Gaps in the existing literature and our work

Despite significant advancements in floorplan generation, there is a notable lack of studies specifically addressing the automated generation of circulation designs within pre-defined floorplans. Much of existing research centers on plan outer boundary, optimizing room size, placement and adjacencies, with limited attention given to circulation paths (Wang et al., Reference Wang, Yang and Zhang2018; Shekhawat et al., Reference Shekhawat, Jain, Bisht, Kondaveeti and Goswami2021a; Bisht et al., Reference Bisht, Shekhawat, Upasani, Jain, Tiwaskar and Hebbar2022; Wu et al., Reference Wu, Fan, Liu and Wonka2018; Wang and Zhang, Reference Wang and Zhang2020). The previous work related to circulation mainly focused on analysis of existing circulation patterns, rather than creating new designs. Works (Naderpour et al., Reference Naderpour, Johnson and Anderson2019), (Lee et al., Reference Lee, Eastman, Lee, Kannala and Jeong2010) and (Tsiamitros et al., Reference Tsiamitros, Mahapatra, Passalidis, Kailashnath and Pipelidis2023) scrutinize the movement of people within a building and identify congestion areas in the building, suggesting alterations to existing designs or proposing new ones. However, they do not provide a method for generating circulations within a floorplan. On the other hand, the method discussed in (Li et al., Reference Li, Jiang, Sun and Zhang2018) directly proposes circulation designs for a given floorplan. However, it is confined to commercial spaces like exhibition halls, museums, etc. While techniques like Space Syntax (Hillier and Hanson, Reference Hillier and Hanson1984; Mustafa and Azeez, Reference Mustafa and Azeez2022; Sabir and Mustafa, Reference Sabir and Mustafa2023) analyze spatial configurations, they do not offer flexible tools for designing circulations with specific constraints (e.g., entry points, corridor width, or privacy). Limited research exists on seamlessly integrating user-defined constraints, such as entry points and corridor thickness, into circulation design. Many studies target specific building types, necessitating more versatile approaches applicable to residential, educational, healthcare, and public facilities (Mustafa and Azeez, Reference Mustafa and Azeez2022; Mustafa and Rafeeq, Reference Mustafa and Rafeeq2019; Rafeeq and Mustafa, Reference Rafeeq and Mustafa2021).

This paper presents an approach that employs graph algorithms to generate circulation designs for a floorplan represented by a plane graph. This approach allows users to include constraints such as entry points, corridor thickness, and privacy constraints as necessary. Our approach can be used to generate circulations in various building types, such as residential buildings, educational infrastructure, commercial spaces, and public facilities like hospitals, hotels, etc. Section title “Results and discussion” discusses several applications of our work. Furthermore, our algorithms are implemented in Python, and a user-friendly interface (GUI) is developed. This allows users to employ our technique directly and observe the results in a fraction of the time. Our previous work introduced dimensioning in rectangular arrangements (Upasani et al., Reference Upasani, Shekhawat and Sachdeva2020), developed algorithms for generating dimensioned rectangular floorplans based on adjacency graphs and user- defined constraints (Shekhawat et al., Reference Shekhawat, Upasani, Bisht and Jain2021b), and extended this to produce both rectangular and non-rectangular dimensioned floorplans for any given plane graph (Bisht et al., Reference Bisht, Shekhawat, Upasani, Jain, Tiwaskar and Hebbar2022). Building on these foundations, this study focuses on integrating circulation spaces into the floorplans of a given plane graph. It advances the contributions of our earlier publications by adding a new concept of circulation in the automated floorplan generation process.

Generating a floorplan and circulation graph for the given PTPG

First, we obtain a floorplan for the given PTPG G using the method described in (Bisht et al., Reference Bisht, Shekhawat, Upasani, Jain, Tiwaskar and Hebbar2022). To introduce spanning circulation into the obtained floorplan, we use Algorithm 1 to generate a circulation graph for G. Let E = {e ₁, e ₂, …, e_m} and V = {v ₁, v ₂, … ,v_n} be the edge set and vertex set in G, respectively. Assume f to be the number of interior faces in the PTPG G, denoted as F = {F ₁, F ₂, …, F_f}. By applying Algorithm 1, we obtain the circulation graph G_C of G. The graph shown in Figure 3 and its corresponding floorplan will be used as an illustrative example throughout the paper.

Illustration of Algorithm 1:

Algorithm 1 produces a circulation graph G_c for the given PTPG G by introducing new vertices in G that correspond to the corridors required within the floorplan for circulation. It also returns a list of triplets (z, x, y) from G_C, where z is a corridor vertex connecting the rooms x and y. The working of Algorithm 1 is explained in the following steps with an illustrative example:

1. i. First, an exterior edge, say e, is chosen arbitrarily in G. The face in G to which the edge e belongs is labeled as F ₁. Then, the edge e is subdivided by adding a new vertex (introducing the first corridor vertex). For the PTPG taken in Figure 4(a) the exterior edge (0, 4) is selected and subdivided by introducing the first corridor vertex labeled as 8 (see Figure 4(b)). To ensure that the graph remains a PTPG, the newly added vertex is made adjacent to the vertex of the face F ₁ that is not an endpoint of the edge e prior to the subdivision (see Figure 4(b)). This first corridor vertex on the exterior edge later translates to the entrance of the final floorplan.

2. ii. In the next step, a face in G adjacent to F ₁ is selected arbitrarily and labeled as F ₂. The shared edge of F ₁ and F ₂ is subdivided by adding the new corridor vertex (see Figure 5(a)). Further, the graph is triangulated by adding an edge between the newly added corridor vertex and the vertex of F ₂ that is not an endpoint of the shared edge of F ₁ and F ₂ prior to the subdivision. Also, an edge is added between the newly added corridor vertex and the previously added corridor vertex in F ₁ to ensure that the graph remains a PTPG (see Figure 5(b)).

3. iii. In order to generate the complete circulation graph, the process is continued by selecting a face adjacent to one of the previously subdivided faces (a face in which at least one edge is subdivided) and subdividing the corresponding shared edge until all the faces of the input graph are subdivided. Further, the graph is triangulated by adding the new necessary edges while ensuring that the resultant circulation graph is a PTPG. Figure 6 shows the complete construction of a possible circulation graph for the PTPG given in Figure 4(a). The algorithm also returns A = {(8, 0, 4), (9, 0, 1), (10, 1, 3), (11, 1, 4), (12, 1, 5), (13, 5, 2), (14, 4, 5)}, the list containing the triplets (z, x, y) such that z is a corridor vertex connecting the rooms x and y in the circulation graph.

Figure 4. (a) A PTPG G, (b) Adding the first corridor vertex numbered 8 by subdividing the exterior edge (0, 4) of the face (0 −1 −4) in G.

Figure 5. (a) Adding the next corridor vertex numbered 9 by selecting the face (0 − 1 − 3) as F ₂ and subdividing the edge (0,1), (b) Triangulating the graph shown in (a) by adding new edges (9, 3) and (9, 8) such that the resultant graph is a PTPG.

Figure 6. Choosing a neighboring face to a subdivided face at each step from (a) to (d) and introducing a new corridor vertex, ensuring that all faces in the graph undergo subdivision.

Now that we have grasped the algorithm’s concept, let us look at an explicit definition of the algorithm.

Algorithm 1: SpanCirc(G, e)

From circulation graph to floorplan with required circulation space

To obtain the floorplan featuring spanning circulation, corridors are incorporated into the floorplan $ \unicode{x1D53D} $ with the help of the circulation graph G_C, which was derived in the previous step. Let $ {V}^{\prime } $ = {c ₁, c ₂, …, c_k} be the set of corridor vertices introduced in G to form G_C. Considering each corridor vertex in G_C one at a time, Algorithm 2 generates a corridor space in $ \unicode{x1D53D} $ between the rooms associated with the corridor vertex. The working of Algorithm 2 will be explained using the example illustrated in Figure 7.

Figure 7. (a) A PTPG G, (b) an RFP $ \unicode{x1D53D} $ associated with the PTPG G, and (c) a circulation graph G_C of PTPG G.

Illustration of Algorithm 2:

In the process of corridor generation within the given floorplan $ \unicode{x1D53D} $ , the following sequential steps are undertaken to ensure the systematic generation of corridors:

1. 1. Wall Shifting:

   In the first iteration, we start with the first triplet (8, 0, 4) extracted from the list A (a list of triplets (z, x, y) returned by Algorithm 1, where z is a corridor vertex in G_C connecting rooms x (Room 1) and y (Room 2)). Initially, the common wall w shared by rooms 0 and 4 is identified within $ \unicode{x1D53D} $ . The coordinates of w, its orientation (horizontal in this case), and the arrangement of rooms 0 and 4 with respect to it are recorded from $ \unicode{x1D53D} $ . Following this, we refer to Table 3 to determine the walls of rooms 0 and 4 that need to be shifted to create a corridor between these rooms, along with the direction of their shifts. In this case, room 0 is positioned to the south of w, and room 4 is positioned to the north of w in $ \unicode{x1D53D} $ . Therefore, Table 3 prescribes shifting the top wall (T) of room 0 to the negative y direction (south) and bottom wall (B) of room 4 to the positive y direction (north) to create a corridor between rooms 0 and 4 in $ \unicode{x1D53D} $ .

   [NOTE: To create a corridor of thickness t between a pair of rooms (a, b), the coincident walls of both the rooms are shifted by t/2 in opposite directions. In this case, the value of each shift is considered as 0.1 throughout the example, aiming to generate a circulation space of thickness 0.2 in $ \unicode{x1D53D} $ .]

2. 2. Preventing unnecessary gaps due to wall shifting

   It can be observed if rooms 0 and 4 have other neighbors such that their common walls with rooms 0 or 4 have the same orientation as w (wall shared by rooms 0 and 4 in $ \unicode{x1D53D} $ ) then the shifts made to the walls of rooms 0 and 4 can cause an unnecessary gap in the floorplan. To prevent this gap, we find the neighbors of rooms 0 and 4 in $ \unicode{x1D53D} $ and determine which of their walls need to be shifted.

   1. i. Iterative neighbor identification

      To identify the neighbors of rooms 0 and 4, the following approach is used: Initially, the common neighbor of vertices 0 and 4 in the original graph G is identified. Vertex 1 is found as a common neighbor. Subsequently, the walls shared by room 1 with rooms 0 and 4 in F are located, say, w ₁ and w ₂, respectively. A pairwise comparison of the orientations of walls w, w ₁, and w ₂ is conducted, and the wall that aligns with the orientation of w is identified (in this case, w ₂). As w ₂ is the wall shared by room 1 and room 4; this implies that the gap is a result of shifting the bottom wall of room 4 to the north. All necessary shifts to eliminate this gap are then supposed to be in the north direction. To identify which wall of room 1 needs to be shifted we use Table 4 (in this case, 1 is the neighboring room referred to in Table 4).

      Following this, the wall w ₂ of room 1 is relabeled as w, and the common neighbor of vertices 1 and 4 is identified in graph G (vertices 1 and 0 are not considered here, since the gap in $ \unicode{x1D53D} $ is caused by shifting the bottom wall of 4 to the north). Here, although 0 is a common neighbor, it is disregarded as it has already been visited once. Now, the common walls w ₁ and w ₂ of the common neighbor of 1 and 4 are located, and their orientations are compared with that of w, as similar to the previous step. This process is repeated until one of the two conditions, C ₁ or C ₂, is met:

      • • The orientation of common wall w does not match with those of walls w ₁ and w ₂. (C1)

      • • There are no more common neighbors that are not visited. (C2)

        Upon identification of neighboring rooms, Table 4 is utilized to determine the walls necessitating adjustments and the results are recorded as pairs (r, w). In this instance, three such pairs were identified: (1, T), (5, T), (6, B). This indicates that, to avoid the gap resulting from shifting the bottom wall of room 4 northward, the top walls of rooms 1 and 5 must be shifted northward. Shifting the top wall of room 5 northward would lead to an overlap between rooms 5 and 6, which is prevented by shifting the bottom wall of room 6 northward.

      Nevertheless, employing the iterative method to identify neighbors for any room pairs (a, b) yields all pairs (r, w) that prevent the gap caused by the creation of a corridor between a and b and also addresses any potential overlap resulting from avoiding unnecessary gaps. Moreover, when rooms a and b have two common neighbors, we initiate the process by selecting any one of the common neighbors first. After identifying all neighbors on one side until one of the two conditions, C ₁ or C ₂ is met, we proceed to the other common neighbor on the opposite side.

      To track the shifts, two tables, Shift Table 1 and Shift Table 2, are maintained (see Figure 8). Both of these tables are updated at the end of every iteration. Shift Table 1 records the shift values of walls, while Shift Table 2 stores triplets in the form of (r, w, c), indicating that the wall w of room r has been shifted during the generation of the corridor corresponding to a corridor vertex c.

      Initially, the shift value for every wall is set to 0 in Shift Table 1. As corridors are generated based on triplets (z, x, y) from the list A, the updates to the shift values of (r, w), for the wall w of any room r, requiring adjustments is determined as max{current, 0.1} or min{current, −0.1} (depending on the direction of shift) at the conclusion of each iteration. Following the final iteration, Shift Table 1 is utilized to shift the walls and generate a spanning circulation within the floorplan.

   2. ii. Preventing redundant shifts:

      After identifying the pairs (r, w) to address potential gaps or overlaps, the shift value for wall w from this pair will be updated in Shift Table 1, but only if Shift Table 2 does not already include the triplet (r, w, c). In other words, the revision occurs if the wall w of room r has not been previously shifted during the creation of a corridor associated with a corridor vertex c.

      Table 3. In this table, T=Top wall, B=Bottom wall, L=Left wall, R=Right wall, N=North, S=South, E=East, W=West. This table illustrates the orientation of the common wall w between two rooms and indicates the walls in both rooms that require shifting based on their arrangement around the common wall w.

      

      Table 4. This table outlines the necessary wall shift in the neighboring room when the orientation of either wall w ₁ or w ₂ aligns with the orientation of w.

      

   This 2-step process is followed in each iteration while creating a corridor corresponding to the triplet (z, x, y) from the list A. The following Figures 9, 10, and 11 illustrates several successive iterations. Figure 12 shows the results of final iteration for the input given in Figure 7 and a spanning circulation in the given floorplan $ \unicode{x1D53D} $ .

   

   Figure 8. Iteration 1: (a) Considering the first triplet (8, 0, 4) with corridor vertex 8 from G_C, (b) Shift Table 1 at the end of iteration 1, and (c) Shift Table 2 at the end of iteration 1.

   

   Figure 9. Iteration 2: (a) Considering the triplet (9, 0, 1) with corridor vertex 9 from G_C, (b) Shift Table 1 at the end of iteration 2, and (c) Shift Table 2 at the end of iteration 2.

