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Treemapping

In information visualization and computing, treemapping is a method for displaying hierarchical data using nested figures, usually rectangles.

Treemap of Singapore's exports by product category, 2012. The Product Exports Treemaps are one of the most recent applications of these kind of visualizations, developed by the Harvard-MIT Observatory of Economic Complexity.

Treemaps display hierarchical (tree-structured) data as a set of nested rectangles. Each branch of the tree is given a rectangle, which is then tiled with smaller rectangles representing sub-branches. A leaf node's rectangle has an area proportional to a specified dimension of the data.[1] Often the leaf nodes are colored to show a separate dimension of the data.

When the color and size dimensions are correlated in some way with the tree structure, one can often easily see patterns that would be difficult to spot in other ways, such as whether a certain color is particularly relevant. A second advantage of treemaps is that, by construction, they make efficient use of space. As a result, they can legibly display thousands of items on the screen simultaneously.

Tiling algorithms edit

To create a treemap, one must define a tiling algorithm, that is, a way to divide a region into sub-regions of specified areas. Ideally, a treemap algorithm would create regions that satisfy the following criteria:

  1. A small aspect ratio—ideally close to one. Regions with a small aspect ratio (i.e., fat objects) are easier to perceive.[2]
  2. Preserve some sense of the ordering in the input data (ordered).
  3. Change to reflect changes in the underlying data (high stability).

Unfortunately, these properties have an inverse relationship. As the aspect ratio is optimized, the order of placement becomes less predictable. As the order becomes more stable, the aspect ratio is degraded.[example needed]

Rectangular treemaps edit

To date, fifteen primary rectangular treemap algorithms have been developed:

Treemap algorithms[3]
Algorithm Order Aspect ratios Stability
BinaryTree partially ordered high stable
Slice And Dice[4] ordered very high stable
Strip[5] ordered medium medium stability
Pivot by middle[6] ordered medium medium stability
Pivot by split[6] ordered medium low stability
Pivot by size[6] ordered medium medium stability
Split[7] ordered medium medium stability
Spiral[8] ordered medium medium stability
Hilbert[9] ordered medium medium stability
Moore[9] ordered medium medium stability
Squarified[10] ordered low low stability
Mixed Treemaps[11] unordered low medium stability
Approximation[12] unordered low medium stability
Git[13] unordered medium stable
Local moves[14] unordered medium stable

Convex treemaps edit

Rectangular treemaps have the disadvantage that their aspect ratio might be arbitrarily high in the worst case. As a simple example, if the tree root has only two children, one with weight   and one with weight  , then the aspect ratio of the smaller child will be  , which can be arbitrarily high. To cope with this problem, several algorithms have been proposed that use regions that are general convex polygons, not necessarily rectangular.

Convex treemaps were developed in several steps, each step improved the upper bound on the aspect ratio. The bounds are given as a function of   - the total number of nodes in the tree, and   - the total depth of the tree.

  1. Onak and Sidiropoulos[15] proved an upper bound of  .
  2. De-Berg and Onak and Sidiropoulos[16] improve the upper bound to  , and prove a lower bound of  .
  3. De-Berg and Speckmann and van-der-Weele[17] improve the upper bound to  , matching the theoretical lower bound. (For the special case where the depth is 1, they present an algorithm that uses only four classes of 45-degree-polygons (rectangles, right-angled triangles, right-angled trapezoids and 45-degree pentagons), and guarantees an aspect ratio of at most 34/7.)

The latter two algorithms operate in two steps (greatly simplified for clarity):

  1. The original tree is converted to a binary tree: each node with more than two children is replaced by a sub-tree in which each node has exactly two children.
  2. Each region representing a node (starting from the root) is divided to two, using a line that keeps the angles between edges as large as possible. It is possible to prove that, if all edges of a convex polygon are separated by an angle of at least  , then its aspect ratio is  . It is possible to ensure that, in a tree of depth  , the angle is divided by a factor of at most  , hence the aspect ratio guarantee.

Orthoconvex treemaps edit

In convex treemaps, the aspect ratio cannot be constant - it grows with the depth of the tree. To attain a constant aspect-ratio, Orthoconvex treemaps[17] can be used. There, all regions are orthoconvex rectilinear polygons with aspect ratio at most 64; and the leaves are either rectangles with aspect ratio at most 8, or L-shapes or S-shapes with aspect ratio at most 32.

For the special case where the depth is 1, they present an algorithm that uses only rectangles and L-shapes, and the aspect ratio is at most  ; the internal nodes use only rectangles with aspect ratio at most  .

Other treemaps edit

Voronoi Treemaps
[18] based on Voronoi diagram calculations. The algorithm is iterative and does not give any upper bound on the aspect ratio.
Jigsaw Treemaps[19]
based on the geometry of space-filling curves. They assume that the weights are integers and that their sum is a square number. The regions of the map are rectilinear polygons and highly non-ortho-convex. Their aspect ratio is guaranteed to be at most 4.
GosperMaps
[20] based on the geometry of Gosper curves. It is ordered and stable, but has a very high aspect ratio.

History edit

 
Hard disk space usage visualized in TreeSize, software first released in 1996

Area-based visualizations have existed for decades. For example, mosaic plots (also known as Marimekko diagrams) use rectangular tilings to show joint distributions (i.e., most commonly they are essentially stacked column plots where the columns are of different widths). The main distinguishing feature of a treemap, however, is the recursive construction that allows it to be extended to hierarchical data with any number of levels. This idea was invented by professor Ben Shneiderman at the University of Maryland Human – Computer Interaction Lab in the early 1990s. [21][22] Shneiderman and his collaborators then deepened the idea by introducing a variety of interactive techniques for filtering and adjusting treemaps.

These early treemaps all used the simple "slice-and-dice" tiling algorithm. Despite many desirable properties (it is stable, preserves ordering, and is easy to implement), the slice-and-dice method often produces tilings with many long, skinny rectangles. In 1994 Mountaz Hascoet and Michel Beaudouin-Lafon invented a "squarifying" algorithm, later popularized by Jarke van Wijk, that created tilings whose rectangles were closer to square. In 1999 Martin Wattenberg used a variation of the "squarifying" algorithm that he called "pivot and slice" to create the first Web-based treemap, the SmartMoney Map of the Market, which displayed data on hundreds of companies in the U.S. stock market. Following its launch, treemaps enjoyed a surge of interest, especially in financial contexts.[citation needed]

A third wave of treemap innovation came around 2004, after Marcos Weskamp created the Newsmap, a treemap that displayed news headlines. This example of a non-analytical treemap inspired many imitators, and introduced treemaps to a new, broad audience.[citation needed] In recent years, treemaps have made their way into the mainstream media, including usage by the New York Times.[23][24] The Treemap Art Project produced 12 framed images for the National Academies (United States), shown the Every AlgoRiThm has ART in It exhibit in Washington, DC and another set for the collection of Museum of Modern Art in New York.

See also edit

References edit

  1. ^ Li, Rita Yi Man; Chau, Kwong Wing; Zeng, Frankie Fanjie (2019). "Ranking of Risks for Existing and New Building Works". Sustainability. 11 (10): 2863. doi:10.3390/su11102863.
  2. ^ Kong, N; Heer, J; Agrawala, M (2010). "Perceptual Guidelines for Creating Rectangular Treemaps". IEEE Transactions on Visualization and Computer Graphics. 16 (6): 990–8. CiteSeerX 10.1.1.688.4140. doi:10.1109/TVCG.2010.186. PMID 20975136. S2CID 11597084.
  3. ^ Vernier, E.; Sondag, M.; Comba, J.; Speckmann, B.; Telea, A.; Verbeek, K. (2020). "Quantitative Comparison of Time‐Dependent Treemaps". Computer Graphics Forum. 39 (3): 393–404. arXiv:1906.06014. doi:10.1111/cgf.13989. S2CID 189898065.
  4. ^ Shneiderman, Ben (2001). "Ordered treemap layouts" (PDF). Infovis: 73.
  5. ^ Benjamin, Bederson; Shneiderman, Ben; Wattenberg, Martin (2002). "Ordered and quantum treemaps: Making effective use of 2D space to display hierarchies" (PDF). ACM Transactions on Graphics. 21 (4): 833–854. CiteSeerX 10.1.1.145.2634. doi:10.1145/571647.571649. S2CID 7253456.
  6. ^ a b c Shneiderman, Ben; Wattenberg, Martin (2001). "Ordered treemap layouts". IEEE Symposium on Information Visualization: 73–78.
  7. ^ Engdahl, Björn. "Ordered and quantum treemaps: Making effective use of 2D space to display hierarchies". {{cite journal}}: Cite journal requires |journal= (help)
  8. ^ Tu, Y.; Shen, H. (2007). "Visualizing changes of hierarchical data using treemaps". IEEE Transactions on Visualization and Computer Graphics. 13 (6): 1286–1293. doi:10.1109/TVCG.2007.70529. PMID 17968076. S2CID 14206074.
  9. ^ a b Tak, S.; Cockburn, A. (2013). "Enhanced spatial stability with Hilbert and Moore treemaps". Transactions on Visualization and Computer Graphics. 19 (1): 141–148. doi:10.1109/TVCG.2012.108. PMID 22508907. S2CID 6099935.
  10. ^ Bruls, Mark; Huizing, Kees; van Wijk, Jarke J. (2000). "Squarified treemaps". In de Leeuw, W.; van Liere, R. (eds.). Data Visualization 2000: Proc. Joint Eurographics and IEEE TCVG Symp. on Visualization (PDF). Springer-Verlag. pp. 33–42..
  11. ^ Roel Vliegen; Erik-Jan van der Linden; Jarke J. van Wijk. (PDF). Archived from the original (PDF) on July 24, 2011. Retrieved February 24, 2010.
  12. ^ Nagamochi, H.; Abe, Y.; Wattenberg, Martin (2007). "An approximation algorithm for dissect-ing a rectangle into rectangles with specified areas". Discrete Applied Mathematics. 155 (4): 523–537. doi:10.1016/j.dam.2006.08.005.
  13. ^ Vernier, E.; Comba, J.; Telea, A. (2018). "Quantitative comparison of dy-namic treemaps for software evolution visualization". Conferenceon Software Visualization: 99–106.
  14. ^ Sondag, M.; Speckmann, B.; Verbeek, K. (2018). "Stable treemaps via local moves". Transactions on Visualization and Computer Graphics. 24 (1): 729–738. doi:10.1109/TVCG.2017.2745140. PMID 28866573. S2CID 27739774.
  15. ^ Krzysztof Onak; Anastasios Sidiropoulos. "Circular Partitions with Applications to Visualization and Embeddings". Retrieved June 26, 2011.
  16. ^ Mark de Berg; Onak, Krzysztof; Sidiropoulos, Anastasios (2013). "Fat Polygonal Partitions with Applications to Visualization and Embeddings". Journal of Computational Geometry. 4 (1): 212–239. arXiv:1009.1866.
  17. ^ a b De Berg, Mark; Speckmann, Bettina; Van Der Weele, Vincent (2014). "Treemaps with bounded aspect ratio". Computational Geometry. 47 (6): 683. arXiv:1012.1749. doi:10.1016/j.comgeo.2013.12.008. S2CID 12973376.. Conference version: Convex Treemaps with Bounded Aspect Ratio (PDF). EuroCG. 2011.
  18. ^ Balzer, Michael; Deussen, Oliver (2005). "Voronoi Treemaps". In Stasko, John T.; Ward, Matthew O. (eds.). IEEE Symposium on Information Visualization (InfoVis 2005), 23-25 October 2005, Minneapolis, MN, USA (PDF). IEEE Computer Society. p. 7..
  19. ^ Wattenberg, Martin (2005). "A Note on Space-Filling Visualizations and Space-Filling Curves". In Stasko, John T.; Ward, Matthew O. (eds.). IEEE Symposium on Information Visualization (InfoVis 2005), 23-25 October 2005, Minneapolis, MN, USA (PDF). IEEE Computer Society. p. 24..
  20. ^ Auber, David; Huet, Charles; Lambert, Antoine; Renoust, Benjamin; Sallaberry, Arnaud; Saulnier, Agnes (2013). "Gosper Map: Using a Gosper Curve for laying out hierarchical data". IEEE Transactions on Visualization and Computer Graphics. 19 (11): 1820–1832. doi:10.1109/TVCG.2013.91. PMID 24029903. S2CID 15050386..
  21. ^ Shneiderman, Ben (1992). "Tree visualization with tree-maps: 2-d space-filling approach". ACM Transactions on Graphics. 11: 92–99. doi:10.1145/102377.115768. hdl:1903/367. S2CID 1369287.
  22. ^ Ben Shneiderman; Catherine Plaisant (June 25, 2009). "Treemaps for space-constrained visualization of hierarchies ~ Including the History of Treemap Research at the University of Maryland". Retrieved February 23, 2010.
  23. ^ Cox, Amanda; Fairfield, Hannah (February 25, 2007). "The health of the car, van, SUV, and truck market". The New York Times. Retrieved March 12, 2010.
  24. ^ Carter, Shan; Cox, Amanda (February 14, 2011). "Obama's 2012 Budget Proposal: How $3.7 Trillion is Spent". The New York Times. Retrieved February 15, 2011.

