- Email: [email protected]
A comment on a minmax location problem
Operations Research Letters 23 (1998) 41–43
A comment on a minmax location problem Gerhard J. Woeginger ∗;1 Institut fur Mathematik, TU Graz, Steyrergasse 30, A-8010 Graz, Austria Received 3 July 1997; received in revised form 1 January 1998
Abstract In a recent paper Hamacher and Schobel (Oper. Res. Lett. 20 (1997) 165–169) study a minmax location problem in the Euclidean plane that draws its main diculty from the restriction that the new facility must not be placed within a so-called forbidden region. Hamacher and Schobel derive a polynomial time algorithm for this problem that runs in O(I 3 ) time for inputs of size I . In this short note we argue that this location problem can be solved in O(I log I ) time by applying standard techniques from computational geometry. Moreover, by providing a matching lower bound in the algebraic computation tree model of c 1998 Elsevier Science B.V. All computation, we show that the time complexity O(I log I ) is in fact the best possible. rights reserved. Keywords: Location theory; Center problems; Restricted location problems
1. Introduction Hamacher and Schobel [3] investigate the following location problem: The input consists of a set P of n points p1 ; : : : ; pn in the Euclidean plane that describes the places where the already existing facilities are located. Moreover, there is a so-called forbidden region F that is a convex polyhedral region that is bounded by f edges. The goal is to compute a new location z ∗ ∈ R2 \int(F) that minimizes the function max kz ∗ − pi k2 ;
16i6n
(1)
where k · k2 denotes the Euclidean distance. In other words, the goal is to nd a circle with the smallest ∗
E-mail: [email protected]. Supported by the START program Y43-MAT of the Austrian Ministry of Science. 1
possible radius that encloses all points in P and whose center z ∗ is not contained in the forbidden region F. From now on, this problem will be abbreviated by LOCFORB. Hamacher and Schobel [3] derive an O(I 3 ) time algorithm for LOCFORB, where I = n+f denotes the size of the input. For the unrestricted version of this problem (where there is no forbidden region) Megiddo’s celebrated paper [5] contains an O(I ) linear time solution algorithm. In this short note, we rst derive an O(I log I ) time algorithm for LOCFORB. The algorithm is based on straightforward applications of techniques from computational geometry (cf. Section 2). Moreover, we will show that there cannot exist an algorithm with better time complexity by providing a matching lower bound (I log I ) in Ben-Or’s algebraic computation tree model [1], (cf. Section 3).
c 1998 Elsevier Science B.V. All rights reserved. 0167-6377/98/$ – see front matter PII: S 0 1 6 7 - 6 3 7 7 ( 9 8 ) 0 0 0 3 3 - 9
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G.J. Woeginger / Operations Research Letters 23 (1998) 41–43
2. A fast algorithm In our O(I log I ) time algorithm for LOCFORB, we will use three simple ingredients from computational geometry: The rst ingredient is the farthest neighbor Voronoi diagram (see e.g. [7] or [2]), a concept that dates back to the early days of computational geometry. For given points p1 ; : : : ; pn , the farthest neighbor Voronoi diagram is a straight-line subdivision of the Euclidean plane into convex polyhedral cells C1 ; : : : ; Cn where some of these cells may be empty. For every i, 16i 6 n, and for every point x ∈ Ci , max kx − pj k2 = kx − pi k2
16j6n
(2)
holds, i.e. for every x ∈ Ci , point pi is the point in P at farthest distance from x. The farthest neighbor Voronoi diagram of n points can be computed in O(n log n) time [7]. Since its skeleton is a planar graph with O(n) faces, it has only O(n) edges and O(n) vertices. The second ingredient is to determine the intersection of a straight-line segment ab with the boundary of a convex polyhedral region with f edges and boundary vertices q1 ; : : : ; qf . By performing a kind of binary search along the boundary, this intersection is easily computed in O(log f) time. The third ingredient is to compute for a given point p and a given straight-line segment ab that point on ab that is closest to p. It is straightforward to verify that this closest point is either point a or point b or the projection of point p onto the line through ab, and hence it can be computed in O(1) constant time. Now let us return to the location problem LOCFORB. The algorithm is as follows: 1. Apply Megiddo’s algorithm [5] to compute the optimum solution z + of the unrestricted version in O(n) time. If z + ∈= F, then z + also solves the restricted version to optimality and we are done. If z + ∈ F holds, the convexity of the objective function (1) yields that the optimum solution z ∗ of the restricted version lies on the boundary of F. 2. Compute the farthest neighbor Voronoi diagram for p1 ; : : : ; pn in O(n log n) time. 3. Compute the intersection of every edge of the Voronoi diagram with the boundary of the for-
