Hmin Hpar Let Ĥ = H \ {Hmin Hpar} If  a half-plane hj* H, such that li*  hj* is bounded in c Then Return {hi*, hj*} ElseReturn unbounded along ray li*  ( Ĥ) End If Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

17. The Half-Plane Intersection Problem Given a set of n half-planes H = {h1, h2, …, hn}, compute their intersection. The Divide-and-Conquer Approach • If n = 1 Then • Split the n half-planes of H into H1 and H2 of sizes • Recursively compute the intersection of H1 and H2, and let K1 and K2 be the results. • (Non-trivial) Intersect polygons K1 and K2 into a single convex polygon K, and Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

18. Intersecting Two Convex Polygons Given two convex polygonsK1 and K2, compute K = K1 K2. What information do we already know and can use? • Computing the intersection of line-segments. Run time: • One edge in K1 can intersect at most edges in K2. • Two convex polygons cannot intersect in more than pairs. Based on the above… • The total runtime to compute K = K1 K2: • This can be too slow if we put it into the main Divide-and-Conquer algorithm; which will give • Can we go faster? Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

19. Intersecting Two Convex Polygons Theorem: The intersection of two convex polygons K1 and K2 containing a total of n points can be computed in O(n) time. Approach: Plane-sweep (from left to right) What can we derive from this? • The vertical sweep-line intersects a convex polygon in at most • Thus, the sweep-line • Status-Structure: • Event Q: K2 K1 Sweep direction Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

20. The Half-Plane Intersection Problem Theorem: The intersection of a set H of n half-planes in 2D can be computed in O(n log n) time. From the Divide-and-Conquer algorithm: T(n) = Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann