Krai Mffilflnfi H HHHffifnin 6- 4 - ■ Hk ■ nsj now KHMKl ■ 'V EnS am ■QQJ 51 *#_ LIBRARY OF THE UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN 510.84 l£6r no. 480-489 cop. a. Digitized by the Internet Archive in 2013 http://archive.org/details/approximationmet489maru ^^ i>>ij ^ Bounds Hallways " ^--^ A r A u h K u *V Moser's Hallway of Figure 1 1 2/2 Tf/2+2/TT 2/2* tt/2+2/tt Inverse L and L Hallway, Fig. 5-ia) 1 2/2-1 i/2 2/2~-l tt/2 U- shape Hallway, Fig. 5(h) ttA 2.381 2.321+ 3.03 2.96 S-shape Hallway, Fig. 5(c) tt/U 1.95 1.82 2.U8 2.32 Hallway of Figure 5(d) tt/U 1.82 tt/2 2.32 2.00 A: denotes area A .: Maximum area which can go through without rotation. A : An upper bound of area with rotation. A,: A Lower bound of area with rotation. J K: denotes ratio K : r A /A r r K : u A /A u r K,:: A /A r = 1 -6- or x' « p intersects an edge of y. This implies that at least one of 0' j+1 x^ or xl does not define Q to be angularly simple, which is a contradiction. Y Q.E.D. Lemma 1 implies that if Q is angularly simple at points x., i = 1, 2, ..., m, then fi is angularly simple at any point which is the linear convex sum of the x.'s, i.e. at l m m E co.x. , where E u. = 1. i"l X X i=l 1 Theorem 1: Let fi be an angularly simple polygon. Then the set of points Y which defines Q, to be angularly simple is a convex set. Y Proof: Let us assume that such a set is not convex. Then there exists two points, x. and x^ in the set such that some points on the line segment x.,x^ are not contained in the set. Any such point is obviously defined by the linear convex sum of x and x^. Since fi is angularly simple at j K y x. and x, , by our assumption, any point on such a line segment defines Q j -K- y to be angularly simple by Lemma 1, which is a contradiction. Q.E.D. -13- Hereafter we consider our class of sofas to be angularly simple polygons. Other properties of the angularly simple polygons and the generalization of angularly simple polygons to cover any polygonal regions have been studied by Maruyama [9]. Since any angularly simple polygon, ft , contains a point (or a set of points) x_ such that a vector from x n to a point on y c — v.„ -««,« «. „ww„w. X. ^ ^ Q an trace y in one direction, we can use such a vector sequence to denote angularly simple curves as well. Such a vector sequence is illustrated in Figure 7» where the region covered by the vectors shows the polygon and the curvature formed by connecting the tops of vectors indicates the polygonal Jordan curve. Let us define such a sequence as follows: A pattern sequence , S, is an ordered set of elementary patterns, S = s_, s , s , ..., s , ..., s n _n s.eR where m denotes the dimension of the elementary patterns and n is called the circularity of S. To make S denote an unique polygon we assume that any two adjacent elementary patterns, s. and s. , have an angular difference of 2-rr/n. In practice, we choose n>U8. For our present pur- pose of describing ft it is sufficient to consider that the dimension, Y m, of s . is one, since we assume that each s. denotes the distance between ' 1 ' 1 x n and the intersection point of y and the vector whose direction corres- ponds to 27Ti/n, for i = 0, 1, 2, ..., n-1. When we consider rotations and translations of a polygon whose representation is in basic point coordinates (possibly with line equalities), it is usually necessary to change the point coordinates (and line equalities). However, a transformation of a polygon which is _ll+_ (a) An Angularly Simple Polygon (b) A Pattern Sequence Corresponding to Polygon (a) Figure 7. An Angularly Simple Polygon and Its Pattern Sequence -15- denoted by a pattern sequence, S, is simple; (i) a translation of an angularly simple polygon, Q , corres- Y ponds simply to a translation of x of the pattern sequence, S, and (ii) a rotation of Q at x corresponds to circular shifting of indices of elementary patterns, i.e. it is adequate to consider the rotation index , r, which will be defined later. Before we define the rotational transformation of S, let us define the canonical pattern sequence. A pattern sequence, S, is called a canonical pattern sequence if the