View of AN STUDY OF TREE LAYOUT ALGORITHM TO CORPORATE MANAGEMENT

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Vol.03, Issue 09, Conference (IC-RASEM) Special Issue 01, September 2018 Available Online: www.ajeee.co.in/index.php/AJEEE

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AN STUDY OF TREE LAYOUT ALGORITHM TO CORPORATE MANAGEMENT Dr. Amit Singh Rathor

Professor (MBA DEPT.) & Group Director OKGEI Kota Rajasthan India

Abstract - This work presents a hierarchical tree layout algorithm based on iterative rearrangement of sub trees. Using a greedy heuristic, all sub trees of a common parent are rearranged into a forest such that gaps between them are minimized. This heuristic is used to build a rearranged tree from bottom-up, starting with forests of the single leafs, and ending with the complete tree. Different cost measures for arrangement operations are discussed, which are based on the shape of a sub tree. This shape can be characterized by the sub tree’s leftmost and rightmost vertices, which determine how gapless this sub tree can be combined with another one. The layout algorithm is used to display an organizational hierarchy. Such a hierarchical layout aids leadership when organizational structures are complex. In particular, it can be used to monitor the performance of organizational units undergoing change, e.g. restructuring. This improves the effectiveness of leadership instruments.

Keywords: Graph layout Leadership Second-line supervisor Change process Restructuring Organigram

1. INTRODUCTION

Complex hierarchical structures entail sophisticated organisational interdependencies across different organisational units. This is shown in the various ways in which power is expressed and in various economic or technical dependencies. Guaranteeing efficient performance is thus a considerable leadership challenge. This is particularly true when units are located at different sites or in different countries.

Formal accountability requires that managers have access to appropriate instruments for monitoring the achievements of any organisational goals and for identifying weaknesses. Economic success in large companies requires that

‘the right hand knows what the left hand is doing’. The organisational units should work in a coordinated fashion and information exchange needs to be smooth in order to improve management effectiveness and efficiency. These requirements need not be guaranteed for companies undergoing a change such as a merger or an acquisition

During the process of merging companies several problems may occur that weaken the economic performance of the company. For example, the responsibilities may not be clearly defined such that different organisational units with similar tasks exist. In addition, problems with distributing given resources can occur. The new tructure may be unclear to many employees who complain of lack of information on the changes going on. This increases

dissatisfaction among the employees and turnover.

In such a situation introducing an animated organigram can provide a tool for representing achievement and fulfillment of organisational criteria. The organigram displays each organisational unit as a vertex and represents dependencies or the hierarchy itself by edges that connect corresponding units.

The impact of satisfying performance criteria becomes immediately apparent, especially when colour coding is used.

Employee access to such a scheme enables comparison and – when desirable – competition among different units and hierarchical levels, and strengthens corporate identity.

The present paper introduces the algorithm applied to find the graphical representation of the organigram. It allows for simple creation of a hierarchical tree layout of a corporate structure. Since an organisational structure will regularly have the form of a tree2 the present paper deals with tree layouts representing a corporate hierarchy. Finding an appropriate graph layout may involve a high degree of complexity depending on the optimization goals targeted.

Hierarchical methods that let the vertices of a particular layer lie on a curve with the same distance to the preceding vertices, are based on NP hard optimization problems Eades and Sugiyama (1990) and Eades and Wormald (1990).

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2 Different methods exist for designing a corporate tree layout depending on the optimization goals or, aesthetic criteria (Eades, Gutwenger, Hong, & Mutzel, 2010) required, e.g.

minimum number of edge crossings (Reingold & Tilford, 1981; Valls, Martí, &

Lino, 1996) or bends (Tamassia, 1987), maximum resolution (maximization of distance between pairs of vertices) (de Fraysseix, Pach, & Pollack, 1990; Eades et al., 2010) or symmetry (Gansner &

North, 1998). The algorithm introduced below deals with tree layout and does not allow any crossings. It hence belongs to the planar graph layout algorithms. It allows for symmetry and high resolution and thus guarantees that each organisational unit can be identified rapidly.

Several constraints are needed to ensure that all levels are treated fairly.

First, the layout should give a clear overview of the whole organisation. This is especially necessary in order to control the effectiveness of leadership instruments at superior levels (‘leadership of leaders’). Second, the organigram should make it easy to find out which units are at the same hierarchical level.

Thus, the performance of the managers at a particular level can easily be compared.

This aids competition between units/managers at the same level. Third, the layout should fill the whole screen no matter what size screen is used. Thus, no employee should be disadvantaged owing to screen size or type.

In contrast to existing methods, the presented algorithm is able to meet the aesthetic requirements of the graph layout the company has set. In particular, the tree can be unbalanced in the sense that subtrees with a large number of leaves may exist. This normally results in a tree layout such that major parts of a given space for the tree layout are not occupied. This is a problem not only but most notably in circular layouts when subtrees with only few hierarchical levels but many leaves are to be displayed. Due to the small radius little space is available for the leaves at lower levels, whereas the subtrees with a large number of levels will use only a part of the available space.

Thus, the layout will appear dense at low levels and unbalanced as a whole.

The algorithm presented is a hierarchical method that arranges

subtrees iteratively, minimizing an appropriate cost function by means of a greedy heuristic. The layout requires the following characteristics:

 placement of vertices in concentric ellipses

 each ellipse corresponds to one hierarchy level

 the closer an ellipse is to the center, the higher the hierarchy level

 the root is located in the center

 edges must not intersect

 (optional): more space between vertices belonging to different organisational units, than space between vertices belonging to the same organisational unit (i.e.

group vertices of each unit together and leave empty space between units).

Balloon layout is not applicable, as it would violate the constraint that individuals on the same hierarchical level are represented by vertices sharing the same distance to the root.

The paper is organized as follows:

Section 2 discusses the practical relevance of displaying the corporate hierarchy in an (animated) organigram to support leadership tasks, particularly with respect to the advantages of such a graphical tool for subordinates. Section 3 introduces the algorithm for designing the tree layout. In Section 4 we apply this algorithm to an organigram of a large company undergoing a change and that uses it to monitor a feedback process.

2. PRACTICAL RELEVANCE FOR LEADERSHIP SUPPORT

Since in a complex and multi-layer hierarchy the enforcement of any leadership measure becomes difficult, the corporate layout can be used to increase pressure on supervisors to enforce subordinate participation owing to the fact that direct orders from higher levels in the hierarchy are relatively inefficient.

The graphic then reveals where specific directives are necessary.

Studies have shown that the possibility of exerting a specific influence on subordinate units, decreases as the hierarchical distance between the supervisor and subordinate increases (Weibler, 2009). This arises mainly from the limited possibilities of supervisors to acquire requisite unit information. Thus,

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3 the effectiveness and efficiency of influence is high where the organisational distance between the supervisor and the subordinate is small (Weibler, 2009, p.

272). This is especially the case for the direct and the indirect supervisor.3 Consequently, a complex hierarchical structure weakens executives’ ability to enforce measures targeted at other managers (e.g. when enforcing a uniform leadership style).

The tree layout makes the impact of order compliance visible and reveals quality of relationships between supervisors and subordinates. For example, several factors may explain why the subordinates of one direct supervisor achieve a particular performance criterion earlier or better than those of another supervisor at the same level, e.g. the existence of a low-quality relationship between the direct supervisor and the subordinates, or an unfavorable relationship between the direct and the indirect supervisor etc. It is well-known that a high-quality relationship between the direct supervisor and subordinate improves the employees’ organisational identification and thus increases satisfaction (see Tangirala et al. (2007) and the literature therein), and/or decreases depersonalization towards customers (Tangirala et al., 2007).

Studies have also shown that the relationship between direct and indirect supervisors affects the performance of the employees. In particular, a favorable relationship between the direct and indirect supervisor enhances the relationship between direct supervisor and employee (Graen et al., 1977), raises employee satisfaction (Pelz, 1952) and increases performance due to greater perceived control (Anderson, Tolsen, Fields, & Thacker, 1990). Thus, detection of weaknesses in such relationships is highly desirable.

Since detection of any vulnerabilities in the organisation is more difficult for highly complex and large structures than for small organisations, the corporate layout can also be used as a tool to support information management.

Thus, it allows one to evaluate immediately which units are affected and need to be targeted with counter measures. Moreover, a graphical layout supports the control of subordinate units.

Since it yields fast information on

performance criteria and allows for comparison between units of the same level, the layout is of particular use to indirect supervisors who need to check the performance of leaders and subordinates and the enforcement of a uniform leadership style (see Weibler (1994)). Furthermore, subordinates expect the indirect supervisor to compensate for or harmonize divergent interests (Weibler, 1994, pp. 150). To do this the indirect or superior supervisors have to receive information on which specific measures are most effective in meeting the set criteria. The tree layout helps provide such information. Implementation of the measures thus highlighted improves trust and satisfaction and helps reduce employee turnover (Costigan, Insinga, Berman, Kranas, & Kureshov, 2011;

Weibler, 1994)

A further reason for using a corporate tree layout is that the employees also benefit from a graphical representation of organisational units and their achievement of specific performance criteria. On the one hand, the layout may strengthen competition among organisational units and improve corporate identity. On the other hand, it can also give an employee the possibility to attract a higher-level supervisor’s attention and gain recognition even if this employee belongs to the so-called ‘out- group’ of the supervisor concerned assuming that the graphical software enables one to zoom in on a specific unit.

This idea is supported by a study (Weibler, 1994) that shows that the employee can influence the indirect supervisor by extraordinary performance.

However, this information has to be communicated to the indirect supervisor.

For out-group employees direct supervisors are less likely to inform supervisors at higher levels. Thus, a scheme such as the corporate tree layout may serve as an objective information carrier.

Representing performance criteria in a corporate layout may also be advantageous for the direct supervisor.

The corporate layout represents the organisational network in such a way that supervisors are seen to act as links connecting employees lower in the hierarchy to the upper management (Tangirala et al., 2007; Likert, 1961). This means that employees may attribute

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4 greater responsibility for any unfavorable events to indirect supervisors (due to power of directive and control). Studies (Meindl, 1990) have shown that this relieves the direct supervisor to some extent.

The corporate layout increases transparency with regard to performance criteria, particularly with respect to management tasks. Depending on the criteria it can also be used to reveal technical or economic dependencies or bottlenecks which may inhibit performance, e.g. or turnover rates.

3. LAYOUT ALGORITHM 3.1. Basic considerations

Consider the synthetic tree layout with 5 hierarchical levels (including the root) as represented in Fig. 1a. When a parent node has many subsequent hierarchical levels and many leaves, it drives out another parent that has only a low number of subsequent hierarchical levels and few leaves. Thus, the resulting tree layout will normally have a large unoccupied area. Arranging the subtrees as identified by their left and right node can avoid such undesirable layouts. Fig.

1b displays the subtrees from level 2 to level 5 side by side (their common parent node is not drawn, but would be placed in the center of an additional row below the displayed figure). Using an appropriate objective function the layout algorithm arranges the subtrees in such a way that the exploitation of the available space for displaying the tree is improved. For example, the leaves at a particular level can be distributed between the free space stemming from a subtree that has many leaves at a higher level.

3.2. Calculation of placement in model coordinates

The coordinates of the vertices in the model are represented using a polar coordinate system, comprising the distance from a vertex to the root in the center and the angle between a vertex, the root and a third fixed point. The edges themselves are sufficiently determined by their adjacent vertices. This has the advantage that the model representation is independent of the size and aspect ratio of the later output device. Thus the layout of the tree can be precalculated for an output device of arbitrary size. The

question is to determine the coordinates of the vertices.

