GPLAN: Computer-Generated Dimensioned Floorplans for given Adjacencies
Krishnendra Shekhawat, Nitant Upasani, Sumit Bisht, Rahil Jain
GGPLAN: Computer-Generated DimensionedFloorplans for given Adjacencies
Krishnendra Shekhawat, Nitant Upasani, Sumit Bisht, Rahil Jain
Department of Mathematics, BITS Pilani, Pilani Campus, India-333031
12 3 4 578 9 6
Room 1Room 3Room 4Room 5Room 6Room 7Room 8Room 9Room 2 a b c Fig. 1.
The working of GPLAN: Illustrating the input format of GPLAN and outputproduced by it a) Input adjacency graph and dimensional constraints b) Multipledimensionless floorplans satisfying all the adjacency requirements c) A dimensionedfloorplan obtained from the dimensionless floorplans satisfying all given dimensionalconstraints bstract.
In this paper, we present GPLAN, software aimed at con-structing dimensioned floorplan layouts based on graph-theoretical andoptimization techniques. GPLAN takes user requirements as input in thefollowing two forms:i. Adjacency graph: It allows user to draw an adjacency graph on a GUI(graphical user interface) corresponding to which GPLAN producesa set of dimensioned floorplans with a rectangular boundary, whereeach floorplan is topologically distinct from others.ii. Dimensionless layout: Here, user can draw any layout with rectangu-lar or non-rectangular boundary on a GUI and GPLAN transforms itinto a dimensioned floorplan while preserving adjacencies, positions,shapes of the rooms.The above approaches represent different ways of inserting adjacenciesand GPLAN generate dimensioned floorplans corresponding to the givenadjacencies. The larger aim is to provide alternative platforms to userfor producing dimensioned floorplans for all given (architectural) con-straints, which can be further refined by architects.
Keywords:
Algorithm, Adjacency Graph, Floorplans, Graph Theory, LinearOptimization, Layouts.
From 1970s, a lot of research has been done in the domain of computer-aidedarchitectural design, where the prime focus is to automatically generate floor-plan layouts, so that these layouts are regarded as initial layouts by archi-tects/designers and can be further modified and adjusted by them. After a lot ofdevelopments in this direction, there still lies a huge gap between the proposedresearch and its practical aspects. One of the major reason for this gap is theinterdisciplinary nature of the floorplanning problem, i.e., the problem cannotbe handled with a specific approach which may be based on architecture, mathe-matics, artificial intelligence, machine learning, computer science, etc. The largeraim of this work is to bridge this gap by considering each constraint one by oneand by identifying and applying a specific technique based on the nature of asub-problem. In this paper, we attempt to handle those sub-problems which aremathematical in nature.
The automated generation of architectural layouts using graph theory beganwith the generation of rectangular floorplans (RFP). The first attempt in thisdirection was made by Levin [1] in the early 1960s. Then in the coming years,many researchers proposed graph-theoretical approach for the enumeration orconstruction of rectangular layouts [2, 3]. In 1975, the first computer algorithmwas given by Sauda [4] for enumerating all topologically distinct RFPs havingp to eight rooms. In the 1980s, comprehensive studies were presented for theexistence and construction of a rectangular dual (dimensionless RFP) whoseprime focus was VLSI design [5–8]. During this time, Roth et al. [9] and Rinsma[10,11] developed efficient graph algorithms for the construction of dimensionlessand dimensioned RFPs with some restrictions on the input graph.During the early 1990s, researchers realised that there are graphs for whichRFPs do not exist [12,13]. In 1995, Giffin et al. [14] gave a linear time algorithmfor constructing an orthogonal floorplan (OFP) for a given planar triangulatedgraph (PTG) with module area requirements. Using the concept of orderly span-ning trees, in 2003, Liao et al. [15] gave a linear time algorithm for constructing adimensionless OFP for any n -vertex PTG which require fewer module types, i.e.,the algorithm uses only I -modules, L -modules and T -modules, but Z -modulescould not be incorporated. In 2010, Marson et al. [16] restricted their work tosliceable floorplans and generated layouts having aspect ratios close to one, with-out considering the adjacency constraints. In 2011, Jokar and Sangchooli [17]used the concept of face area of a graph for the construction of dimensionlessOFPs for given PTGs. In 2012, Eppstein et al. [18] gave a method for finding anarea universal RFP for the given adjacency requirements whenever such lay-out exists. They also gave the necessary and sufficient condition for an RFP tobe area universal, i.e., an RFP is area universal if and only if it is one-sided.In 2013, Alam et al. [19] gave the construction of an area-universal OFP for agiven PTG. In 2018, Wang et al. [20] presented the automated regeneration ofwell-known existing dimensionless RFPs while considering underlying adjacencygraph of the existing floorplan. The proposed prototype is called GADG (graphapproach to floor plan generation). In the same year, Shekhawat [21] enumer-ated all possible maximal RFPs without considering dimensions of the rooms. In2020, Upasani et al. [22] developed a prototype for generating dimensioned RFPsfor any drawn rectangular arrangement, satisfying adjacency, size and symmet-ric requirements. They do not take adjacency graph as an input. More recently,Wang and Zhang [23] extended GADG [20] for generating dimensioned OFPscorresponding to user-specified design requirements.Shape grammar can be seen as a parallel and an efficient approach for au-tomation where the idea is to generate designs through the execution of shaperules [24]. In 1995, Harada et al. [25] presented an interactive model for generat-ing floorplans using shape grammars. In 2003, Wonka et al. [26] introduced splitgrammars for incorporating flexible design requirements to model buildings us-ing a variety of styles. In 2005, Duarte [27] implemented a shape-grammar basedtechnique to recreate Alvaro Siza’s designs. Recently a lot of work has been doneto generate building models and their 3D representations by extending the useof split grammars [28–30].In the recent times, many new approaches have evolved for supervised au-tomation in floorplan design. In 2010, Merell et al. [31] introduced a supervisedlearning algorithm (based on bayesian networks) for generating residential floor- an RFP is area universal if any assignment of areas to rectangles can be realized bya rectangular module. lans. In this direction, in 2019, Wu et al. [32] also used a data-driven techniquefor constructing interior layouts with fixed outer boundaries. As an extension,Hu et al. [33] presented the generation of floorplans using a graph neural network(GNN) while considering room adjacencies in the form of layout graphs (layoutgraphs enable human users to provide sparse design constraints). This work istrained on the data-set provided by [32].As an alternative approach, in 2013, Rodrigues et al. [34] presented an evo-lutionary strategy for incorporating complex topological and geometric user re-quirements while successfully generating a feasible layout solution, but the im-plementation of the proposed work has not been discussed. In the same year,Bao et al. [35] constructed good floorplan layouts using simulated annealing,which are characterised by parameters like lighting, heating and circulations. In2018, Wu et al. [36] presented a hierarchical framework and used mixed inte-ger quadratic programming for the dimensioning of layouts with fixed exteriorboundaries.Considering the automation where user can insert his choices, in 2019, Nisz-tuk et al. [37] built a tool for automated floorplan generation, covering the ma-jority of adjacency and size constraints, but it is limited to rectangular roomsonly and generates empty spaces in the layouts. Recently, Shi et al. [38] usedreinforcement learning based on a heuristic search technique called Monte CarloTree Search to generate a closest feasible dimensionless RFP corresponding toany adjacency graph inserted by the user.Because of the stochastic nature of algorithms presented in the above papers,their time-complexity is very high and are thus not suitable for complex buildingdesign. Also, most of the above-discussed work is restricted to a single layoutfor the given constraints. Clearly, multiple layouts allow the possibility to build-ing practitioners for analysing, comparing these designs and choosing the mostappropriate one. In 2012, Regateiro et al. [39] produced multiple dimensionedfloorplans without considering adjacency relation using block algebra, but areunable to generate all possible solutions. Also, the proposed work does not givedetails about its implementation. In 2018, Zawidzki [40] first generated a set ofcandidate layouts based on adjacency constraints using a depth-first backtrack-ing search algorithm and then included other customised objectives based onuser satisfaction to give an optimal architectural layout. However, it took eighthours to generate 30 candidate solutions only. Nisztuk et al. [37] also producedmultiple solutions using a greedy approach and thus have very high computa-tional time which clearly shows the efficiency of graph algorithms over greedysearch techniques. The gaps in the existing literature can be listed as follows:i. Dimensionless floorplans (with rectangular boundary): Corresponding togiven adjacencies, there exist linear time algorithms for constructing dimen-sionless rectangular or orthogonal floorplans [8,13,15,38,41], but we did notnd a computer-based tool that can generate a floorplan for any given planartriangulated adjacency graph. In this work, we present a software GPLANthat provides a GUI to the user for drawing an adjacency graph, and thengenerates a floorplan layout while satisfying all the adjacency requirements.It first prefers to generate a rectangular layout if it exists; otherwise, anorthogonal layout is generated. It is also possible to check the existence ofan RFP corresponding to given graph using GPLAN.ii. Dimensioned floorplans: For a given adjacency graph, there are algorithmsfor generating dimensioned RFP [9, 11] and dimensioned OFP [23] but ageneralized optimization technique for producing a feasible floorplan for anygiven dimensions of the rooms is not available. GPLAN provides a GUIwhere user can insert dimensions of all the rooms and using optimizationtechniques, it produces dimensioned floorplans satisfying given adjacenciesas well as dimensions.iii. Multiple floorplans: GPLAN is capable of generating all possible topologi-cally distinct floorplans corresponding to given adjacency constraints, whichis not common in the existing literature. In particular, the generation of mul-tiple OFPs has not been done previously.iv. Time complexity: A lot of existing work is capable of producing residentialbuilding layouts because they have a small number of rooms [31, 36, 37], butfor the complex building structures with a large number of rooms, we needefficient algorithms. GPLAN generates a variety of layouts in a few seconds.v. Irregular floorplans (IFP): Irregular floorplans are floorplans with non-rectangularboundary (see Figure 2c). In the recent times, some work has been done forbuilding IFPs for given adjacencies [32], but it is limited to a small numberof rooms, and proposed algorithms cannot be generalised to any adjacencygraph. In particular, there exist efficient algorithms for constructing dimen-sionless RFPs and OFPs corresponding to given adjacency relations, butthere does not exist such an algorithm for IFPs. Therefore, to produce di-mensioned IFP, GPLAN generates a GUI where a user can draw any dimen-sionless IFP and can give dimensional constraints as input. It then producesa dimensioned IFP while preserving adjacency relations of underlying di-mensionless IFP along with the positions and shapes of the rooms.vi. Re-generation of floorplans: There exists a limited work for the re-generationof floorplans [20,42]. A well-known architectural floorplan F (RFP or OFP)can be re-generated using GPLAN by extracting the underlying adjacencygraph G of F and then by generating a floorplan corresponding to G . Inthis case, GPLAN also produces floorplans which are topologically distinctto F . A well-known IFP can also be reproduced by drawing it on a GUI,but GPLAN does not generate topologically distinct IFPs. In this section, we present a few important terminologies which are used fre-quently in literature and also throughout this paper. efinition 1.
Floorplans.
A floorplan is a partition of a finite-sized polygon P into a finite set of dimensioned polygons { P , P . . . P n } called rooms. Anirregular floorplan (IFP) has non-overlapping rooms with no restrictions on outerboundary P . Orthogonal floorplans (OFPs) are a particular case of IFPs where P is a rectangle. Moreover, when the contained polygons P , P . . . P n are allinternally disjoint rectangles, and the envelope P is convex, the floorplan is saidto be a rectangular floorplan (RFP).Two rooms in a floor plan are adjacent if they share a wall or a section of it,where a wall of a room refers to the edges forming its perimeter. a b c Fig. 2.
Floorplans Typology a) Rectangular floorplan (RFP) b) Orthogonal floorplan(OFP) c) Irregular floorplan (IFP)
Definition 2.
Graphs.
A graph is a set of vertices and edges denoted by G ( n, m ),where n and m denote the number of vertices and edges respectively. A graphis said to be planar if it can be embedded in the plane without crossing ofedges; otherwise, it is a non-planar graph (the graph in Figure 3 is planar whilethe graph in Figure 4a is non-planar). A plane graph is a planar graph with anembedding that divides the plane into connected components called faces/regions(the graph in Figure 3 has 20 internal faces and 1 exterior face).In architectural terms, an adjacency graph is a graph that provides a specificneighbourhood between the given rooms. For each floorplan, there exists a graphknown as the weak dual graph, which can be constructed by replacing each roomwith a vertex and adding an edge to the vertices that correspond to the adjacentrooms (see Figure 2a where red edges show the weak dual graph of the floorplan). Definition 3.
Separating Triangle.
A separating triangle a-b-c is a cycle oflength three in a graph G such that G - { a, b, c } is disconnected. For example,in Figure 4c, the cycle 1-5-3 is a separating triangle because the graph becomesdisconnected on the removal of vertex 6. Definition 4.
Properly Triangulated Planar Graph (PTPG) [5], [7]
A connectedplanar graph is triangulated if all of its faces (except the exterior) are triangular;exterior face can or cannot be a triangle. This graph is called planar triangulatedraph (PTG). The graph in Figure 4c is a PTG. A PTG with no separating tri-angle and with exterior face of the length at least 4 is called properly triangularplanar graph (PTPG). For example, the adjacency graph shown in Figure 3 is aPTPG, whereas the graphs shown in Figure 4 are not PTPG.
62 3451879 10 1112
Fig. 3.
A properly triangulated planar graph (PTPG)
Fig. 4.
Graphs that are not PTPGs a) A non-planar graph b) A non-triangulatedgraph c) A PTG with a separating triangle ( (cid:52)
Definition 5.
Regular Edge Labelling (REL) [41]
A regular edge labelling of aPTPG G having the exterior face of length 4 is a partition of the interior edges of G into two subsets T , T of directed edges such that for each interior vertex u ,the edges incident to u appear in a counterclockwise order around u as follows:a set of edges in T leaving u , a set of edges in T entering u , a set of edges in T entering u and a set of edges in T leaving u .Let N, E, S, W be the four exterior vertices in a clockwise order. All interioredges incident to N are in T and entering N . All interior edges incident to E re in T and entering E . All interior edges incident to S are in T and leaving S . All interior edges incident to W are in T and leaving W . For example, a RELfor the PTPG in Figure 3 is shown in Figure 5. N 1 E2 S3W 4 0 T T Fig. 5.
Regular edge labelling
Definition 6.
Shortcut [7].
A graph G is said to be bi-connected if after deletingany vertex of G , it remains connected, i.e., G has no cut vertices. The graph inFigure 6a is 1-connected having vertices 3 and 4 as cut-vertices, while the graphin Figure 6b is bi-connected.A shortcut in a planar bi-connected graph G is an edge that is incident totwo vertices on the outer boundary of G but is not part of the outer boundary.For example, in Figure 6b, 0-1-2-3-4-5 forms the outer boundary of the graphand edges (1,5) and (2,4) are shortcuts. Definition 7.
Corner implying Path (CIP) [7].
A corner implying path (CIP)in a planar bi-connected graph G is a path u , u , . . . , u n on the outer boundaryof graph G with the property that ( u , u n ) is a shortcut and u , u , . . . , u n − are not the endpoints of any shortcut. For example, in Figure 6b, 1-0-5 is a CIPbecause edge (1,5) is a shortcut and 0 is not an endpoint of any shortcut. This section talks about the working of GPLAN which has been developed inPython for constructing dimensioned floorplans for the given adjacency relations.In the coming subsections, we will talk about the existence and construction offloorplans (RFP and OFP) corresponding to a given adjacency graph. In Section4, we discuss dimensioned irregular floorplans.For GPLAN, the input for adjacencies can be given in the following two ways(refer to Figure 7a):
12 3 4 5 678 a b
Fig. 6.
Biconnectivity, Shortcut and Corner Implying Paths
1. In the form of an adjacency graph, as shown in Figure 7b (construction offloorplans (RFP and OFP) corresponding to given adjacencies is discussedin Section 3),2. In the form of a dimensionless layout, as shown in Figure 7c (construction ofdimensioned IFP corresponding to given adjacencies is discussed in Section4).Other than the adjacency constraints, GPLAN takes dimensional constraints asinput, which is shown in Figure 7d.
A floorplan can be seen as a planar graph whose weak dual graph (see Definition2) is always a PTG (see Definition 4), i.e., for a floorplan to exist correspondingto a given adjacency graph, it must be connected, planar and triangulated. Inthe case of a 4-joint in a floorplan, the weak dual graph has a cycle of length4, but 4-joint is a limiting case of 3-joint as shown in Figure 8. Hence we haverestricted to floorplans having only 3-joints. Since, a weak dual graph is alwaysa PTG, the input graph for GPLAN must be a PTG. In GPLAN, if user insertsa non-planar or a non-triangular graph as an input, then it generates an erroras shown in Figure 9.A PTG can be 1-connected or bi-connected. It can be seen in the literaturethat the construction of floorplans exists for bi-connected PTGs only ( [8, 9, 15,17, 20]) because they may be easy to handle and floorplans corresponding to bi-connected PTGs are comparatively architecturally significant. For a comparison,refer to Figure 10.Hence, we are considering bi-connected PTG as an input for GPLAN, i.e.,it generates an error if the given graph is not bi-connected PTG as shown inFigure 11.