first elementary pattern corresponds to the direction of the X-coordinate, and the ordering of the elementary patterns corresponds to the counter-clockwise rotation of corresponding vectors. Henceforth, we assume that each pattern sequence is canonical. This assigns the orientation of the corresponding polygon (without con- fusion, we sometimes use polygon when refering to angularly simple polygons). For the rotation of a pattern sequence, S, at a point x n , it is convenient to assume that the unit rotation angle, 6 , is 2Tr/n (or possibly, an integer multiple of 2-rr/n). For example, if S' v x Q ) - s , s l9 s 2 , ..., B±t ..., s n _ 2 , s n _. then the clockwise rotation of S(x ) through 5 degrees is given by > ± > * 2 9 ■"•» i' •*• ' n-1* and the counter-clockwise rotation of S(x~) through 0 S(x Q ) has been rotated in a clockwise direction through an angle of r degrees. If r<0 , S(x n ) -16- has been rotated in a counter-clockwise direction through an angle of r§ degrees. Thus, in general, we have the following expression for a pattern sequence: r r r+1 r+2 r+i r+n-1 if r + i < then r+i becomes n + r + i (r+i (mod n) for all i.) Thus far we have described rotation of S(x n ) around x~. Rotation around any point is accomplished simply by the change of the rotation index with an appropriate translation of x n of S(x_). To generate a sequence, S, for a specified x_, we project a ray starting from x along each direction 27ri/n, for i = 0, 1, ,.., n-1. Then we measure the distance, s . , by detecting the intersection between the ray and an edge of the given hallway, if any intersection exists within the distance V, called the visibility distance from x». For the representation of hallways we use the usual chain representation, i.e. points and line segments are connected in such a way that the clockwise sequence describes the free space as its right hand side. -17- III. STRATEGY OF THE SOFA PROGRAM AND GSPS The procedure for the solution of the arbitrary two-dimensional sofa problem, which will be described, is intuitive. "It is as simple.- as' the "paper-cut" approach in which one takes a sufficiently large round paper , S (x n ) (r and x n not important), and cuts away the minimum amount of paper necessary to enable the paper to. move through the hallway. In other words one trys to maximize the remaining paper area which can still go through the hallway. To get an idea of this procedure, let us consider an example which is illustrated in Figure 8. Two canonical pattern sequences, S (x ) and S_(x 9 ), denote papers which are possibly maximal at locations x^ and x~ , respectively, where the circularity, n, is 2k. o i — s n ' s i ' s ? ' •••» s -i' •••' s 23 2 : S ' S l* S 2 ' **** S i* ••'» s p3 The paper, S ( w x + (l -w)x ), which can be located at both x and x and whose area becomes maximal, is obtained by intersecting the paper S (x ) and the reoriented paper S~(x ) of S (x ) , namely: S_^(Xp) - s' , s' , s \» s q > •••» s i_3» •••» S 2 20 - s^, s^ 2 , s^, sj, ..., s^_ 3 , ..., s^ Q ■ e" c" q" b" q" s" 1. The term "a paper" is used to reference a two-dimensional object which may or may not go through a given hallway. An edge trimmed paper which can go through a given hallway is called a sofa for the given hallway. -18- ^s§p^ (b) So(wx 1 +(l-w)x 2 ) = S (x 1 )n S_ 3 (x 2 ) Figure 8. Sofas at Point x and x and Their Intersection, S (ojx + (l - u))x p ) -19- Thus S r ( u x 1 + (1 - u)x 2 ) = S ( Xl ) f) S_ 3 (x 2 ) = min(s.,sV) for i = 0, 1, . .., 23 where u) = or 1, and r=0 if w=l and r=-3 if w=0. Let us consider the case where the distance between x and x„, denoted by d(x ,x ) = X (where A is a unit translation distance), is small and the circularity, n, of the pattern sequence is sufficiently large. Then both the unit rotation angle, <5 , and the unit translation distance, A, are sufficiently small. Hence, application of such trans- formations can be thought of as "continuous" transformations of S from x to x p . So, in general, we have: S r (a3X 1 + (1 - o))x 2 ) = S r ( X]L ) S r (x 2 ) where |r. - r | ^1, r = r id + r (l - oo), d(x ,x ) = X and ^ wj^ 1. The above intersection