The layout algorithm presented here is based on iteratively aggregating subtrees, starting with single leaves on the deepest level as subtrees. In each iteration, subtrees are aggregated in order

a) Tree

Arrangement with non-overlapping bounding boxes

(b) Simple Arrangement Fig. 1. Synthetic data set.

to optimise the shape of the aggregated tree. Being invariant, each tree constructed in such a way must be a homomorph to the corresponding original tree. The equivalence operations allowed are thus the ordering of subtrees, including vertical flips. In order to calculate efficient, non-overlapping layouts, the layout algorithm requires information about the shape of each subtree. This information is provided by descriptions of the left and right sides of a tree’s convex hull. For the left side, this description is a vector containing the

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5 coordinate of the leftmost vertex in each level of the subtree. Likewise, the right side is described by a vector of the coordinates of the rightmost vertex in each level of a subtree.

This representation can be illustrated using the dark blue4 tree on the left in Fig. 1b as an example. This tree has four levels. On the highest and second highest levels (i.e. in the fourth and third row), it contains only a single vertex. In the second-lowest level (i.e. in the second row), it spans the space of four vertices, although it might actually contain fewer vertices or gaps. In the lowest level, it contains 13 vertices or gaps. The leftmost vertices of this tree from the highest to the lowest level are the vertices in columns seven, six and one.

Thus, the vector of leftmost vertices is Left

= (7,7,6,1). The vector of rightmost vertices is Right = (7,7,9,13), the width- vector of the tree is Width = Right Left + 1

= (1,1,4,13). The corresponding vectors for the rightmost, dark red tree in this figure are Left = (1,1,NA,NA), Right = (1,1,NA,NA), Width = (1,1, 0,0), as this tree has only single vertices in the two uppermost levels, but no vertices in lower levels.

3.3. Algorithms for aggregating subtrees

The problem is specified as follows: given a set of subtrees, each specified by the coordinates of its leftmost and rightmost elements on each level, the objective is to find the optimal arrangement of the subtrees in an aggregated tree. An arrangement is defined either by the clockwise ordering of subtrees in the arrangement, allowing for calculation of the coordinates of the subtrees, or directly these coordinates of the subtrees. A quality measure for arrangements should consider the tree width as well as its symmetry.

Aggregation is carried out using (a) iterative nearest neighbor merging, (b) branch and bound. In (a) for each iteration, the pair among all subtrees leading to the biggest cost reduction is chosen and merged. The overall computational complexity is cubic in the number of subtrees, as the distance between all subtrees must be calculated and updated. In (b) the leftmost elements of the aggregated tree are defined by the first elements in the ordering. Thus, only

lower and upper bounds on the rightmost elements in the tree are required, in order to derive lower and upper bounds on the tree widths on each level. A lower bound on the tree width is obtained by expressing the subtree widths on each level, by relaxing the shape requirements for all subtrees. An upper bound on the tree width is obtained by calculating a bounding box around each subtree. The sum of these bounding box widths can serve as an upper bound for the width of the aggregated tree.

3.3.1. Postprocessing

3.3.1.1. Local smoothing within subtrees.

After the order of subtrees in a tree has been determined, the coordinates of the subtrees can be post-processed such that they are more evenly distributed.

However, this smoothing must not change the boundaries of the tree. Thus, it can only affect inner subtrees.

This post-processing is done iteratively, by first determining the gaps between subtrees and consequently shifting subtrees to less dense regions.

3.3.1.2. Global smoothing.

Another variant of smoothing is to move all nodes in one level, regardless of subtree borders. While this allows a more equivalent distribution of nodes, it can also lead to edge intersections between neighboring nodes when plotting nodes of one level on a circular path instead of a line. Therefore, the maximum offset between the original position and the smoothed position of a node can be specified.

3.3.1.3. Additional gap between subtrees

In order to increase the visibility of the boundaries of subtrees, an additional gap between subtrees can be introduced. This gap is considered after a subtree has been constructed, and before this subtree is aggregated with its siblings.

3.3.1.4. Non-equal distance between levels

As the number of nodes as well as the available circumference differ between levels, the increase of radius between levels can be adjusted to reduce variability among the node densities of different levels.