   

   Figure 10. Iteration 3: (a) Considering the triplet (10, 3, 1) with corridor vertex 8 from G_C, (b) Shift Table 1 at the end of iteration 3, and (c) Shift Table 2 at the end of iteration 3.

   

   Figure 11. Iteration 4: (a) Considering the triplet (11, 1, 4) with corridor vertex 11 from G_C, (b) Shift Table 1 at the end of iteration 4, and (c) Shift Table 2 at the end of iteration 4.

   

   Figure 12. Final Iteration: (a) Considering the triplet (14, 4, 5) with the last corridor vertex 14, (b) Shift Table 1 at the final iteration, and (c) An RFP of G featuring spanning circulation obtained by using the Shift Table 1 shown in (b).

   Now, that we have gained an intuitive idea of the algorithm, let us look at it in a formal manner with all the required details.

   Algorithm 2: FpCirc(G_c, F, A, (p, q), t)

   

   Algorithm 3: CorridorLoop(a, b)

   

   The above Algorithm 3 is used as a sub-routine in Algorithm 2 in Line 12 to check, which other rooms’ walls have to be shifted to accommodate for the creation of corridor space between rooms x and y.

Generating minimal circulation space

In this stage, we produce minimal circulation within the floorplan, precisely the most essential corridors needed to access all areas. This optimization is crucial for addressing corridor redundancy, preventing certain rooms from needlessly losing space and shrinking in size, especially after multiple coordinate shifts outlined in Algorithm 2.

Notably, the display of the initial corridor, recognized as the door, can be excluded since it is already known and thus can be ignored in the representation.

We generate minimal circulation with the help of the well-known problem of the Minimum set-cover (Cormen et al., Reference Cormen, Leiserson, Rivest and Stein2001).

(1) $$ \underset{A\in Y}{\cup }A=X $$

The set-covering problem is classified as NP-hard, indicating that only sub-optimal solutions can be obtained using approximation algorithms. The following algorithm is one of several greedy approximation algorithms proposed to address this problem.

Working of Algorithm 4:

First, let us intuitively understand the algorithm before delving into its details. Initially, the set Y = ϕ, and at each step of the algorithm, we add a subset S_i (i $ \in $ {1, 2, …, m}) to Y, covering the maximum number of uncovered elements in X, with ties being broken by taking the lower i. Let us look at an example to understand this algorithm further.

The sets X and S = {S ₁, S ₂, S ₃, S ₄} are defined as shown in Figure 13 (each element of X is represented by a red dot, and the boundaries of the sets S_i are marked by dotted-lines). Let U be the set of uncovered elements in X and d_j = |S_j $ \cap $ U| for every S_j $ \in $ S. In the first iteration of the algorithm, we observe that U = X and the value of d(j) is the maximum for j = 3, with a value of 8; hence, we include S ₃ into the set cover Y. In the next step, we update set U and the values of d(j). In the second iteration, d(j) is the maximum for j = 2, with a value of 6; hence, we include S ₂ into Y. Next, we update set U and include S ₁ based on the updated d(j) values. When we recompute U, we note that it is empty (meaning all elements of set X are covered), so we stop the algorithm. Hence, the required Minimum set-cover for this problem instance is Y = {S ₁, S ₂, S ₃} (marked by blue dotted-lines). Note that in this case, the algorithm coincidentally gives an optimal set cover.

Algorithm 4: Greedy-Set-Cover(X, S) (Cormen et al., Reference Cormen, Leiserson, Rivest and Stein2001)

Figure 13. An instance of minimum set cover to illustrate Algorithm 4.

By virtue of being a Greedy Algorithm, we must prove that the Algorithm 4 indeed returns a set-cover that is not significantly larger than the optimal set cover. A comprehensive proof of this algorithm can be found in Appendix title “Greedy set cover”.

Conversion to minimum set-cover problem:

In this step, we transform our minimization problem into an instance of the Minimum set-cover problem. Given G_C as the circulation graph and V_C as the set of corridor vertices, V represents the set of rooms in $ \unicode{x1D53D} $ . As earlier mentioned in this section, we can omit the first corridor vertex as it is identified as a door. Consequently, the updated set of corridor vertices is defined as:

(2) $$ {V}_C^{\prime }={V}_C\backslash \mathit{\min}\left\{j|j\in {V}_C\right\} $$

For every corridor vertex j in $ {V}_C^{\prime } $ and room v in V, we define:

(3) $$ {\displaystyle \begin{array}{c}{P}_j=\left\{v\in V|v\;\mathrm{belongs}\ \mathrm{to}\ \mathrm{same}\ \mathrm{face}\;\mathrm{as}\;j\;\mathrm{in}\;{G}_C\right\}\\ {}\;\mathrm{and}\;{Q}_v=\left\{j\in {V}_C|\left(j,v\right)\in E\left({G}_C\right)\right\}\end{array}} $$

(4) $$ P=\left\{{P}_j|j\in {V}_C^{\prime}\right\}\;\mathrm{and}\;Q=\left\{{Q}_v|v\in V\right\} $$

By considering the set of rooms V as the set X and the set P as the set S, we do some pre-processing and then use Algorithm 4 as a sub-routine in the following algorithm.