bidden region F. This can be done in O(n log f) time. Since every one of the O(n) edges intersects the boundary in at most two points, there are O(n) intersection points. 4. Sort the intersection points along the boundary of F in O(n log n) time. This yields a partition of the boundary of F by the intersection points into O(n + f) straight-line segments such that every segment is entirely contained in one of the Voronoi cells. 5. For every one of these O(n + f) straight-line segments s proceed as follows: If the segment s is contained in the Voronoi cell Ci , then compute the auxiliary point AUX(s) on the segment that is at minimum distance d(s) from point pi . The solution z ∗ of LOCFORB is one of these auxiliary points AUX(s) and the radius of the enclosing circle equals d(s). Correctness and time complexity of this algorithm are obvious by the above discussion. Theorem 1. There exists an O(n log n + n log f + f) = O(I log I ) time solution algorithm for the minmax location problem LOCFORB. 3. A lower bound In this section, we will derive an (I log I ) lower bound on the time complexity of any solution algorithm for LOCFORB in the algebraic computation tree (ACT). The ACT is one of the standard models of computation in Theoretical Computer Science (cf. e.g. [1,6,8]). The atomic operations that can be performed within one time-unit√ in the ACT are the arithmetic operations +, −, ∗, =, · and branching operations that compare a variable against 0. In the UNIFORM-GAP-ON-A-CIRCLE problem, one is given n points a1 ; : : : ; an on the rst quadrant of the unit circle x2 + y2 = 1 together with a real number ”, 0¡”¡1. (Note that the points a1 ; : : : ; an are not necessarily sorted along the unit-circle.) The question is to decide whether the straight line distance between neighboring points on the unit circle is uniformly equal to ”. Lee and Wu [4] have shown that answering the UNIFORM-GAP-ON-A-CIRCLE problem requires (n log n) worst case time in the ACT model.
G.J. Woeginger / Operations Research Letters 23 (1998) 41–43
We will show that any solution algorithm with running time T (I ) for LOCFORB yields an O(n + T (n)) solution algorithm for UNIFORM-GAP-ON-A-CIRCLE. Clearly, the result of Lee and Wu [4] then yields that T (I ) is (I log I ). Indeed, let us start with an instance a1 ; : : : ; an and ” of UNIFORM-GAP-ON-A-CIRCLE. We proceed as follows. • First, we determine in O(n) time the point amin with smallest x-coordinate in a1 ; : : : ; an . • Starting with b1 = amin , we then compute a sequence of points bi , 1 6 i 6 n, on the rst quadrant of the unit circle such that the distance of bi−1 to bi is ” and such that bi−1 always lies to the left (with respect to the x-coordinate) of bi . • Next, we determine for every i, 1 6 i 6 n, the line ‘i∗ through point bi and the origin. • Point ci , 1 6 i 6 n, is the (unique) intersection point of the line ‘i∗ with the circle x2 + y2 = 4 that lies in the third quadrant. • Finally line ‘i , 1 6 i 6 n, is de ned as the line through ci that is perpendicular to line ‘i∗ . Observe that every single point bi and ci and every single line ‘i∗ and ‘i can be computed in constant time by applying operations allowed in the algebraic computation tree model. Furthermore, observe that the distance from point bi to point ci , 1 6 i 6 n, always equals 3. Moreover, by construction the UNIFORM-GAPON-A-CIRCLE instance has answer YES if and only if {a1 ; : : : ; an } = {b1 ; : : : ; bn } holds. If bn is not in the rst quadrant of the unit circle, then the instance of UNIFORM-GAP-ON-A-CIRCLE has answer NO and we stop. Otherwise, we de ne the following instance of LOCFORB: The set P of already existing facilities is de ned as P = {a1 ; : : : ; an }. The forbidden region F is de ned as the intersection of n + 2 halfplanes that all contain the origin; the rst n of these halfplanes have boundary lines ‘i , the other two halfplanes have boundary lines x = 10 and y = 10, respectively. Since the lines ‘i are sorted by slope, the precise shape and the boundary of F can be computed in O(n) time. This completes the description of the instance of LOCFORB. Claim 2. If the instance of the UNIFORM-GAP-ON-ACIRCLE problem has answer NO; then the constructed instance of LOCFORB has a feasible solution where the objective function takes a value strictly less then 3.