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6 3.3.1.5. Transformation into Cartesian screen coordinates

The algorithms described above yield the position of each node in the tree in polar coordinates: the depth in the tree corresponds to the radius, while the horizontal position corresponds to the angle. As a result, these coordinates can be transformed into Cartesian screen coordinates.

3.4. Algorithms

The following code presents the steps in the layout algorithm: first, Formula 1 calculates the offset of a leftmost placement of one object on the right of another object. Then Formula 2 finds the shape of the combination of several objects, by calculating the vectors of the leftmost and rightmost vertices in the aggregated object. Algorithm 1 then calculates the cost of an aggregated object, which is again characterized by its leftmost and rightmost elements. In the heuristic in Algorithm 2, an ordering of subtrees is sought that minimizes these costs.

3.4.1. Formula 1

In order to illustrate the idea of the first algorithm, we now extend the example from above, where the shapes of the dark blue and dark red trees in Fig. 1b were calculated. We assume, that the ordering requires the dark red tree to be to the right of the dark blue tree. The algorithm then produces the offset of the dark red tree on the right relative to its predecessor, the dark blue tree on the left.

This offset should be minimal, in order to minimize gaps between the trees. As can be seen from the figure, the vector RightOfLeft of rightmost elements of the left tree and the vector LeftOfRight of leftmost elements of the tree on the right are relevant for calculating this offset. In particular, as the tree on the right only contains vertices in the two upper levels, only these two elements of each of these two vectors are relevant. The offset Offset is now calculated as

where NA-values in RightOfLeft have to be replaced by 1, and NAvalues in LeftOfRight have to be replaced by 1. In the example above, this results in

3.4.2. Formula 2

Once the offset between subtrees is calculated, the shape of the resulting aggregated tree can be determined by the following formula:

where AggregatedLeft is a vector of the leftmost elements in the tree resulting from the combination of K subtrees, Leftk is the vector of leftmost elements in the kth subtree, and Offsetk is the offset of the kth subtree, where the first subtree has an offset of zero.

Likewise, the vector of rightmost elements AggregatedRight can be calculated as In the example above, the tree resulting from the combination of the blue and red trees would have the leftmost elements AggregatedLeft = (7,7,6,1) and the rightmost elements AggregatedRight = (8,8,9,13).

3.4.3. Algorithm 1

This algorithm provides four different cost measures for an aggregated object: The first measure is the area occupied by hull of the object, including area in possible gaps between objects. In the example above, this is Cost = (8 7 + 1) + (8 7 + 1) + (9 6 + 1) + (13 1 + 1) = 2 + 2 + 4 + 13 = 21.

The second measure is the width of a bounding box placed around the aggregated object. This corresponds to the difference between the rightmost and leftmost vertices within all vertices of the aggregated tree (i.e. over all levels). In the example from above this would be Cost = 13 1 + 1 = 13.

The third measure is the variance of the coordinates of the leftmost and rightmost vertices, and the fourth their skewness. Continuing the example from above, these costs are Cost = variance (7,7,6,1,8,8,9,13) = 9.734 and Cost = skewness (7,7,6,1,8,8,9,13) = 0.304, respectively.

The pseudocode is provided in Algorithm 1.

3.4.4. Algorithm 2

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7 This greedy heuristic aims at finding a cost-minimizing ordering of a given set of subtrees into a single, aggregated tree.

Inputs comprise: matrix SubLeft, which consists of the horizontally stacked vectors of leftmost vertices of its subtrees, matrix SubRight, which is a matrix of the horizontally stacked vectors of the rightmost vertices if its subtrees, the cost function to be used.

The greedy heuristic works by first calculating for each possible pair of subtrees the costs of combining them.

Then, in each iteration the cost- minimizing combination is chosen, the two subtrees are combined and the affected combination costs are recalculated, before the next iteration is performed. The algorithm stops after all subtrees have been merged into a single, combined tree.