Working of Algorithm 5:

With the aid of an example, we will explore the process of transforming our problem instance into a minimum set-cover problem and utilizing Algorithm 4 as a sub-routine to achieve our objective (refer to Figure 14).

Figure 14. The circulation graph G_C (from Algorithm 1) and the RFP $ \unicode{x1D53D} $ (from (Shekhawat, Reference Shekhawat2018)).

Our objective is to connect (cover) all rooms using corridors, the set of rooms V can be seen as the set X of the Minimum Set-Cover problem. To determine the coverage of each corridor (set of rooms it connects in F_C), we defined the set P_j for each corridor vertex j (refer to Equation 3). Since P_j $ \subseteq $ V $ \forall j\in {V}_C^{\prime } $ (refer to Equation 2), we can deduce that these P_j’s serve as the subsets S_i in the Minimum Set-Cover problem. Additionally, to ensure no room is left uncovered, we introduced an auxiliary set Q_v for every room vertex v $ \in $ V, which contains the corridor vertices it is adjacent to in G_C (refer to Equation 3).

In our example, we have the following sets:

(5) $$ X=V=\left\{0,1,2,3,4,5,6,7\right\} $$

(6) $$ {V}_C^{\prime }=\left\{9,10,11,12,13,14\right\} $$

(7) $$ {P}_9=\left\{0,1,3,4\right\},{P}_{10}=\left\{0,1,3,2\right\},{P}_{11}=\left\{0,1,4,5\right\}, $$

(8) $$ {P}_{12}=\left\{1,2,4,5\right\},{P}_{13}=\left\{1,2,5,7\right\},{P}_{14}=\left\{1,4,5,6\right\} $$

(9) $$ {Q}_0=\left\{9\right\},{Q}_1=\left\{9,10,11,12\right\},{Q}_2=\left\{10,12,13\right\},{Q}_3=\left\{9,10\right\} $$

(10) $$ {Q}_4=\left\{11,14\right\},{Q}_5=\left\{11,12,13,14\right\},{Q}_6=\left\{14\right\},{Q}_7=\left\{13\right\} $$

(11) $$ P=\left\{{P}_9,{P}_{10},{P}_{11},{P}_{12},{P}_{13},{P}_{14}\right\},Q=\left\{{Q}_0,{Q}_1,{Q}_2,{Q}_3,{Q}_4,{Q}_5,{Q}_6,{Q}_7\right\} $$

From the sets defined in the equations 5 to 11, it can be observed that the rooms 0, 6 and 7 are made accessible by only one corridor each as |Q ₀| = |Q ₆| = |Q ₇| = 1. Therefore, the corresponding corridor vertices 9, 14 and 13 will definitely be there in the minimum set of non-redundant corridors. We observe that P ₉ $ \cup $ P ₁₃ $ \cup $ P ₁₄ = V = X, i.e., the corridor vertices 9, 13, and 14 covers the whole set X, and thus directly gives us the minimum set of non-redundant corridors to be $ {C}^{\prime } $ = {9, 13, 14}. In the event that some rooms in X were left uncovered, {9, 13, 14} would not be the minimum set of non-redundant corridors. To find the minimum set of non-redundant corridors, we would have run the subroutine 4 on the remaining uncovered rooms and P.

Once the set $ {C}^{\prime } $ of non-redundant corridors is obtained, Algorithm 2 is used to calculate shift values for the rooms only to insert the corridors corresponding to the corridor vertices in set $ {C}^{\prime } $ . This process results in a floorplan with the minimal set of corridors, ensuring accessibility to all rooms (refer to Figure 15).

Figure 15. The circulation graph G_c (from Algorithm 1) with the set of non-redundant corridors marked in red and the floorplan $ \unicode{x1D53D} $ (from (Shekhawat, Reference Shekhawat2018)) featuring minimal circulation.

In Figure 15, we can observe that there are only 3 corridor spaces. To ensure accessibility throughout the floorplan, certain rooms must be designated as public. This is achieved by designing rooms with entrances on walls facing a corridor, thereby enabling seamless navigation. Rooms with walls facing multiple corridors include more than one door, serving as transition points to maintain connectivity across the floorplan. For instance, in the corridor surrounded by rooms 0, 1, 3, and 4, all four rooms feature a door on the wall facing this corridor. To facilitate movement within this floorplan, room 5 will have two doors: one on the top wall (facing the corridor surrounded by 1, 4, 6, and 5) and another on the left wall (facing the corridor surrounded by 1, 2, 5, and 7). Similarly, room 4 will have two entrances on the bottom wall, with one opening in the corridor surrounded by the rooms 0, 1, 3, and 4, and the other in the corridor surrounded by the rooms 1, 4, 6, and 5. Alternatively, for seamless movement throughout this floorplan, room 1 can have three doors, one on each wall facing a corridor, while other rooms have exactly one door opening into any one of the connected corridors.

Algorithm 5: MinCirc(P, Q, V)

Note: The minimal circulation space is generated independently of room designation. Once the circulation layout is established, some rooms are made ‘public’ to ensure accessibility for all other rooms. This is achieved by designing entrances on every wall that faces a corridor.