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Proof. Suppose that the UNIFORM-GAP-ON-A-CIRCLE instance has answer NO. Then one of the points bi , say bk , is not contained in the set P = {a1 ; : : : ; an } of facilities. Consider the circle C with center ck and radius 3. Since point ck , the origin, and point bk all three lie on the line ‘k∗ , the circle C contains the entire unit-circle while touching it in point bk . Since bk ∈= P holds, a circle around ck with slightly smaller radius will also yield a feasible solution for LOCFORB. Claim 3. If the instance of the UNIFORM-GAP-ON-ACIRCLE problem has answer YES; then the constructed instance of LOCFORB does not have a feasible solution where the objective function takes a value less then 3. Proof. Suppose that the UNIFORM-GAP-ON-A-CIRCLE instance has answer YES. Then P = {b1 ; : : : ; bn } holds. It is easy to see that the optimum location z ∗ must lie on the boundary of F. If it lies on the line x = 10 or y = 10, then the value of the objective function is ¿ 9. If z ∗ lies on one of the lines ‘i , then the distance of point bi ∈ P to z ∗ is at least three. Theorem 4. In the algebraic computation tree model; every solution algorithm for the minmax location problem LOCFORB has time complexity (I log I ). References [1] M. Ben-Or, Lower bounds for algebraic computation trees, Proc. 15th Annual Symp. on the Theory of Computing, 1983, pp. 80 – 86. [2] H. Edelsbrunner, Algorithms in Combinatorial Geometry, Springer, Berlin, 1987. [3] H.W. Hamacher, A. Schobel, A note on center problems with forbidden polyhedra, Oper. Res. Lett. 20 (1997) 165–169. [4] D.T. Lee, Y.F. Wu, Geometric complexity and some location problems, Algorithmica 1 (1986) 193–211. [5] N. Megiddo, Linear-time algorithms for linear programming in R3 and related problems, SIAM J. Comput. 12 (1983) 759 –776. [6] J. Seiferas, A variant of Ben-Or’s lower bound for algebraic decision trees, Inform. Process. Lett. 26 (1988) 273–276. [7] M.I. Shamos, D. Hoey, Closest-point problems, Proc. 16th IEEE Symp. on Foundations of Computer Science, 1975, pp. 151–162. [8] A.C.C. Yao, Lower bounds for algebraic computation trees with integer inputs, SIAM J. Comput. 20 (1991) 655–668.
A comment on a minmax location problem Gerhard J. Woeginger ∗;1 Institut fur Mathematik, TU Graz, Steyrergasse 30, A-8010 Graz, Austria Received 3 July 1997; received in revised form 1 January 1998
Abstract In a recent paper Hamacher and Schobel (Oper. Res. Lett. 20 (1997) 165–169) study a minmax location problem in the Euclidean plane that draws its main diculty from the restriction that the new facility must not be placed within a so-called forbidden region. Hamacher and Schobel derive a polynomial time algorithm for this problem that runs in O(I 3 ) time for inputs of size I . In this short note we argue that this location problem can be solved in O(I log I ) time by applying standard techniques from computational geometry. Moreover, by providing a matching lower bound in the algebraic computation tree model of c 1998 Elsevier Science B.V. All computation, we show that the time complexity O(I log I ) is in fact the best possible. rights reserved. Keywords: Location theory; Center problems; Restricted location problems
1. Introduction Hamacher and Schobel [3] investigate the following location problem: The input consists of a set P of n points p1 ; : : : ; pn in the Euclidean plane that describes the places where the already existing facilities are located. Moreover, there is a so-called forbidden region F that is a convex polyhedral region that is bounded by f edges. The goal is to compute a new location z ∗ ∈ R2 \int(F) that minimizes the function max kz ∗ − pi k2 ;
16i6n
(1)
where k · k2 denotes the Euclidean distance. In other words, the goal is to nd a circle with the smallest ∗
E-mail: [email protected]. Supported by the START program Y43-MAT of the Austrian Ministry of Science. 1
possible radius that encloses all points in P and whose center z ∗ is not contained in the forbidden region F. From now on, this problem will be abbreviated by LOCFORB. Hamacher and Schobel [3] derive an O(I 3 ) time algorithm for LOCFORB, where I = n+f denotes the size of the input. For the unrestricted version of this problem (where there is no forbidden region) Megiddo’s celebrated paper [5] contains an O(I ) linear time solution algorithm. In this short note, we rst derive an O(I log I ) time algorithm for LOCFORB. The algorithm is based on straightforward applications of techniques from computational geometry (cf. Section 2). Moreover, we will show that there cannot exist an algorithm with better time complexity by providing a matching lower bound (I log I ) in Ben-Or’s algebraic computation tree model [1], (cf. Section 3).