operation may be interpreted as "min" operation and the following tree search strategy may be thought of as the "max" operation. Many studies have been done on both combinatorial and heuristic search algorithms [2, 10, 12, lU, etc.], and a comprehensive survey of them has most recently been done by Pohl [13]. While Howden [6] used a straight -forward exhaustive search strategy for the solution of the sofa problem by computer, we use the following heuristic tree search strategy. -20- Our partial ternary tree (sometimes 5-ary is required depending on the complexity of the given hallway) is developed to a depth of L levels in the following way. Nodes are divided into two classes: active and terminal. Nodes coming from an active node are examined for bounding. This stops further wasteful exploration by using the property that the area of paper is monotonically non-increasing (because of the intersection operation which was defined above). As soon as the paper area becomes smaller than the bound B at a node v , the tree exploration from such a node is terminated. Then the path from such a terminal node to the root node is eliminated (or pruned) from the partial tree which is currently being developed. When the partial search tree is completed to L levels by the repetition of the above generation and pruning, the paper will be moved down in the tree until the paper encounters a node that leads to more than two active nodes. It is possible that the paper cannot be moved down in the tree by the above process. In such a case the paper will be moved one level down the tree in such a way that the next node which has been chosen leads to a better solution. If the developed L level partial search tree has no active nodes to be explored in the next, then our procedure will stop and we conclude that a "sofa" which is larger than the present bound cannot be moved through the given hallway. To expand an active node, v , whose paper orientation is <5r, where 6 = 2ir/n, we attach to v three successor nodes, v' , , v' and v' ' r ' r-1 r r+1 whose orientations correspond to rotation indices r-1, r and r+1, respectively. Here the distance between v and any of v' , V' or v' is the unit transla- r * r-1 r r+1 2 tion distance A. Thus the node v' of the rotation index r means simply the Some successor nodes can have only rotational transformations. -21- unit translation A of a paper at node v in the direction 6r coupled with the intersection operation. The nodes v' , and v* ., indicate the unit r-1 r+1 translation of the paper at v in the directions 6(r-l) and 6(r+l), respec- tively. That is, the former contains the unit angle rotation of the paper in the clockwise direction and the latter rotates the paper through the unit angle in the counter-clockwise direction. Thus by our approach, we treat translation and rotation of a paper simultaneously. This is the major advantage of our representation of the paper. (With a slight change of the above strategy, one can deal with rotation independently.) An example of a U-level ternary search tree with bounding is shown in Figure 9. Those doubly circled nodes are terminal nodes and others are active nodes. Those marked A (x ) and A (x ) are actually active nodes for the expansion of the next search tree whose next root node will be A (x p ). Figure 10 shows the data structure of our tree search approach which corresponds to the example of Figure 9. If the level of the partially developed tree is one, L = 1, then the procedure discussed above is simple a "mini-max" strategy which turns out to be strictly a local optimization. If L >1 then the procedure con- tains some global optimization as well as local optimization. We repeat the above L-level partial search tree generation and pruning process until the paper reaches the other end of the given hallway. Then the resulting paper, S , is stored as the present maximal "sofa" (k') for the given problem as well as the new bound B . A slightly larger (~k) (k) paper than S v , S' , is fed into the hallway next and the paper-cut process is repeated unless there is no gain of the sofa obtained since the previous iteration. Because