Limitations of the approximation algorithm

As previously explained, the minimization of the corridor space is a problem equivalent to the Minimum set-cover problem. Based on the understanding of complexity classes, we know that this problem belongs to the NP-hard class of problems, implying that no polynomial time algorithm exists to solve this problem. Due to the inherent complexity of NP-hard problems, we could only approximate the solution using approximation algorithms such as Algorithm 4. As a result, there may be instances where the algorithm returns a sub-optimal solution to the optimization problem.

As shown in Figure 16, there could be cases where the obtained corridor space might not be the optimal minimized corridor space but a sub-optimal one. This limitation is an intrinsic characteristic of the minimization problem. Nevertheless, we are actively exploring various alternatives to reduce such sub-optimal cases. One such alternative is the application of the above-mentioned greedy algorithm (Algorithm 4) consecutively, aiming to further eliminate redundant corridors.

Figure 16. (a) The input to the GUI for which we want the minimal set of corridors with entry between rooms 2 & 9 (b) The expected minimal corridor space (c) The sub-optimal minimized set of corridors given by Algorithm 4 (redundant corridor between rooms 2 & 7).

Customization of the circulation space according to user-specified privacy constraints

In any building, it is common for the users to designate certain rooms as private, accessible only to a specific group of individuals, while the others are considered as public. Typically, in residences, bedrooms, storage rooms, and occasionally kitchens are kept private, whereas other rooms made open for public access. In the case of office buildings or factories, areas like server rooms, storage spaces, and specific security zones are restricted to authorized personnel. Consequently, these restrictions influence the layout of corridors within such buildings.

Consider the following scenario (see Figure 17): Rooms A and B each share a wall with Room C. If Room C is designated as public space, one can move from A to B (or vice versa) through Room C. Conversely, if Room C is private, the only route to move between A and B would be through a corridor around Room C.

Figure 17. (a) If Room C is public, we can move from A to B through C without a corridor (b) If Room C is private, we have to create a corridor similar to the one marked by the grey stripes and use it or other adjacent public rooms to move between A and B.

Building on this fundamental idea, next, we recall the definition of private and public rooms. Following this, we describe the algorithm designed to handle these constraints while generating the floorplans with the required circulation space. In addition to other inputs, the user also provides a list of private and public rooms.

In other words, when going from Room A to B, one will not visit any private rooms unless Room A or B is private. There will always exist a path from Room A to Room B via corridors and other public rooms. There is no restriction on the number of entry doors to any arbitrary room, but it can be assumed that a public room can have more than one door.

Illustration of Algorithm 6:

To understand the working of Algorithm 6, we will look at the approach used to address privacy constraints with the help of an example (refer to Figure 18).

Figure 18. (a) The input graph. (b) The spanning circulation (like mentioned in previous algorithm the entry corridor, here 0 − 1, is not explicitly shown since it is known to be a door). (c) The corridor placement if the room 5 alone is made public.

Our central idea is to position corridors only around private rooms, ensuring that movement from one room to another does not necessitate passing through a private room, as defined. Additionally, given that free-movement is permitted in public rooms, there is no need for a corridor around them. In fact, a public room can be regarded as a corridor in itself. This approach helps prevent a reduction in room area, as constructing corridors would otherwise result in a smaller room area for a given plot size.

To obtain the required circulation space, first, we obtain the spanning circulation for the given floorplan using Algorithm 2. Next, we get the list of public and private rooms from the user. By analyzing the circulation graph of the spanning circulation, we obtain the set $ {C}^{\prime } $ of corridors that do not surround any public rooms. Subsequently, we use Algorithm 2 to calculate shift values for the rooms to insert the corridors corresponding to the corridor vertices in set $ {C}^{\prime } $ . The resulting floorplan satisfies the privacy constraints defined by the user (refer to Figure 19).

Figure 19. (a) The circulation graph corresponding to the spanning circulation. The corridor vertices excluding the encircled ones are the ones that will be considered for the final floorplan. (b) The final floorplan with the corridor placement according to the constraint that only room 5 is public and the rest are private.

To demonstrate the use of public rooms, here are a few paths facilitating movement between different rooms in the floorplan shown in Figure 19:

1. $ 2\to 5\to 4 $

2. $ 7\to 5\overset{\mathrm{via}\hskip0.3em \mathrm{corridor}}{\to }3 $

3. $ 3\overset{\mathrm{via}\hskip0.3em \mathrm{corridor}}{\to }5\to 6 $

4. $ 0\overset{\mathrm{via}\hskip0.3em \mathrm{corridor}}{\to }5 $

It can be observed that in all of the paths, the private rooms serve as either the source or destination but never as an intermediate room. Now that we have a general understanding of Algorithm 6, let us formally define the Algorithm.

Algorithm 6: PublicPrivate(G, (p, q))

Algorithm validation and complexity

Mathematical Validation: Algorithm 1:

To mathematically validate Algorithm 1, we confirm it preserves the properties of a PTPG while generating a circulation graph.