External links edit

  • Treemap Art Project produced exhibit for the National Academies in Washington, DC
  • "Discovering Business Intelligence Using Treemap Visualizations", Ben Shneiderman, April 11, 2006
  • Comprehensive survey and bibliography of Tree Visualization techniques
  • Vliegen, Roel; van Wijk, Jarke J.; van der Linden, Erik-Jan (September–October 2006). (PDF). IEEE Transactions on Visualization and Computer Graphics. 12 (5): 789–796. doi:10.1109/TVCG.2006.200. PMID 17080801. S2CID 18891326. Archived from the original (PDF) on 24 July 2011.
  • History of Treemaps by Ben Shneiderman.
  • Hypermedia exploration with interactive dynamic maps Paper by Zizi and Beaudouin-Lafon introducing the squarified treemap layout algorithm (named "improved treemap layout" at the time).
  • Indiana University description
  • Live interactive treemap based on crowd-sourced discounted deals from Flytail Group
  • Treemap sample in English from The Hive Group
  • Several treemap examples made with Macrofocus TreeMap
  • Visualizations using dynamic treemaps and online treemapping software by drasticdata

treemapping, information, visualization, computing, treemapping, method, displaying, hierarchical, data, using, nested, figures, usually, rectangles, treemap, singapore, exports, product, category, 2012, product, exports, treemaps, most, recent, applications, . In information visualization and computing treemapping is a method for displaying hierarchical data using nested figures usually rectangles Treemap of Singapore s exports by product category 2012 The Product Exports Treemaps are one of the most recent applications of these kind of visualizations developed by the Harvard MIT Observatory of Economic Complexity Treemaps display hierarchical tree structured data as a set of nested rectangles Each branch of the tree is given a rectangle which is then tiled with smaller rectangles representing sub branches A leaf node s rectangle has an area proportional to a specified dimension of the data 1 Often the leaf nodes are colored to show a separate dimension of the data When the color and size dimensions are correlated in some way with the tree structure one can often easily see patterns that would be difficult to spot in other ways such as whether a certain color is particularly relevant A second advantage of treemaps is that by construction they make efficient use of space As a result they can legibly display thousands of items on the screen simultaneously Contents 1 Tiling algorithms 1 1 Rectangular treemaps 1 2 Convex treemaps 1 2 1 Orthoconvex treemaps 1 3 Other treemaps 2 History 3 See also 4 References 5 External linksTiling algorithms editTo create a treemap one must define a tiling algorithm that is a way to divide a region into sub regions of specified areas Ideally a treemap algorithm would create regions that satisfy the following criteria A small aspect ratio ideally close to one Regions with a small aspect ratio i e fat objects are easier to perceive 2 Preserve some sense of the ordering in the input data ordered Change to reflect changes in the underlying data high stability Unfortunately these properties have an inverse relationship As the aspect ratio is optimized the order of placement becomes less predictable As the order becomes more stable the aspect ratio is degraded example needed Rectangular treemaps edit To date fifteen primary rectangular treemap algorithms have been developed Treemap algorithms 3 Algorithm Order Aspect ratios StabilityBinaryTree partially ordered high stableSlice And Dice 4 ordered very high stableStrip 5 ordered medium medium stabilityPivot by middle 6 ordered medium medium stabilityPivot by split 6 ordered medium low stabilityPivot by size 6 ordered medium medium stabilitySplit 7 ordered medium medium stabilitySpiral 8 ordered medium medium stabilityHilbert 9 ordered medium medium stabilityMoore 9 ordered medium medium stabilitySquarified 10 ordered low low stabilityMixed Treemaps 11 unordered low medium stabilityApproximation 12 unordered low medium stabilityGit 13 unordered medium stableLocal moves 14 unordered medium stableConvex treemaps edit Rectangular treemaps have the disadvantage that their aspect ratio might be arbitrarily high in the worst case As a simple example if the tree root has only two children one with weight 1 n displaystyle 1 n nbsp and one with weight 1 1 n displaystyle 1 1 n nbsp then the aspect ratio of the smaller child will be n displaystyle n nbsp which can be arbitrarily high To cope with this problem several algorithms have been proposed that use regions that are general convex polygons not necessarily rectangular Convex treemaps were developed in several steps