we use a heuristic search strategy rather than an exhaustive type strategy, the iteration of the paper-cut process as well -22- Level The Root Node of The Current Partial Tree Level 1 Level 2 Level 3 Level 4 The Root Node of The Next Partial Tree Figure 9. Four-level Ternary Search Tree With Bounding: those Doubly Circled Nodes are Terminal Nodes and Others are Active Nodes -23- A (k) / \ >1 A r (x 17 ) , 4 A ( r k i 2 (x 19 ),4 Figure 10. Double Linked List Used for the Search Strategy (the state corresponds to the partial search tree of Fig. 9). -fill- Initialize A Pattern Sequence S: A = 0, A = 0, B = max Generate L-level Tree using Bound B, Each Node is either Terminated or Active yes Advance Paper, prune the Tree and Up-date A and S no no B = A, A = Increment S Figure 11. A Simplified Flow Chart of GSPS -25- as the incrementation of the obtained sofa for the next iteration become 3 quite important in finding an optimal trajectory. From the above argument , we have the simplified flow chart of GSPS (General Sofa Problem Solver) which is illustrated in Figure ll. The following is a description of the flow chart. Initialization of the Pattern Sequence (or Paper) We can choose any one of the following starting papers: (i) A sufficiently large sofa. (ii) The lower or the upper bound sofa. (iii) A sofa which can go through the given hallway without rotation (this can be found easily). (iv) A sufficiently small sofa. Of course the number of iterations required for the convergence to the solution by GSPS depends upon which of the papers we choose as the initial paper, upon the initial bound for the sofa area, and upon the means to increment the sofa for the next iteration (this will be discussed next). To reduce the number of iterations, it is preferable to choose a smaller, lower bound sofa as an initial paper, if such a shape of the bound sofa is easily estimatable. 3. An optimal trajectory T is described by a sequence of pairs of elements, x and. r: T = (x ± ,r ± ), ... , [x ± ,r ± ) 11 mm Or it is determined by a pair of sequences of x and r. k, GSPS will find both an optimal trajectory (or path) and an optimal shape of a sofa. However, if a trajectory is given, then GSPS will find an optimal shape of a sofa for a given hallway. -26- Incrementation of Sofa Let us assume that after the k-th iteration we have a pattern sequence: _(k) . (k) (k) (k) ' ' i ' ' n-1 whose area is denoted "by A : A (k) = sin(2,/n) • (s< k, s (k > + *f .<*>.!*>) / 2 . n-1 . _ i l+l i=0 Then the paper which will be provided for the (.k+ljst iteration is the one (k) whose area is slightly larger that A . For such an incrementation of the sofa, we may consider the following approaches, (i) Equi-increment : V J «- s. ; + e , for all i (e small). s l l (ii) Isomorphic -increment: s!^ k) + c • sf , for all i, c >1. I l " (.iiij Differential-increment: s'< k) * s! k) ♦ c(s< k) - s^" 1 '), for all 1, 0< ■ < 1. II ix — One may consider some other incrementation approaches as well as the mixed approaches of the above three. If we know the lower bound sofa and if we have chosen our starting paper relatively far from the bound sofa, then it seems that the best incrementation approach is to consider the difference between the lower bound sofa and the present paper. However, if the lower bound sofa is unknown or not accurately estimatable, then this approach -27- be used. We will see that any one of the above three can be used satis- factorily, and we will also consider the combination of the above three, namely: .'