c 1998 Elsevier Science B.V. All rights reserved. 0167-6377/98/$ – see front matter PII: S 0 1 6 7 - 6 3 7 7 ( 9 8 ) 0 0 0 3 3 - 9
42
G.J. Woeginger / Operations Research Letters 23 (1998) 41–43
2. A fast algorithm In our O(I log I ) time algorithm for LOCFORB, we will use three simple ingredients from computational geometry: The rst ingredient is the farthest neighbor Voronoi diagram (see e.g. [7] or [2]), a concept that dates back to the early days of computational geometry. For given points p1 ; : : : ; pn , the farthest neighbor Voronoi diagram is a straight-line subdivision of the Euclidean plane into convex polyhedral cells C1 ; : : : ; Cn where some of these cells may be empty. For every i, 16i 6 n, and for every point x ∈ Ci , max kx − pj k2 = kx − pi k2
16j6n
(2)
holds, i.e. for every x ∈ Ci , point pi is the point in P at farthest distance from x. The farthest neighbor Voronoi diagram of n points can be computed in O(n log n) time [7]. Since its skeleton is a planar graph with O(n) faces, it has only O(n) edges and O(n) vertices. The second ingredient is to determine the intersection of a straight-line segment ab with the boundary of a convex polyhedral region with f edges and boundary vertices q1 ; : : : ; qf . By performing a kind of binary search along the boundary, this intersection is easily computed in O(log f) time. The third ingredient is to compute for a given point p and a given straight-line segment ab that point on ab that is closest to p. It is straightforward to verify that this closest point is either point a or point b or the projection of point p onto the line through ab, and hence it can be computed in O(1) constant time. Now let us return to the location problem LOCFORB. The algorithm is as follows: 1. Apply Megiddo’s algorithm [5] to compute the optimum solution z + of the unrestricted version in O(n) time. If z + ∈= F, then z + also solves the restricted version to optimality and we are done. If z + ∈ F holds, the convexity of the objective function (1) yields that the optimum solution z ∗ of the restricted version lies on the boundary of F. 2. Compute the farthest neighbor Voronoi diagram for p1 ; : : : ; pn in O(n log n) time. 3. Compute the intersection of every edge of the Voronoi diagram with the boundary of the for-
bidden region F. This can be done in O(n log f) time. Since every one of the O(n) edges intersects the boundary in at most two points, there are O(n) intersection points. 4. Sort the intersection points along the boundary of F in O(n log n) time. This yields a partition of the boundary of F by the intersection points into O(n + f) straight-line segments such that every segment is entirely contained in one of the Voronoi cells. 5. For every one of these O(n + f) straight-line segments s proceed as follows: If the segment s is contained in the Voronoi cell Ci , then compute the auxiliary point AUX(s) on the segment that is at minimum distance d(s) from point pi . The solution z ∗ of LOCFORB is one of these auxiliary points AUX(s) and the radius of the enclosing circle equals d(s). Correctness and time complexity of this algorithm are obvious by the above discussion. Theorem 1. There exists an O(n log n + n log f + f) = O(I log I ) time solution algorithm for the minmax location problem LOCFORB. 3. A lower bound In this section, we will derive an (I log I ) lower bound on the time complexity of any solution algorithm for LOCFORB in the algebraic computation tree (ACT). The ACT is one of the standard models of computation in Theoretical Computer Science (cf. e.g. [1,6,8]). The atomic operations that can be performed within one time-unit√ in the ACT are the arithmetic operations +, −, ∗, =, · and branching operations that compare a variable against 0. In the UNIFORM-GAP-ON-A-CIRCLE problem, one is given n points a1 ; : : : ; an on the rst quadrant of the unit circle x2 + y2 = 1 together with a real number ”, 0¡”¡1. (Note that the points a1 ; : : : ; an are not necessarily sorted along the unit-circle.) The question is to decide whether the straight line distance between neighboring points on the unit circle is uniformly equal to ”. Lee and Wu [4] have shown that answering the UNIFORM-GAP-ON-A-CIRCLE problem requires (n log n) worst case time in the ACT model.