Remark 1.
If a graph is non-planar or non-triangulated, using the existing al-gorithms, it can be made planar [43] (by deleting a few edges) and triangula-tion [44] (by adding a few edges). A 1-connected PTG can be made bi-connected bc d Fig. 7.
GPLAN interface and input constraints ig. 8.
Fig. 9.
GPLAN gave an error if the input graph is non-planar or not triangulated
Fig. 10. sing the biconnectivity algorithm given in [45]. The following algorithm will beincorporated into GPLAN in the near future.
Fig. 11.
Since most of the buildings are rectangular [46], we first prefer to construct a RFPfor the given adjacency graph. If RFP does not exist, an OFP is constructed.A dimensionless RFP is known as a rectangular dual, which only exist foradjacency graphs that are PTPGs (see Definition 4). In 1985, the followingtheorem [6] was proposed.
Theorem 1.
A bi-connected PTPG G has a rectangular dual if and only if ithas no more than four corner implying paths (CIPs).It is clear from Theorem 1 that checking the number of CIPs for a graphhaving a large number of vertices is not an easy calculation by hand. Hence, weincorporated RFPchecker to GPLAN so that user can draw any bi-connectedPTG on the GUI and can check if there exists an RFP for the required graph.It also specifies the reason for the non-existence of an RFP. For an illustration,refer to Figure 12.
From Figure 2a, we can see that the construction of a weak dual graph fromits floorplan is very easy, but the converse is not true, i.e., for a given PTG,constructing its corresponding floorplan automatically requires an efficient algo-rithm and its implementation. We first discuss the construction of an RFP andthen move to the construction of an OFP.To generate RFPs for bi-connected PTPGs, we extend the rectangular dualfinding algorithm proposed by [41] by defining the process of 4-completion usingcorner implying paths (CIPs). ig. 12.
Using RFPchecker, illustrating the cases for which RFPs do not exist ant and He [41] propose transforming a bi-connected PTG into a PTPG(satisfying Theorem 1) by adding four new vertices
N, E, S, W and connectingthem to the exterior boundary of the input graph. To transform a bi-connectedPTG to a PTPG, we select four vertices u , u , u , u on the exterior face in aclockwise order. Let P i ( i = 0 , , ,
3) be the path on the exterior face between u i and u i +1 (if i >
3, then reduce i by 4, i.e., u is same as u ). We connect N, E, S and W to every vertex in P , P , P and P respectively and add four new edges( N, W ) , ( N, E ) , ( S, E ) and (
S, W ) to have the required PTPG (see Figure 13b).The vertices u i ( i = 0 , , ,
3) are selected in such a way that the additionof new edges doesn’t lead to the formation of separating triangles. This can bedone by choosing u i ( i = 0 , , ,
3) as corner vertices which are obtained using thedefinition of CIP, and using the method proposed in [7]. Let the number of CIPsbe k ( k ≤ k pathsand pick additional 4 − k vertices from the outer boundary to obtain the fourcorner vertices. Figure 13a shows an input graph G and its CIPs. We choosevertex 1 , , , N, E, W and S to form a PTPG as shown in Figure 13.Kant and He [41] do not consider the special case when the input graphis a triangle. For this case, we can choose 3 outer edges as P , P , P and P consisting of only one vertex which can be chosen arbitrarily. Figure 14 showsthe 4-completion for a triangle with vertex 1 chosen as a path P .Once we have a bi-connected PTPG, using the algorithms given in [41], wecan construct the corresponding RFP. The major steps involved in the construc-tion of a RFP are illustrated in Figure 15. GPLAN automatically generates aRFP corresponding to any bi-connected PTPG as illustrated in Figure 16. Remark 2.
At this stage, we are not considering the functionality of given spaces,but the user has a choice to insert the room names based on their functions asillustrated in Figure 16.
It is clear from Theorem 1 that there does not exist a RFP corresponding to abi-connected planar triangulated graph G if any of the following holds:i. G has more than four corner implying paths (see Figure 17a),ii. G has separating triangles (see Figure 17b),iii. The exterior face of G is triangular (see Figure 17c).In all these cases, we need to add extra vertices to G and then triangulate G sothat the modified G has at most four corner implying paths, has no separatingtriangles and has exterior face non-triangular. In this case, we obtain an RFPfor modified G and then merge the rooms corresponding to the newly addedvertices to build an OFP for G . For a better understanding refer to Figure 18where the graph in Figure 18a has a separating triangle (cid:52) : 1 u : 3 u : 5u : 7 Corner implying paths:[8,1,2][2,3,4][4,5,6][6,7,8] P : [1,2,3] P : [3,4,5] P : [5,6,7]P : [7,8,9] b N EW S a Fig. 13. a) A PTG with corner implying paths b) A PTPG derived from given PTGby using 4-completion considering u i ( i = 0 , , ,
3) as corner vertices P : [1,2]P : [2,3]P : [3,1]P : [1] Fig. 14.
12 3 45 6789NW S E
12 3 45 6789NS
12 3 45 6789W E
12 3 45 6789NW S E
Given bi-connected PTPG 4-completionRegular edge labeling
N1 5
62 4
783 S9
W 12 5439 678 E21 93 84 75 6Vertical st-graph Horizontal st-graphVertical rectangular dual Horizontal rectangular dualRequired rectangular dual
Fig. 15.
Construction of a RFP corresponding to a given bi-connected PTPG ig. 16.