*> ♦ s<*\l + Cl ) - c^" 1 ' + . . (k) For the new bound B of the (k+l)st tree search interation, the present rv) A , which is not incremented, is used. Paper-cut Method We apply the following conjecture for our paper-cut process. If the given hallway is symmetric, then the solution for the sofa problem with Moser's objective ■ function, i.e. the maximal region which can go through the hallway, is also symmetric. This conjecture gives us a little gimmic to simplify our GSPS and makes it easier to implement as well as enabling a faster convergence of the solution. -28- IV. SOME COMPUTATIONAL RESULTS Solutions for those hallways which are illustrated in Figure 5, as well as a solution to Moser's hallway of Figure 1, by GSPS are illus- trated in Figures 12 through 16. Since the solutions for the sofa problems with Moser's objective function are non-trivial, it may be preferable to indicate the obtained solution as the ratio between the solution area, A, and A , the sofa area which can go through the given hallway without an application of any rotational transformation (this is also the obtained area) . Table II shows the solutions (the unit translation, A, and the unit rotation 6 = 2ir/n) for Moser's sofa problem using Mini-Max strategy (L = l). The table shows that the larger n is, the larger A we get; this agrees with our intuitive knowledge since with larger n we get a more accurate representation of angularly simple polygons by a pattern sequence, especially if some elementary patterns are hug?. Also for larger A we get larger A. This result seems to contradict Howden's statement ([6] p. 300): "indicating that accuracy (of approximately three units) is more dependent on x, the fineness G, than on the size of A6". However, Howden's method and ours are quite different - the sofa will be operated on in such a way so that it can move through the given hallway and will therefore have less constraints from the hallway for larger A. Of course A should be less than a certain amount, e.g. 8/20, otherwise the translation of the sofa becomes so discrete that a solution by our GSPS does not make sense. After testing our GSPS for different L :> 1, we found that L = h is enough for iterating the paper-cut prpcess. Thus we set L = h 9 n = k8, and A = 1/10. A solution for Moser's sofa problem with an equi-incrementation of -29- Table II. Results Obtained by Mini-Max Strategy For Moser's Sofa Problem ^"^■^""^ n X ^^^^ 2k 1+8 1/20 1.1 1.38 2/20 1.35 1.61 U/20 1.38 1.73 6/20 1.75 1.86 Unit translation distance Circularity (the number of elementary patterns). Thus the unit rotation angle is 6 = 2U/n. Computer area/A , A Upper bound = 2.828 Lower bound = 2.207 = 1 -30- feed-back for the sofa is shown in Figure 12. Figure 12(a) shows the shape of the sofa after the first iteration. A = 1.88 which is about 85% of the lower bound, A» = tt/2 + 2/tt, indicated in Table I. After the Uth iteration, the sofa is about 90% of Aj, which is a good approximate solution for the given problem. Figure 13 shows the solution for the hallway of Figure 5(a). After the 3rd iteration we get A 3 " =IT ; = 1.27 - 8l% of the lower bound, Aj = V2. A solution for the hallway of Figure 5(b) is illustrated is Figure lU, and we get about 95% of the lower bound A^ = 2.3^ (2.8 times the area of the sofa which can go through the hallway without rotation). From these results we may conclude that for a smooth hallway GSPS works very well. Figure 15 shows the solution for the hall- way of Figure 5(c) in which we achieved 88% of the lower bound, Aj? = 1.82. Figure l6 shows the solution for the hallway of Figure 5(d) whose lower bound is Ajs = tt/2, and we get 77% of the lower bound. This percentage sounds low, but it is fairly good considering the severe constraints. The shape is, still, quite similar to the lower bound sofa which consists of two connected circles. Solutions for the sofa problems in Figures 5(e) and (f) are not illustrated since their solutions are quite similar to those of Figures 12 and 13, respectively. The ratio between the area obtained and the sofa which can move through the hallway without rotational transformations are different, as are the areas. From these solutions we conclude that GSPS gives a fairly good approximate solution for two-dimensional sofa problems (including an optimal trajectory for such a sofa) with Moser's objective function in a reasonable amount of time. The average run time for a single hallway is 10 to 15 seconds per iteration and about 1.5 times this for a doubly -31- connected hallway. The procedure was written in PL/1 language and implemented on an IBM 360/75 at the University of Illinois. -32- (a) After the First Iteration A (D . K (D . 