Lecture 3: Geometric Algorithms(Convex sets, Divide & Conquer Algo.)

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1 Advanced Algorithms Fall 2015 Lecture 3: Geometric Algorithms(Convex sets, Divide & Conuer Algo.) Faculty: K.R. Chowdhary : Professor of CS Disclaimer: These notes have not been subjected to the usual scrutiny reserved for formal ublications. They may be distributed outside this class only with the ermission of the faculty. 3.1 Comutational Geometry Comutational geometry has evolved from the classical disciline of the design and analysis of algorithms. It is concerned with comutational comlexity of geometric roblems that arise in discilines like, attern recognition, comuter grahics, comuter vision, robotics, and very-large scale integrated (VLSI) layout, oeration research, statistics, etc. In contrast to the classical aroach to roving mathematical theorems about geometry-related roblems, this disciline emhasizes the comutational asect of these roblem and attemts to exloit the underlying geometric roerties, e.g., the metric sace, to derive efficient algorithmic solutions. The classes of roblems that we address in this chater include roximity, intersection, searching, oint location, convex set, convex hull, Voronoi diagrams, CAD/CAM, etc. 3.2 Divide and Conuer This is a basic roblem solving techniue and has roven to be very useful for geometric roblems as well. This techniue mainly involves artitioning of the given roblem into several subroblems, recursively solving each subroblem, and then combining the solutions to each of the subroblems to obtain the final solution to the otimal roblem. We demonstrate its working by considering the roblem of comuting the common intersection of n half lanes in the lane. Given is the set S of n half-lanes, h i, reresented by, a i x+b i y c i,i = 1,2,...,n. (3.1) Examle 3.1 Let for i = 1, h 1 is: 5x+4y 20. This gives all oints in half lane as shown in figure 3.1. It is simle to understand that, common intersection of half-lanes, denoted by, C(S) = n h i (3.2) i=1 is a convex set, which may or may not be bounded. If it is bounded, it is convex olygon. In figure 3.2 the shaded area is common intersection Figure 3.1: Half lane 5x+4y 20.

2 3-2 Lecture 3: Geometric Algorithms(Convex sets, Divide & Conuer Algo.) The divide and conuer algorithm comutes the following stes: Algorithm 1 : Algorithm common-intersection-d&c(s); 1: if S 3 then 2: comute the intersection CI(S) exlicitly 3: Return (C(S)) 4: end if 5: Divide S into two arox. eual subsets S 1 and S 2 6: CI(S 1 ) = Common-Intersection-D&C(S 1 ) 7: CI(S 2 ) = Common-Intersection-D&C(S 2 ) 8: CI(S) = Merge (CI(S 1 ), CI(S 2 )) 9: Return (CI(S)) The key ste is the merge of two common intersections. Because CI(S 1 ) and CI(S 2 ) are convex, the merge ste basically calls for the comutation of the intersection of two convex olygons, which can be solved in time roortional to the size of the olygons. The running time of D&C algorithm is O(nlogn) due to following recurrence formula, where n = S. T(3) = O(1) T(n) = 2T ( n 2) +O(n)+M ( n 2, n 2), where,m ( n 2, n 2) = O(n)denotesthemergetime(ste 5). 3.3 Convex Set Figure 3.2: Divide-and-conuer scheme for common intersecting half-lanes(half lanes are on the side of arrows). Definition 3.2 Convex Set. In Euclidean sace, a convex set is the region such that, for every air of oints within the region, every oint on the straight line segment that joins the air of oints is also within the region. Let S be a vector sace over the real numbers R, or, more generally, some ordered field. This includes Euclidean saces. A set C in S is said to be convex if, for all and in C and all α in the interval [0,1], the oint (1 α)+α also belongs to C. In other words, every oint on the line segment connecting and is in C. This imlies that a convex set in a real or comlex toological vector sace is ath-connected, thus connected. Furthermore, C is strictly convex if every oint on the line segment connecting and other than the endoints is inside the interior of C. x y x Figure 3.3: Convex set, Non-convex set. y In geometric terms, a body C in the Euclidean sace is convex if and only if the line segment joining any two oints in C lies totally in C. But, this theorem is not suitable for comutational urose as there are