each step improved the upper bound on the aspect ratio The bounds are given as a function of n displaystyle n nbsp the total number of nodes in the tree and d displaystyle d nbsp the total depth of the tree Onak and Sidiropoulos 15 proved an upper bound of O d log n 17 displaystyle O d log n 17 nbsp De Berg and Onak and Sidiropoulos 16 improve the upper bound to O d log n displaystyle O d log n nbsp and prove a lower bound of O d displaystyle O d nbsp De Berg and Speckmann and van der Weele 17 improve the upper bound to O d displaystyle O d nbsp matching the theoretical lower bound For the special case where the depth is 1 they present an algorithm that uses only four classes of 45 degree polygons rectangles right angled triangles right angled trapezoids and 45 degree pentagons and guarantees an aspect ratio of at most 34 7 The latter two algorithms operate in two steps greatly simplified for clarity The original tree is converted to a binary tree each node with more than two children is replaced by a sub tree in which each node has exactly two children Each region representing a node starting from the root is divided to two using a line that keeps the angles between edges as large as possible It is possible to prove that if all edges of a convex polygon are separated by an angle of at least ϕ displaystyle phi nbsp then its aspect ratio is O 1 ϕ displaystyle O 1 phi nbsp It is possible to ensure that in a tree of depth d displaystyle d nbsp the angle is divided by a factor of at most d displaystyle d nbsp hence the aspect ratio guarantee Orthoconvex treemaps edit In convex treemaps the aspect ratio cannot be constant it grows with the depth of the tree To attain a constant aspect ratio Orthoconvex treemaps 17 can be used There all regions are orthoconvex rectilinear polygons with aspect ratio at most 64 and the leaves are either rectangles with aspect ratio at most 8 or L shapes or S shapes with aspect ratio at most 32 For the special case where the depth is 1 they present an algorithm that uses only rectangles and L shapes and the aspect ratio is at most 2 2 3 3 15 displaystyle 2 2 sqrt 3 approx 3 15 nbsp the internal nodes use only rectangles with aspect ratio at most 1 3 2 73 displaystyle 1 sqrt 3 approx 2 73 nbsp Other treemaps edit Voronoi Treemaps 18 based on Voronoi diagram calculations The algorithm is iterative and does not give any upper bound on the aspect ratio Jigsaw Treemaps 19 based on the geometry of space filling curves They assume that the weights are integers and that their sum is a square number The regions of the map are rectilinear polygons and highly non ortho convex Their aspect ratio is guaranteed to be at most 4 GosperMaps 20 based on the geometry of Gosper curves It is ordered and stable but has a very high aspect ratio History edit nbsp Hard disk space usage visualized in TreeSize software first released in 1996Area based visualizations have existed for decades For example mosaic plots also known as Marimekko diagrams use rectangular tilings to show joint distributions i e most commonly they are essentially stacked column plots where the columns are of different widths The main distinguishing feature of a treemap however is the recursive construction that allows it to be extended to hierarchical data with any number of levels This idea was invented by professor Ben Shneiderman at the University of Maryland Human Computer Interaction Lab in the early 1990s 21 22 Shneiderman and his collaborators then deepened the idea by introducing a variety of interactive techniques for filtering and adjusting treemaps These early treemaps all used the simple slice and dice tiling algorithm Despite many desirable properties it is stable preserves ordering and is easy to implement the slice and dice method often produces tilings with many long skinny rectangles In 1994 Mountaz Hascoet and Michel Beaudouin Lafon invented a squarifying algorithm later popularized by Jarke van Wijk that created tilings whose rectangles were closer to square In 1999 Martin Wattenberg used a variation of the squarifying algorithm that he called pivot and slice to create the first Web based treemap the SmartMoney Map of the Market which displayed data on hundreds of companies in the U S stock market Following its launch treemaps enjoyed a surge of interest especially in financial contexts citation needed A third wave of treemap innovation came around 2004 after Marcos Weskamp created the Newsmap a treemap that displayed news headlines This example of a non analytical treemap inspired many imitators and introduced treemaps to a new broad audience citation needed In recent years treemaps have made their way into the mainstream media including