1#88 (b) After the Fourth Iteration A W = K W - 1.98 Figure 12. A Solution for Moser's Sofa Problem With Moser's Objective Function By GSPS Equi-Increment -33- Figure 13 A Solution for the Hallway of Figure 5(a) With Moser's Objective Function By GSPS; (3) Third Iteration A = Isomorphic Increment (3) (3) After the Third Iteration A VJy =K v '= 1.27 -3U- (a) After the First Iterati on A (l) = 2.09, K (l) = 2.66 (b) After the Second Iteration A (2) = 2.2, K (2) = 2.8 Figure Ik. A Solution for the Hallway of Figure 5(b) With Moser's Objective Function By GSPS Mixed (Equi-Isomorphic) Increment -35- (a) After the First Iteration A^ 1 ' = 1.57, IT 1 ' = 2.0 (b) After the Second Iteration (?) (?) A K ! = 1.60, K v ; = 2.03 Figure 15. A Solution for the Hallway of Figure 5(c. With Moser's Objective Function By GSPS Differential Increment -36- (a) After the First Iteration A (1) = 0.97, K (1) = 1.23 (t>) After the Third Iteration A XDJ = 1.21, K KJJ = 1.5U Figure 16. A Solution for the Hallway of Figure 5(h) With Moser's Objective Function By GSPS Mixed (Equi-Isomorphic) Increment -37- V. CONCLUSION By restricting the class of sofas to a class of angularly simple polygons, we have developed the most easily transformable representation of such a polygon, called a sequential pattern sequence. However, as we have pointed out, the restriction of objects to a class of angularly simple polygons is the strongest restriction and such a polygon may not represent exactly a solution for some sofa problems. Still, the shapes of the sofas obtained are quite similar to those of the lower bound sofas found analytically. Through the restriction of sofas to angularly simple polygons and the heuristic tree search strategy which is applied by the character of non-increasing sofas, we have developed the two-dimensional sofa pro- blem solver, GSPS. As we can see from our computation examples, the system is fast enough to give us "good" approximate, or near optimal, solutions for the sofa problem. A little modification of GSPS leads to the most generalized sofa problem solver, with not only Moser's objective function but with some other predefined objective function, such as Howden's objective function. The idea of angularly simple polygons and their representation leads us not only to the computer solution of the sofa problem but also to the solution of the two-dimensional hiden line problem, the path finding problem [ti] in a geometrically constrained space with limited sight, the dynamic sofa problem L9] s and form perception in psychology [l], among others. We feel positive that there are more applications of this class of objects as well as applications of the generalized angularly simple polygons [9] with a combination of artificial intelligence., Finally, we feel that "analog" approaches rather than numerical approaches for solving problems gives some clues to solving other problems by computer. -38- REFERENCES [l] F. Attneave and M. D. Arnoult, "The Quantitative Study of Shape and Pattern Perception", Psychological Bulletin , Vol. 53, No. 6, November 1956, PP. 1*52-1*71. [2] J. E. Doran and D. Michie, "Experiments with Graph Traverser Programs", Proceedings of the Royal Society (A) , Vol. 29^, September 1966, PP. 235-259. [3] H. Freeman, "On the Encoding of Arbitrary Geometric Configurations", IRE Transactions on Electronic Computers , Vol. EC-10, June 1961, PP. 260-268. [h] H. Freeman and L. Garder, "A Pictorial Jigsaw Puzzle: The Computer Solution of a Problem in Pattern Recognition", IEEE Transactions on Electronic Computers , Vol. EC-13, April I96U, PP. 118-127. [5] M. Goldberg, "A Solution of Problem 66-11: Moving Furniture Through a Hallway", SIAM Review , Vol. 11, No. 1, January 1969 , PP. 118-127 • [6] W. E. Howden, "The Sofa Problem", Computer Journal , Vol. 11, No. 3, November 1968, PP. 299-301. [7] A. Kaufmann, Graphs, Dynamic Programming and Finite Games , Academic Press, New York, 1967. [8] K. Maruyama, "A Procedure for Detecting Intersections and Its Application", Department of Computer Science, University of Illinois, Urbana, Illinois Report No. UU9, May 1971. [9] K. Maruyama, unpublished paper, May 1971. [10] D. Michie, "Strategy-Building with