An RFP generated using GPLAN a b c
Fig. 17.
Input graphs for which RFPs do not exist
436 12 5
43 12 5 a b c d
Fig. 18.
Construction of an OFP corresponding to a given PTG and a new edge has been added to triangulate the modified graph. 18c representsan RFP corresponding to the graph in 18b. Now, in 18c, room 6 is the extraroom which needs to be merged with either room 1 or room 2 to have an OFPas shown in Figure 18d.
Fig. 19.
OFPs for the graphs in Figure 17
The steps for the construction of an OFP for a bi-connected PTG havebeen incorporated into GPLAN. For an illustration, OFPs corresponding to eachgraph in Figure 17 are shown in Figure 19.
Here, the idea is to construct all topologically distinct floorplans correspond-ing to a given PTG. Two floorplans are topologically distinct if they have thesame underlying weak dual graph, but their horizontal and vertical adjacenciesare different. For example, refer to Figure 20 where two topologically distinctfloorplans are illustrated.For a given graph, GPLAN iterates over different possible boundary pathsand finds different RELs possible for that boundary path using the concept
Fig. 20.
Topologically distinct RFPs of flippable item [18] and hence generates all possible RFPs for the obtainedPTPG. For the input graph in Figure 15, GPLAN iterates over all possible 154boundaries and generates 1300 topologically distinct RFPs in 282.45 seconds.A few of topologically distinct RFPs generated using GPLAN are illustrated inFigure 21.For a bi-connected PTG G , GPLAN finds all possible ways to add extravertices to G so that G has at most four corner implying paths (CIPs) and hasno separating triangles (STs). For a graph with k > k C ways to add extra vertices so that the graph has at most 4 CIPs. Similarly, therecan be three possible ways for the removal of a ST from the graph. In this way,GPLAN finds all possible ways to convert a bi-connected PTG to a bi-connectedPTPG for which an RFP exists. For each PTPG thus obtained, it generates allpossible RFPs using the method described in Section 3.3 and for each RFP, itthen merges the extra room to obtain all possible OFPs. For the input graph inFigure 17b, GPLAN generates 256 topologically distinct OFPs in 26.56 seconds,a few of which are illustrated in Figure 22. For incorporating the dimensional constraints into each topological solution, weimplement an algorithm which is based on the network flow model proposed byUpasani et. al [22]. This algorithm is based on an iterative linear optimisationframework employed on the horizontal and vertical st-graphs (obtained duringthe construction of an RFP, Figure 15). However, in this paper, we propose animproved objective function which further minimises the total area of the RFPin comparison to [22]. Against the total width/height optimised in [22], we usethe difference between width/height and their maximum bounds as the objectivefunction, which has given more optimised results in terms of area of the floorplan.The optimisation problem for both the st-graphs is stated as follows: Fig. 21.
Out of 1300 solutions, which can be obtained using GPLAN, a few RFPscorresponding to the input graph in Figure 15
M inimize : (cid:88) w ( e j,i ) − d maxi such that : (cid:88) w ( e ji ) = (cid:88) w ( e ik ) ∀ i ∈ V ( G )min ( d i ) ≤ (cid:88) w ( e ji ) ≤ max ( d i ) ∀ i ∈ V ( G ) (1)where d maxi is the maximum dimension (width/height) of room i , (cid:80) w ( e ji ) de-notes the total inflow and (cid:80) w ( e ik ) denotes the total outflow from vertex i .Figure 23 enlists all equality and inequality constraints, along with the objec-tive functions associated with these st-graphs. Conforming to these constraints,GPLAN optimises width and height separately using the dual-simplex methodto generate a feasible dimensioned floorplan. Higher efficiency of the simplexmethod, when compared to any other stochastic optimisation algorithms, helpsto incorporate dimensions in all topological solutions within a reasonable time. Fig. 22.
Out of 256 solutions, which can be obtained using GPLAN, a few OFPscorresponding to the input graph in Figure 17b
Taking dimensional requirements as input is challenging for orthogonal roomsas the constructs of width and height are difficult to define in this case (see Fig.24b). Thus, for dimensioning, we consider RFP before the merging of extra rooms(as show in Figure 24c). Customizing dimensions for rectangular partitions ofan orthogonal room is convenient, as all such parts are simply integrated afternetwork flow optimisation to yield a dimensioned OFP. A dimensioned OFPgenerated using GPLAN is shown in Figure 24d.
An irregular floorplan (IFP) has been shown in Figure 25a. Construction of anIFP for an adjacency graph is more challenging than the construction of anRFP or an OFP because, in both RFP and OFP, the boundary of the floorplanis fixed, i.e., rectangular, but in case of an IFP, the boundary of the layout isvariable. In the literature, there does not exist any algorithm that talks aboutthe existence and construction of an IFP corresponding to a given PTG. Hence,in this work, we take a dimensionless IFP as an input, which can be drawn on aGUI, and then generate a dimensioned IFP while satisfying the dimensions givenby the user and preserving the adjacencies and topology of the rooms (which arealso given by the user in terms of a dimensionless IFP). NS
12 3 45 6789
W E
12 3 45 6789
NW S E
Minimize (cid:1) ( w(e j,i ) - d imax ) (cid:2) w N1 (cid:2) (cid:2) w (cid:2) (cid:2) w (cid:2) (cid:2) w + w (cid:2) (cid:2) w N5 (cid:2) (cid:2) w N6 (cid:2) (cid:2) w (cid:2) (cid:2) w + w (cid:2) (cid:2) w (cid:2) N1 = w = w w N5 = w w N6 = w +w w +w = w +w w = w = w w = w w +w = w Minimize (cid:1) ( w(e j,i ) - d imax ) (cid:2) w N1 (cid:2)
11 4.5 (cid:2) w (cid:2) (cid:2) w (cid:2) (cid:2) w + w (cid:2) (cid:2) w N5 (cid:2) (cid:2) w N6 (cid:2) (cid:2) w (cid:2) (cid:2) w + w (cid:2) (cid:2) w (cid:2) N1 = w = w w N5 = w w N6 = w +w w +w = w +w w = w = w w = w w +w = w Vertical st-graph Horizontal st-graph (For Width constraints) (For Height constraints)
Fig. 23.