G.J. Woeginger / Operations Research Letters 23 (1998) 41–43
We will show that any solution algorithm with running time T (I ) for LOCFORB yields an O(n + T (n)) solution algorithm for UNIFORM-GAP-ON-A-CIRCLE. Clearly, the result of Lee and Wu [4] then yields that T (I ) is (I log I ). Indeed, let us start with an instance a1 ; : : : ; an and ” of UNIFORM-GAP-ON-A-CIRCLE. We proceed as follows. • First, we determine in O(n) time the point amin with smallest x-coordinate in a1 ; : : : ; an . • Starting with b1 = amin , we then compute a sequence of points bi , 1 6 i 6 n, on the rst quadrant of the unit circle such that the distance of bi−1 to bi is ” and such that bi−1 always lies to the left (with respect to the x-coordinate) of bi . • Next, we determine for every i, 1 6 i 6 n, the line ‘i∗ through point bi and the origin. • Point ci , 1 6 i 6 n, is the (unique) intersection point of the line ‘i∗ with the circle x2 + y2 = 4 that lies in the third quadrant. • Finally line ‘i , 1 6 i 6 n, is de ned as the line through ci that is perpendicular to line ‘i∗ . Observe that every single point bi and ci and every single line ‘i∗ and ‘i can be computed in constant time by applying operations allowed in the algebraic computation tree model. Furthermore, observe that the distance from point bi to point ci , 1 6 i 6 n, always equals 3. Moreover, by construction the UNIFORM-GAPON-A-CIRCLE instance has answer YES if and only if {a1 ; : : : ; an } = {b1 ; : : : ; bn } holds. If bn is not in the rst quadrant of the unit circle, then the instance of UNIFORM-GAP-ON-A-CIRCLE has answer NO and we stop. Otherwise, we de ne the following instance of LOCFORB: The set P of already existing facilities is de ned as P = {a1 ; : : : ; an }. The forbidden region F is de ned as the intersection of n + 2 halfplanes that all contain the origin; the rst n of these halfplanes have boundary lines ‘i , the other two halfplanes have boundary lines x = 10 and y = 10, respectively. Since the lines ‘i are sorted by slope, the precise shape and the boundary of F can be computed in O(n) time. This completes the description of the instance of LOCFORB. Claim 2. If the instance of the UNIFORM-GAP-ON-ACIRCLE problem has answer NO; then the constructed instance of LOCFORB has a feasible solution where the objective function takes a value strictly less then 3.
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Proof. Suppose that the UNIFORM-GAP-ON-A-CIRCLE instance has answer NO. Then one of the points bi , say bk , is not contained in the set P = {a1 ; : : : ; an } of facilities. Consider the circle C with center ck and radius 3. Since point ck , the origin, and point bk all three lie on the line ‘k∗ , the circle C contains the entire unit-circle while touching it in point bk . Since bk ∈= P holds, a circle around ck with slightly smaller radius will also yield a feasible solution for LOCFORB. Claim 3. If the instance of the UNIFORM-GAP-ON-ACIRCLE problem has answer YES; then the constructed instance of LOCFORB does not have a feasible solution where the objective function takes a value less then 3. Proof. Suppose that the UNIFORM-GAP-ON-A-CIRCLE instance has answer YES. Then P = {b1 ; : : : ; bn } holds. It is easy to see that the optimum location z ∗ must lie on the boundary of F. If it lies on the line x = 10 or y = 10, then the value of the objective function is ¿ 9. If z ∗ lies on one of the lines ‘i , then the distance of point bi ∈ P to z ∗ is at least three. Theorem 4. In the algebraic computation tree model; every solution algorithm for the minmax location problem LOCFORB has time complexity (I log I ). References [1] M. Ben-Or, Lower bounds for algebraic computation trees, Proc. 15th Annual Symp. on the Theory of Computing, 1983, pp. 80 – 86. [2] H. Edelsbrunner, Algorithms in Combinatorial Geometry, Springer, Berlin, 1987. [3] H.W. Hamacher, A. Schobel, A note on center problems with forbidden polyhedra, Oper. Res. Lett. 20 (1997) 165–169. [4] D.T. Lee, Y.F. Wu, Geometric complexity and some location problems, Algorithmica 1 (1986) 193–211. [5] N. Megiddo, Linear-time algorithms for linear programming in R3 and related problems, SIAM J. Comput. 12 (1983) 759 –776. [6] J. Seiferas, A variant of Ben-Or’s lower bound for algebraic decision trees, Inform. Process. Lett. 26 (1988) 273–276. [7] M.I. Shamos, D. Hoey, Closest-point problems, Proc. 16th IEEE Symp. on Foundations of Computer Science, 1975, pp. 151–162. [8] A.C.C. Yao, Lower bounds for algebraic computation trees with integer inputs, SIAM J. Comput. 20 (1991) 655–668.