1. i. Input Validity: The input graph G(V, E) is a valid PTPG with n vertices, m edges, and f faces satisfying Euler’s formula n − m + f = 2.

2. ii. Planarity and Triangulation: The algorithm maintains planarity by adding vertices through edge subdivision and forming new triangular faces. Each new vertex is connected to adjacent vertices, ensuring the resulting graph G_C remains planar and fully triangulated.

3. iii. Connectivity and Spanning: Every new corridor vertex connects to at least two existing vertices, forming a connected subgraph. The algorithm ensures that every room in the original graph is adjacent to at least one corridor vertex, providing a complete spanning circulation.

4. iv. Termination: The algorithm processes each face once, and since the number of faces f is finite, it always terminates.

Time Complexity Analysis: Algorithm 1:

1. i. Input Initialization: The input graph G(V, E) contains n vertices and m edges. Initializing the graph takes linear time in terms of the vertices and edges. Time Complexity: O(n + m).

2. ii. Processing the First Corridor (Step 2): In this step, one vertex is added by subdividing an edge, and the adjacency between this new corridor vertex and the surrounding vertices is updated. Since this involves a constant number of operations: Time Complexity: O(1).

3. iii. Main Loop: Growing the Circulation (Step 3): The algorithm processes each face adjacent to previously subdivided faces. For each face, a new vertex is inserted, and edges are added to maintain adjacency and triangulation.

   1. 1. Subdividing a face (inserting a vertex and adding edges) takes constant time, O(1).

   2. 2. This loop runs once for each face; hence the total time depends on the number of faces, f. Since a PTPG graph has a linear relationship between its faces and vertices (f = O(n)), the total time for this step is proportional to the number of faces: Time Complexity: O(f) = O(n).

4. iv. Triangulation (Maintaining PTPG): The graph remains a triangulated PTPG throughout the algorithm by adding edges to maintain triangular faces. Each edge might be subdivided once, but since subdivision is constant-time per face, the overall triangulation step takes linear time in the number of edges. Time Complexity: O(m).

5. v. Adjacency Matrix Calculation: Building the adjacency matrix to represent the modified graph structure takes O(n ²) time in the worst case, as each vertex may be adjacent to any other vertex.

Overall Time Complexity: O(n ²) (since m = O(n)) + O(n) (since f = O(n)) + O(n ²) = O(n ²)).

Mathematical Validation: Algorithms 2 and 3

1. 1. Correctness:

   1. i. Algorithm 2 iterates through all corridor vertices, ensuring that each corridor is processed.

   2. ii. Algorithm 3 recursively explores common neighbors, ensuring all affected rooms are considered.

   3. iii. The use of Shift Tables ensures that wall shifts are correctly tracked and applied.

2. 2. Termination:

   1. i. Algorithm 2 terminates when all corridor vertices in V’ are processed.

   2. ii. Algorithm 3 recursively explores common neighbors, ensuring all affected rooms are considered.

   3. iii. The use of Shift Tables ensures that wall shifts are correctly tracked and applied.

3. 3. Invariants:

   1. i. The planarity of the floorplan is maintained throughout the process.

   2. ii. The connectivity of rooms is preserved while adding corridors.

Complexity Analysis: Algorithms 2 and 3

Algorithm 2:

1. i. Let n be the number of rooms and m be the number of corridors.

2. ii. Sorting $ {V}^{\prime } $ takes O(mlogm) time.

3. iii. The main loop iterates m times, once for each corridor.

4. iv. For each iteration: Finding common walls and shift values: O(1)

5. v. Updating room coordinates: O(n)

Overall time complexity: O(mlogm + m * T( Algorithm3) + n), where T( Algorithm3) is the time complexity of Algorithm 3.

Algorithm 3:

1. i. Let d be the maximum degree of a vertex in the graph.

2. ii. Finding common neighbors: O(d)

3. iii. The loop iterates at most d times. For each iteration: Finding common walls: O(1).

4. iv. The recurrence relation: T(d) = d * (O(1) + T(d — 1)).

Solving this, we get: T(d) = O(d!).

However, in practice, the depth of recursion is limited by the planarity of the graph and the orientation condition, so the actual runtime is likely to be closer to O(d ²) or O(d ³).

The mathematical proof for Algorithm 4, along with the time complexity analysis for both Algorithms 4 and 5, is provided in Appendix title “Greedy set cover”.

The validation and complexity analysis for Algorithm 6 follow a similar approach to Algorithm 1, as the circulation graph is constructed in the same manner, with the exception that corridor vertices corresponding to public rooms are removed.

Limitations

While our approach addresses many challenges in automated circulation design, there are areas for future development.

1. i. Incorporating more complex constraints, such as building codes and accessibility requirements, could further improve the real-world applicability of the generated designs.

2. ii. The approach currently focuses on 2D floorplans and does not automatically generate 3D representations or multi-story circulation, which could be studied further.