Dimensioning model based on network flow and linear optimization bcd
Fig. 24. a) A PTG which is not a PTPG b) Dimensionless OFP generated by GPLANcorresponding to given PTG c) Partitioned OFP and dimensional constraints d) Di-mensioned OFP generated by GPLAN
234 5 6 78 1234 5 6 78910 a bcd
Fig. 25.
Construction of a dimensioned irregular floorplan a) Dimensionless IFP asinput b) IFP transformed into a RFP by adding extra rooms (red) and partitioningorthogonal rooms into rectangles (yellow) c) Dimensional requirements d) Resultingdimensioned IFP obtained using GPLAN he input IFP may not always contain rectangular rooms; hence we par-tition the orthogonal rooms into a minimum number of rectangles, using [47].However, to employ the network flow formulation for dimensioning, the exte-rior boundary of floorplan should also be rectangular. Therefore, extra roomsare added to dimensionless IFP, as shown in Figure 25b, such that the outerboundary assumes the shape of a rectangle. The dimensions of IFP are obtainedsimilar to an RFP, i.e., by using the network flow optimisation [22]. However,no dimensional constraints are imposed on these additional rooms, as they aremerely intermediaries, and will be removed from the final dimensioned floorplan,as illustrated in Figure 25d.Since RFPs and OFPs are special cases of IFPs, the dimensioned RFPs andOFPs can also be generated by drawing corresponding dimensionless RFPs andOFPs on the GUI generated by GPLAN.
In this paper, we presented the automated generation of dimensioned floorplanswith rectangular and non-rectangular boundaries, where rectangular boundaryfloorplans are generated corresponding to a given adjacency graph and floorplanshaving variable boundaries are produced based on the initial layout drawn by theuser. It is exciting to see that both the approaches can be used for re-generatingwell-known existing architectural floorplans.In the first case, we need to extract the underlying graph of the existingfloorplan F and then using GPLAN, we can re-generate F while introducing thedimensions given by the user. For an illustration, refer to Villa Trissino floor-plan in Figure 26a, designed by Scamozzi [48] in 1778. Its underlying graph isshown in Figure 26b and the re-generated Villa Trissino floorplan by GPLANis demonstrated in Figure 26c. GPLAN efficiently produces floorplans that aretopologically distinct to Villa Trissino floorplan, some of which are shown inFigure 26d. Clearly, GPLAN is capable of re-generating any floorplan with rect-angular boundary and it also generates topologically distinct floorplans, whichprovides a set of alternatives to the existing floorplans.In the second case, user can draw the existing floorplan on a GUI and GPLANgenerates a dimensioned floorplan corresponding to the dimensional constraints.In this case, the user has more flexibility in choosing the floorplan he wants tore-generate, but alternative floorplans are not possible to generate. Here, theidea is to re-generate complex building structures with new dimensions becausewith time, the dimensional requirements change a lot. The steps of re-generationof Banstead Home School Plan (shown in Figure 27a) are shown in Figure 27,where Figure 27b is drawn by the user and is transformed into Figure 27c usingGPLAN, after which dimensional constraints are provided by the user. Figure27d presents the dimensioned Banstead Home School Plan re-generated usingGPLAN. bcd
567 1 89102 0 34
Fig. 26. a) Villa Trissino floorplan designed by Ottavio [48] b) Underlying adjacencygraph of Villa Trissino taken as input by GPLAN c) Re-generated Villa Trissino floor-plan by GPLAN d) Some floorplans that are topological distinct with Villa Trissinofloorplan bc d
Fig. 27. a) Banstead Home School Plan by Higginbotham [49] b) Drawing the BansteadHome School Plan on a GUI generated by GPLAN c) Partitioning of rectilinear roomsand dimensional constraints d) Re-generated Banstead Home School Plan (with newdimensions) by GPLAN
Conclusion and Limitations
This work presents the automated generation of floorplans based on the followingtwo cases:i. Floorplans with rectangular boundary: For a given PTG G , there alwaysexists a floorplan, and there exist many algorithms for generating either anRFP or an OFP corresponding to G . In this paper, instead of consideringRFP and OFP separately, we proposed an approach which first generatesan RFP if it exists; otherwise, it generates an OFP. By doing this, theuser has more flexibility in considering the adjacency relations, i.e., it is notrequired for the user to have prior knowledge of graphs for which an RFPdoes not exists and the work is not limited to a specific class of graphs.Furthermore, for a given set of adjacencies, the user has a lot of choices interms of topologically distinct layouts. The additional feature of GPLAN isits ability to generate a feasible dimensioned layout for any given dimensions.Furthermore, GPLAN is very efficient in handling the graphs with a largenumber of vertices, at the same time, it can quickly generate a large numberof layouts for the given adjacencies.ii. Floor plans with non-rectangular boundary: These floorplans are compar-atively difficult to handle because of the flexibility in the boundary layoutand this is why there does not exist any promising work for the automatedgeneration of IFP corresponding to the given graphs. At the same time,introducing dimensions to these layouts is also challenging as compared toRFPs and OFPs. Hence, the proposed work can be seen as an alternativeapproach for building dimensioned IFP where the idea is to insert the ad-jacency relation through dimensionless layouts which, by default, also con-siders the geometry of rooms. Then, dimensioned IFPs are produced whilepreserving the given adjacencies, positions and shapes of the drawn rooms.As mentioned in the Section 1, GPLAN can be seen as a beneficial toolfor architects/designers, which is capable