3.4.5. Algorithm 3 – Bottom up construction

The heart of the algorithm is designed as bottom up construction of the tree involving the following steps:

1. Start in the lowest level with single vertices (= leafs) as subtrees.

2. Combine all subtrees having the same parent in the secondlowest level using Algorithm 2.

3. Once all subtrees in the lowest level are aggregated, proceed to the next higher level, and add the parent vertices to their corresponding aggregated trees.

Place them such that they are in the center of their aggregated tree (i.e. in the middle of the bounding box around the aggregated tree constructed out of their children).

4. If the highest level is not yet reached, then go to 2, i.e. combine all subtrees having the same level in the next higher level using Algorithm 2.

(a) Area as cost function ’

(B)Bounding box width as cost function

(c) Variance as cost function

d) Skewness as cost function Fig. 2. Subtree arrangements of iterative nearest neighbor merging with local smoothing only and using different objective functions.

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4. APPLICATION 4.1. Synthetic data

As an example we consider the synthetic tree rsepresented in Fig. 1a.

Using nearest neighbor merging, different arrangements are obtained depending on the cost function chosen. Fig. 2a to d displays the arrangements obtained from using either area as objective function (Fig.

2a), the width of the bounding box (Fig. 2b), or variance (Fig. 2c) and skewness (Fig. 2d). Each arrangement involves local smoothing. Fig. 3 shows the subtree aggregation by means of branch and bound using the area as cost function.

4.2. Real dataset

The layout algorithm is used to display the organigram of a large company undergoing change. An animated circular tree layout is used to monitor any performance criterion.

Organisational units having fulfilled the criterion can be represented by coloured nodes in the tree. To distinguish between subordinate and supervisor performance different colours are suggested. When all the members of the organisation have access to the layout, the tree can be used not only to initiate competition but also to strengthen a shared identity

Fig. 3. Branch and bound (area as cost function).

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9 (a) Algorithm of Reingold and

Tilford

(b) Nearest neighbor merging and variance as cost function

(c) Nearest neighbor merging and variance as cost function with

increase subtree gaps

(d) Branch and bound, variance as cost function

Fig. 4. Organigram of a large company using different layout

methods.

Fig. 4b to d shows the various circular tree layouts found when using different objective functions. For comparison, Fig. 4a shows a classic layout without subtree aggregation, found by using the algorithm of Reingold and Tilford (1981).

The tree layout is designed such that the hierarchical level of any organisatorical unit can be rapidly identified. It also serves to facilitate comparison of performance of organisatorical units at these same hierarchical level. At first glance this might not seem to be particularly valuable. However, in cases of merger or acquisition, the ability to display organisational units at the same level becomes important. In such a context, it becomes particularly important to ascertain the extent to which a new organisatorical unit adapts to a superior unit. Thus, the layout helps monitor the level of cooperation and thus the performance of old and new organisational units facing restructuring.

In a large organisation the supervisors at a given level will be interested in observing subordinate units. The resulting tree layout found from the algorithm presented here can be modified quickly to accommodate such needs, i.e. the corresponding subtrees can be zoomed in on and displayed on the screen immediately.

5. SUMMARY

A hierarchical tree layout algorithm based on arranging subtrees iteratively is introduced. Among different aesthetic criteria, the resulting layout should cope with unbalanced trees in the sense that some parent vertices have a small number of child vertices and others a large number of child vertices. The algorithm is based on the idea of characterizing any subtree by its leftmost and its rightmost vertex at each level. The subtrees should be arranged as densely as possible.

Different cost functions can be applied in order to meet various aesthetic properties. For large trees a greedy heuristic is preferred in order to solve the resulting optimization problem.

The tree layout is applied to management support in change

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10 processes such as mergers or acquisitions where monitoring the performance of the new organisational units and the cooperation of the old and new units becomes of great importance. Using appropriate performance criteria the graphical representation of the organisational hierarchies can reveal weaknesses in the organisation and improve the effectiveness of leadership instruments, especially in highly complex organisations.

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