Graph Traversers", Machine Intelligence, Vol. 1 , N. L. Collins and D. Michie (Eds.), 1967, PP. 3-15. [ll] L. Moser, "Problem 66-11: Moving Furniture Through a Hallway", SIAM Review , Vol. 8, No. 3, July 1966, PP. 38l. [12] A. Newell and G. Ernst, "The Search for Generality", Proceedings of the IFIP Congress , 1965, PP. 17-22. " ~"~" [13] I. Pohl, "Bi-Directional and Heuristic Search in Path Problems", SLAC Report, No. 10 h , May 1969. [ik] I. Scoins, "Linear Graphs and Trees", Machine Intelligence, Vol. 1, N. L. Collins and D. Michie (Eds.), 1967 , PP. 3-15. — [15] J. Sebastian, "A Solution of Problem 66-11: Moving Furniture Through a Hallway", SIAM Review , Vol. 12, 1970, PP. 582-586. [16] I. E. Sutherland, "A Method for Solving Arbitrary Wall Mazes by Computer", IEEE Transactions on Computers , Vol. C-l8, No. 12, December 1969 , PP. 1092-1097. Form AEC-427 (6/68) AECM 3201 U.S. ATOMIC ENERGY COMMISSION UNIVERSITY -TYPE CONTRACTOR'S RECOMMENDATION FOR DISPOSITION OF SCIENTIFIC AND TECHNICAL DOCUMENT ( See Instructions on Reverie Side ) 1. AEC REPORT NO. COO-2118-0026 UIUCDCS-R-71- 1 +89 2. TITLE AN APPROXIMATION METHOD FOR SOLVING THE SOFA PROBLEM 3. TYPE OF DOCUMENT (Cheek one): a- Scientific and technical report 1 I b. Conference paper not to be published in a journal: t Title of conference Date of conference Exact location of conference Sponsoring organization □ c. Other (Specify) 4. RECOMMENDED ANNOUNCEMENT AND DISTRIBUTION (Check one): La a. AEC's normal announcement and distribution procedures may be followed. I | b. Make available only within AEC and to AEC contractors and other U.S. Government agencies and their contractors. "2 c. Make no announcement or distribution. 5. REASON FOR RECOMMENDED RESTRICTIONS: SUBMITTED BY: NAME AND POSITION (Please print or type) Kiyoshi Maruyama Research Assistant Organization Department of Computer Science University of Illinois Urbana t Illinois 6l801 Signature Date December IT, 1971 FOR AEC USE ONLY 7. AEC CONTRACT ADMINISTRATOR'S COMMENTS, IF ANY, ON ABOVE ANNOUNCEMENT AND DISTRIBUTION RECOMMENDATION: 8. PATENT CLEARANCE: I I a. AEC patent clearance has been granted by responsible AEC patent group. I I b. Report has been sent to responsible AEC patent group for clearance. I I c. Patent clearance not required. 1. Report No. UUO-211ti-002b UIUCDCS-R-71-1+89 BIBLIOGRAPHIC DATA SHEET 4. Title and Subtitle AN APPROXIMATION METHOD FOR SOLVING THE SOFA PROBLEM 3. Recipient's Accession No. 5. Report Date Oct. 1971 7. Author(s) Kiyoshi Maruyama 8« Performing Organization Rept. No. 9. Performing Organization Name and Address Department of Computer Science University of Illinois at Urbana-Champaign Urbana, Illinois 618OI 10. Project/Task/Work Unit No. Illiac IV 11. Contract/Grant No. AT (11-1) -2118 12. Sponsoring Organization Name and Address U.S. Atomic Energy Commission 13. Type of Report & Period Covered March 1971-June 1971 14. 15. Supplementary Notes 16. Abstracts A procedure for the solution of the two-dimensional sofa problem is described. A new class of polygons, angularly simple polygons, is defined as a class of permissible sofas. The pattern representation, S (x ), developed for this class of polygons has the advantage of allowing easy polygonal transformations. The procedure called GSPS, described herein, gives a good approximate solution to the sofa problem in reasonable time. Slight modification of the procedure leads to an algorithm for the solution of the general sofa problem. 17. Key Words and Document Analysis. 17a. Descriptors Sofa problem, hallway, objective function, polygonal Jordan curve, polygon, angularly simple, pattern sequence. 17b. Identifiers/Open-Ended Terms 17c. COSATI Field/Group 18. Availability Statement Releasable to the public 19. Security Class (This Report) UNCLASSIFIED 20. Security Class (This Page UNCLASSIFIED 21- No. of Pages kk 22. Price FORM NTIS-35 (10-70) USCOMM-DC 40329-P7I UNIVERSITY OF ILLINOIS-URBANA UN.'vEnS'TYOF'LL.NO.S-URBANA S10 84 IL6H no. C002 no 480-486(1971 5 8 ¥ nch*onU.».ons> S tem1o,Bt-«.vUion 3 0112 088400061 u • m ■i*"i-\ *W-: ■ •■t>A ■ •ij t I I • I .'*# £»! */- KlfRHKfHBBH I . » Jt j I