of generating a set of dimensionedfloorplans for given adjacency relations; at the same time, it can also be used tore-generate existing floorplans. Although GPLAN has its merits, but it has thefollowing limitations which we need to address in the near future:i. It can be seen in Section 3.5 that GPLAN is capable of producing a verylarge number of solutions but it is not feasible for the user/designer to gothrough each solution. Hence, we need to identify and pick good architec-tural layouts from the obtained solution set. The first step in this directionis to restrict solutions on the basis of boundary constraints, i.e., for eachroom adjacent to the exterior, the user will be asked to choose its preferredlocation based on cardinal and inter-cardinal directions. It is possible toincorporate this part to GPLAN in future because GPLAN finds all pos-sible boundary solutions for a given adjacency graph (as discussed in 3.5)to generate topologically distinct floorplans. For example, Figure 28a showsan adjacency graph with 3 vertices and Figure 28b shows the boundary Exterior rooms Directions North East South West 0 Yes 1 Yes Yes 2 Yes Yes
Possible Boundaries
North East South West[2,0] [0,1] [1] [1,2][2,0] [0] [0,1] [1,2][2] [2,0] [0,1] [1,2] c da b Fig. 28. a) An adjacency graph b) User-defined constraints c) Possible boundariesbased on user-defined constraints d) Floor plans restricted to user-defined boundaryconstraints constraints defined by the user. Out of the 12 possible boundaries, 3 bound-aries satisfy the user defined constraints as shown in Figure 28c, and theircorresponding floorplans are shown in Figure 28d. The solutions satisfyinguser-defined boundary constraints will be further sorted on the basis of day-light and other architectural constraints which we need to collect as feedbackfrom the architects while presenting GPLAN to them. Other than the floorplan assessment, we are also planning to introduce circulations to the floor-plans obtained using GPLAN. A graph-theoretical approach for insertingcirculations is given by Baybars [50] and there are some recent works inthis direction, for example [51]. By exploring all available possibilities andunderstanding the architectural requirements, we will try to generate floor-plans with circulations using GPLAN. At large, our objective is to adaptGPLAN for residential buildings while considering functionality and otherarchitectural inputs.ii. Since IFPs are more suitable for complex building structures like hospitalsand universities, a separate study is required for the automated generationof IFPs for the given adjacencies. One of the ideas is to consider the distancematrix along with the adjacency graph for constructing dimensioned IFPswhile considering the boundary layout as input.We acknowledge that architectural design is a multi-disciplinary and multi-constraints problem where producing an optimum solution which satisfies allonstraints and is simultaneously acceptable to architects is near to impossible.Therefore, computers cannot be a replacement to architects; nevertheless, atthe same time, they can provide a variety of good initial layouts. Hence, in thispaper, we presented GPLAN, which can be seen as a major contribution towardsthe automated generation of floorplans and it can be taken to new heights aftergetting inputs from designers/architects.
Acknowledgement
The research described in this paper evolved as part of the research projectMathematics-aided Architectural Design Layouts (File Number: ECR/2017/000356)funded by the Science and Engineering Research Board, India.
References
1. Levin PH. Use of graphs to decide the optimum layout of buildings.
The Architects’Journal , 7:809–815, 1964.2. Steadman P. Graph theoretic representation of architectural arrangement.
Archi-tectural Research and Teaching , pages 161–172, 1973.3. Mitchell WJ, Steadman P, and Liggett RS. Synthesis and optimization of smallrectangular floor plans.
Environment and Planning B: Planning and Design ,3(1):37–70, 1976.4. Sauda EJ. Computer program for the generation of dwelling unit floor plans. In
MArch Thesis University of California Los Angeles , 1975.5. Ko´zmi´nski K and Kinnen E. An algorithm for finding a rectangular dual of aplanar graph for use in area planning for vlsi integrated circuits. In , pages 655–656. IEEE, 1984.6. Ko´zmi´nski K and Kinnen E. Rectangular duals of planar graphs.
Networks ,15(2):145–157, 1985.7. Bhasker J and Sahni S. A linear time algorithm to check for the existence of arectangular dual of a planar triangulated graph.
Networks , 17(3):307–317, 1987.8. Bhasker J and Sahni S. A linear algorithm to find a rectangular dual of a planartriangulated graph.
Algorithmica , 3:247–278, 1988.9. Roth J, Hashimshony R, and Wachman A. Turning a graph into a rectangularfloor plan.
Building and Environment , 17(3):163–173, 1982.10. Rinsma I. Nonexistence of a certain rectangular floorplan with specified areas andadjacency.
Environment and Planning B , 14:163–166, 1987.11. Rinsma I. Rectangular and orthogonal floorplans with required room areas andtree adjacency.
Environment and Planning B , 15:111–118, 1988.12. Rinsma I, Giffin JW, and Robinson DF. Orthogonal floorplans from maximalplanar graphs.
Environment and Planning B , 174:67–71, 1990.13. Yeap KH and Sarrafzadeh M. Floor-planning by graph dualization: 2-concaverectilinear modules.
SIAM Journal on Computin , 22(3):500–526, 1993.14. Giffin JW, Watson K, and Foulds LR. Orthogonal layouts using the deltahedronheuristic.
Australasian J. Combinatorics , 12:127–144, 1995.15. Liao CC, Lu HI, and Yen HC. Compact floor-planning via orderly spanning trees.
Journal of Algorithms , 48(2):441–451, 2003.6. Marson F and Musse SR. Automatic real-time generation of floor plans based onsquarified treemaps algorithm.
International Journal of Computer Games Tech-nology , 2010, 2010.17. Jokar MRA and Sangchooli AS. Constructing a block layout by face area.