3 Lecture 3: Geometric Algorithms(Convex sets, Divide & Conuer Algo.) 3-3 infinitely many ossible airs of oints to be considered. However, other roerties of convexity C be utilized to yield an algorithm. For examle, a cube in R 3 is a convex but its boundary is not, for the boundary does not contain segment, unless and lie in the same two-dimensional face of the cube. The imortance of convexity theory stems from the fact that convex sets freuently arise in many areas of mathematics, and are helful in elementary reasoning. Even infinite-dimensional theory is based on 2- and 3-dimensional reasoning. Any two distinct oints and of real vector sace R determine a uniue line. It consists of all oints of the form (1 α) +α, α ranging over all real numbers. Those oints for which α 0 and those for which 0 α 1 form resectively the ray from through and the segment. The convex subsets of R (the set of real numbers) are simly the intervals of R. Some examles of convex subsets of the Euclidean lane are solid regular olygons, solid triangles, and intersections of solid triangles. Some examles of convex subsets of a Euclidean 3-dimensional sace are the Archimedean solids and the Platonic solids. The Keler-Poinsot olyhedra are examles of non-convex sets. Examle 3.3 The figure 3.4 shows a line segment, while 3.4 shows a circle in 2-dimensional convex set. Solution. the case of line segment, let the oints marked on line are 2, 3, 4, 5. Let us call the oints on line as the set C. Hence, (1 α) +α C. Let α = 0.5, = 2, and = 5. We note that (1 0.5) = 3.5 C. Similarly, we can verify it for circle. Let the circle is x 2 + y 2 = 2, i.e., centered at (0, 0), with radius of 1. Let x = 0.5,y = 0.75 is and x = 0.4,y = 0.8 are the oints and. Let α = 0.6. Then first, (1 α) +α = (1 0.6)0.5+( 0.4) 0.6 = Similarly, for y, (1 α) +α = (1 0.6)0.75+( 0.8) 0.6 = Thus, we note that (1 α) +α Figure 3.4: Points ona line segment as a Convex set, Points in a circle as a convex set. = ( 0.04, 0.18) C, which satisfy the criteria of convexity. Like, this it should satisfy for all (x,y) in the set C. r = 1 Definition 3.4 A function is convex if and only if its eigrah, the region (in hashed) above its grah (see fig. 3.5), is a convex set. Proerties of Convex sets: If S is a convex set in n- dimensional sace, then for any collection of r, r > 1, n- dimensional vectors u 1,...,u r in S, and for any nonnegative numbers λ 1,...,λ r such that λ 1 + +λ r = 1, then we have: r λ k u k S. (3.3) k=1 Figure 3.5: Convex function.

4 3-4 Lecture 3: Geometric Algorithms(Convex sets, Divide & Conuer Algo.) A vector of this tye is known as a convex combination of u 1,...,u r Intersections and unions The collection of convex subsets of a vector sace has the following roerties: 1. The emty set and the whole vector-sace are convex. 2. The intersection of any collection of convex sets is convex(figure 3.6). 3. The union of a non-decreasing seuence of convex subsets is a convex set. For the receding roerty of unions of non-decreasing seuences of convex sets, the restriction to nested sets is imortant: The union of two convex sets need not be convex. For examle, the non-decreasing convex sets are C 1 = {a,b}, and C 2 = {a,b,c}. Then, their union, C 1 C 2 = {a,b,c}, is obviously, a convex set. Definition 3.5 Jordan curve theorem. In toology, a Jordan curve is a non-self-intersecting continuous loo in the lane. The Jordan curve theorem asserts that every Jordan curve divides the lane into an interior region bounded by the curve and an exterior region containing all of the nearby and far away exterior oints, so that any continuous ath connecting a oint of one region to a oint of the other region intersects with that loo somewhere. Hence, any two oints, outside a Jordan curve will intersect even number of times, and two ointer,r, one is outside and other inside will intersect the curve odd number of times (see figure 3.7, and 3.7). Consider the following roblem. Given a simle closed Jordan olygon curve, determine if the interior region enclosed by the curve is convex. This roblem can be readily solved solved by observing that if line segment defined by all airs of vertices of the olygon curve, v i,v j,i j,1 i,j n, where n denotes the total number of vertices, lie totally inside the region, then the region is convex. This would yield a straight algorithm with time comlexity O(n 3 ), as there are O(n 2 ) line segments, and to test if each line segment lies totally in the region takes O(n) time by comaring it against every olynomial segment. Outside Region r Inside region r A B A B Figure 3.6: Intersection of Convex sets A and B, 2-dimensional, 2-dimensional, (c) 3-dimensional, are all convex. It may be noted that the interior angle of each vertex must be strictly less than π, in order for the region to be convex. If it is more than π it is reflex. It may be noted that all the vertices must be convex, is the necessar condition for the region to be convex. A B Figure 3.7: Jordan Curve.

5 Lecture 3: Geometric Algorithms(Convex sets, Divide & Conuer Algo.) 3-5 in P, on the boundary, or outside? Is oint inside the olygon The general roblem we d like to solve is, given a oint (x,y) and a olygon P (reresented by its seuence of vertices), is (x,y) Fortunately the roblem has a simle and elegant answer. Just draw a ray (ortion of a line extending infinitely in one direction) down (or in any other direction) from (x, y), count crossings on ray. Every time the olygon crosses this ray, it searates a art of the ray that s inside the olygon from a art that s outside. For this, all we really need to know is whether there s an even or odd number of them. In even crossings, the oint is outside the olygon. Algorithm 2 : function is-oint-inside-olygon(olygon, oint) return inside/outside; 1: for each line segment of the olygon do 2: if ray down from (x,y) crosses segment then 3: crossings++; 4: end if 5: if crossings is odd then 6: return (inside); 7: else 8: return (outside); 9: end if 10: end for References [1] Mikhail J. Atallah & Marina Blanta, Algorithms and Theory of Comutation Handbook, Secial Toics and Techniues, II ed., CRC Press, [2] Allen B. Tucker, Jr., The comuter Science and Engineering Handbook, CRC Press, 1997.