3. iii. One limitation of the Minimal Circulation methodology is that the designation of public rooms to ensure accessibility is performed after the circulation space is generated. Designating rooms as ‘public’ with multiple doors can affect privacy in environments where it is important, often requiring additional adjustments or re-designations to address this issue effectively. This approach requires users to manually make adjustments to refine the circulation design and ensure that it meets specific accessibility and functional requirements. However, this limitation can be partially addressed using the public-private circulation customization algorithm.

Walkthrough: using our approach for automated circulation design

Here we provide a step-by-step guide on how to use our approach for generating floorplans with optimized circulation. The steps are demonstrated in the examples shown in Figures 20–26 to illustrate the process. Additionally, a working demonstration of the graphic user interface (GUI) is available online on the link provided in Appendix A:

1. 1. Initial input

   The user begins by preparing a PTPG representing the desired room adjacencies. This graph serves as the foundation for the floorplan generation.

2. 2. Generating spanning Circulation: (refer to Figures 20, 21, and 25)

   1. i. Click on the ‘Circulation’ button in the user interface.

   2. ii. Choose ‘Normal’ as the circulation option.

   3. iii. Input the entry door location and desired corridor thickness.

   4. iv. Click ‘Submit’ to generate the floorplan with spanning circulation (see Figure 25 for spanning circulation).

3. 3. Optimizing Circulation Space: To minimize circulation space and optimize room sizes (refer Figures 21 and 22).

   1. i. Choose ‘Get Minimal Circulation’ from the options.

   2. ii. Input the entry door location and corridor thickness.

   3. iii. Submit to generate a floorplan with minimized circulation space.

4. 4. Customizing Circulation (Optional): If the user wishes to eliminate specific corridors (refer Figures 23–26).

   1. i. Click the ‘Circulation’ button again.

   2. ii. Select the ‘Remove Corridor’ option.

   3. iii. For each pair of rooms, input ‘0’ to remove the corridor or ‘1’ to keep it.

   4. iv. Submit to update the circulation design.

      Note: This option may suggest using certain rooms for transit as well as their primary purpose.

5. 5. Implementing Privacy Constraints: For more control over public and private spaces.

   1. i. Provide a list specifying which rooms should be private or public.

   2. ii. Submit to generate a circulation design that includes corridors around private rooms while allowing public rooms to serve dual purposes (transit and primary use).

Figure 20. User draw a PTPG as input graph and chooses the option of circulation and then “normal” when the pop-up appears.

Figure 21. For the input graph in Figure 20, the user selects the Circulation option, specifies the entry door, sets the corridor thickness, and chooses either ‘Get minimal circulation’ for optimized space or ‘Submit’ for spanning circulation.

Figure 22. The GUI displays the floorplan with minimal circulation. Note that every room has at least one of its walls facing a shared corridor. The rooms which share a wall instead of a corridor can have an internal connection or no connection as per user preferences.

Figure 23. For the same input graph. user checks “Remove circulation” and chooses the option of circulation.

Figure 24. The GUI shows a possible spanning circulation and displays a window where user can remove any set of corridors at once. The pair of numbers displayed to the left of a text box denotes the pair of rooms on either side of that corridor. For example, as per image on right, if user wants to remove corridor between rooms 1 and 4, then the user enters “1” in the text box beside the pair “1 4”. There is also a button which allows user to remove all corridors and give back the corresponding floorplan.

Figure 25. The initial spanning circulation.

Figure 26. The Spanning circulation has been updated, removing corridors between room pairs (1,4) and (5,6).

Suppose there’s a need for additional corridors beyond the minimal circulation but lesser than the spanning circulation. In that case, users can remove specific corridors after generating the spanning circulation, if necessary. Figures 23–26 illustrate this option for removing corridors.

Circulation designs for a small office layout

To explore the practical applicability of our software, we evaluate its functionality in generating circulation designs for a small office layout. Figures 27 and 28 illustrate a floorplan of the office, demonstrating how the approach addresses non-rectangular boundaries and generates a spanning circulation layout. This layout is further refined using minimal circulation while designating certain rooms as public (see Figure 29). The configuration of the office layout is represented using the following permutation: [‘Reception (Rec)’, ‘Office1 (OF1)’, ‘Office2 (OF2)’, ‘Meeting Room (MR)’, ‘Office3 (OF3)’, ‘Server Room (SR)’, ‘WC2’, ‘Break Room (BR)’, ‘Store (ST)’, ‘Waiting Room (WR)’, ‘WC1’].

Figure 27. An input graph corresponding to the room list for a small office, selecting entrance of the layout and thickness of the required corridor.

Figure 28. A floorplan for the given input graph featuring spanning circulation.

Figure 29. Optimizing the circulation space: (a) Initial floorplan with redundant corridors (1, 2, and 3), (b) Optimized floorplan after designating the Break Room (BR) and Waiting Room (WR) as public rooms.

The floorplan shown in Figure 28 includes a spanning circulation that can be optimized using the minimal circulation feature. Corridors 1, 2, and 3, as depicted in Figure 29(a), are redundant and can be eliminated by designating the Break Room (BR) and Waiting Room (WR) as public rooms. The resulting optimized floorplan is presented in Figure 29(b).