TheInternational Journal of Advanced Manufacturing Technology , 54:801–809, 2011.18. Eppstein D, Mumford E, Speckmann B, and Verbeek K. Area-universal and con-strained rectangular layouts.
SIAM Journal on Computing , 41(3):537–564, 2012.19. Alam MJ, Biedl T, Felsner S, Kaufmann M, Kobourov S, and Ueckerdt T. Com-puting cartograms with optimal complexity.
Discrete & Computational Geometry ,50(3):784–810, 2013.20. Wang X-Y, Yang Y, and Zhang K. Customization and generation of floor plansbased on graph transformations.
Automation in Construction , 94:405–416, 2018.21. Shekhawat K. Enumerating generic rectangular floor plans.
Automation in Con-struction , 92:151–165, 2018.22. Upasani N, Shekhawat K, and Sachdeva G. Automated generation of dimensionedrectangular floorplans.
Automation in Construction , 113:103149, 2020.23. Wang X-Y and Zhang K. Generating layout designs from high-level specifications.
Automation in Construction , 119:1–12, 2020.24. Mitchel WJ. The logic of architecture (design computation and cognition). MITPress London, 1990.25. Harada M, Witkin A, and Baraff D. Interactive physically-based manipulationof discrete/continuous models. In
Proceedings of the 22nd Annual Conference onComputer Graphics and Interactive Techniques , SIGGRAPH ’95, pages 199–208.Association for Computing Machinery, 1995.26. Wonka P, Wimmer M, Sillion F, and Ribarsky W. Instant architecture. In
ACMSIGGRAPH 2003 Papers , SIGGRAPH ’03, page 669–677, New York, NY, USA,2003. Association for Computing Machinery.27. Duarte JP. A discursive grammar for customizing mass housing: the case of siza’shouses at malagueira.
Automation in Construction , 14:265–275, 2005.28. M¨uller P, Wonka P, Haegler S, Ulmer A, and Van GL. Procedural modeling ofbuildings.
ACM Trans. Graph. , 25(3):614–623, July 2006.29. M¨uller P, Zeng G, Wonka P, and Van GL. Image-based procedural modeling offacades.
ACM Trans. Graph. , 26(3):85–es, July 2007.30. Wu F, Yan D-M, Dong W, Zhang X, and Wonka P. Inverse procedural modelingof facade layouts.
ACM Trans. Graph. , 33(4), July 2014.31. Merrell P, Schkufza E, and Koltun V. Computer-generated residential buildinglayouts. In
ACM SIGGRAPH Asia 2010 papers , pages 1–12. 2010.32. Wu W, Fu X-M, Tang R, Wang Y, Qi Y-H, and Liu L. Data-driven interiorplan generation for residential buildings.
ACM Transactions on Graphics (TOG) ,38(6):1–12, 2019.33. Hu R, Huang Z, Tang Y, van KO, Zhang H, and Huang H. Graph2plan: Learningfloorplan generation from layout graphs. arXiv preprint arXiv:2004.13204 , 2020.34. Rodrigues E, Gaspar AR, and Gomes ´A. An evolutionary strategy enhanced witha local search technique for the space allocation problem in architecture, part 2:Validation and performance tests.
Computer-Aided Design , 45(5):898–910, 2013.35. Bao F, Yan D-M, Mitra NJ, and Wonka P. Generating and exploring good buildinglayouts.
ACM Transactions on Graphics (TOG) , 32(4), 2013.36. Wu W, Fan L, Liu L, and Wonka P. Miqp-based layout design for building interiors.In
Computer Graphics Forum , volume 37, pages 511–521. Wiley Online Library,2018.7. Nisztuk M and Myszkowski PB. Hybrid evolutionary algorithm applied to auto-mated floor plan generation.
International Journal of Architectural Computing ,17(3):260–283, 2019.38. Shi F, Soman RK, Han J, and Whyte JK. Addressing adjacency constraints inrectangular floor plans using monte-carlo tree search.
Automation in Construction ,115:103187, 2020.39. Regateiro F, Bento J, and Dias J. Floor plan design using block algebra andconstraint satisfaction.
Advanced Engineering Informatics , 26(2):361–382, 2012.40. Zawidzki M and Szklarski J. Effective multi-objective discrete optimization oftruss-z layouts using a gpu.
Applied Soft Computing , 70:501–512, 2018.41. Kant G and He X. Two algorithms for finding rectangular duals of planar graphs.In
International Workshop on Graph-Theoretic Concepts in Computer Science ,pages 396–410. Springer, 1993.42. Koning H and Eizenberg J. The language of the prairie: Frank lloyd wright’s prairiehouses.
Environment and Planning B , 8(3):295–323, 1981.43. D´aniel M and Schlotter I. Obtaining a planar graph by vertex deletion.
Algorith-mica , 62:807–822, 2012.44. Biedl T, Kant G, and Kaufmann M. On triangulating planar graphs under thefour-connectivity constraint.
Algorithmica , 19:427–446, 1997.45. Jon Harris. Jgraphed–a java graph editor and graph drawing framework.
Projectreport, Carleton University , 2004.46. Steadman P. Why are most buildings rectangular.
Arq magazine , 10(2):119–130,2006.47. Wu S-Y and Sahni S. Fast algorithms to partition simple rectilinear polygons.
VLSI Design , 1(3):193–215, 1994.48. Bertotti Scamozzi. wikipedia.org/wiki/villa trissino (cricoli), 1778.49. Higginbotham Peter. workhouses.org.uk/.50. Baybars I. The generation of floor plans with circulation spaces.
Environment andPlanning B , 9:445–456, 1982.51. Egor G, Sven S, Martin D, and Reinhard K. Computer-aided approach to publicbuildings floor plan generation. magnetizing floor plan generator.