usage by the New York Times 23 24 The Treemap Art Project produced 12 framed images for the National Academies United States shown the Every AlgoRiThm has ART in It exhibit in Washington DC and another set for the collection of Museum of Modern Art in New York See also editDisk space analyzer Data and information visualization List of countries by economic complexity which includes a list of Products Exports Treemaps Marimekko Chart a similar concept with one level of explicit hierarchy References edit Li Rita Yi Man Chau Kwong Wing Zeng Frankie Fanjie 2019 Ranking of Risks for Existing and New Building Works Sustainability 11 10 2863 doi 10 3390 su11102863 Kong N Heer J Agrawala M 2010 Perceptual Guidelines for Creating Rectangular Treemaps IEEE Transactions on Visualization and Computer Graphics 16 6 990 8 CiteSeerX 10 1 1 688 4140 doi 10 1109 TVCG 2010 186 PMID 20975136 S2CID 11597084 Vernier E Sondag M Comba J Speckmann B Telea A Verbeek K 2020 Quantitative Comparison of Time Dependent Treemaps Computer Graphics Forum 39 3 393 404 arXiv 1906 06014 doi 10 1111 cgf 13989 S2CID 189898065 Shneiderman Ben 2001 Ordered treemap layouts PDF Infovis 73 Benjamin Bederson Shneiderman Ben Wattenberg Martin 2002 Ordered and quantum treemaps Making effective use of 2D space to display hierarchies PDF ACM Transactions on Graphics 21 4 833 854 CiteSeerX 10 1 1 145 2634 doi 10 1145 571647 571649 S2CID 7253456 a b c Shneiderman Ben Wattenberg Martin 2001 Ordered treemap layouts IEEE Symposium on Information Visualization 73 78 Engdahl Bjorn Ordered and quantum treemaps Making effective use of 2D space to display hierarchies a href Template Cite journal html title Template Cite journal cite journal a Cite journal requires journal help Tu Y Shen H 2007 Visualizing changes of hierarchical data using treemaps IEEE Transactions on Visualization and Computer Graphics 13 6 1286 1293 doi 10 1109 TVCG 2007 70529 PMID 17968076 S2CID 14206074 a b Tak S Cockburn A 2013 Enhanced spatial stability with Hilbert and Moore treemaps Transactions on Visualization and Computer Graphics 19 1 141 148 doi 10 1109 TVCG 2012 108 PMID 22508907 S2CID 6099935 Bruls Mark Huizing Kees van Wijk Jarke J 2000 Squarified treemaps In de Leeuw W van Liere R eds Data Visualization 2000 Proc Joint Eurographics and IEEE TCVG Symp on Visualization PDF Springer Verlag pp 33 42 Roel Vliegen Erik Jan van der Linden Jarke J van Wijk Visualizing Business Data with Generalized Treemaps PDF Archived from the original PDF on July 24 2011 Retrieved February 24 2010 Nagamochi H Abe Y Wattenberg Martin 2007 An approximation algorithm for dissect ing a rectangle into rectangles with specified areas Discrete Applied Mathematics 155 4 523 537 doi 10 1016 j dam 2006 08 005 Vernier E Comba J Telea A 2018 Quantitative comparison of dy namic treemaps for software evolution visualization Conferenceon Software Visualization 99 106 Sondag M Speckmann B Verbeek K 2018 Stable treemaps via local moves Transactions on Visualization and Computer Graphics 24 1 729 738 doi 10 1109 TVCG 2017 2745140 PMID 28866573 S2CID 27739774 Krzysztof Onak Anastasios Sidiropoulos Circular Partitions with Applications to Visualization and Embeddings Retrieved June 26 2011 Mark de Berg Onak Krzysztof Sidiropoulos Anastasios 2013 Fat Polygonal Partitions with Applications to Visualization and Embeddings Journal of Computational Geometry 4 1 212 239 arXiv 1009 1866 a b De Berg Mark Speckmann Bettina Van Der Weele Vincent 2014 Treemaps with bounded aspect ratio Computational Geometry 47 6 683 arXiv 1012 1749 doi 10 1016 j comgeo 2013 12 008 S2CID 12973376 Conference version Convex Treemaps with Bounded Aspect Ratio PDF EuroCG 2011 Balzer Michael Deussen Oliver 2005 Voronoi Treemaps In Stasko John T Ward Matthew O eds IEEE Symposium on Information Visualization InfoVis 2005 23 25 October 2005 Minneapolis MN USA PDF IEEE Computer Society p 7 Wattenberg Martin 2005 A Note on Space Filling Visualizations and Space Filling Curves In Stasko John T Ward Matthew O eds IEEE Symposium on Information Visualization InfoVis 2005 23 25 October 2005 Minneapolis MN USA PDF IEEE Computer Society p 24 Auber David Huet Charles Lambert Antoine Renoust Benjamin Sallaberry Arnaud Saulnier Agnes 2013 Gosper Map Using a Gosper Curve for laying out hierarchical data IEEE Transactions on Visualization and Computer Graphics 19 11 1820 1832 doi 10 1109 TVCG 2013 91 PMID 24029903 S2CID 15050386 Shneiderman Ben 1992 Tree visualization with tree maps 2 d space filling approach ACM Transactions on Graphics 11 92 99 doi 10 1145 102377 115768 hdl 1903 367 S2CID 1369287 Ben Shneiderman Catherine Plaisant June 25 2009 Treemaps for space constrained visualization of hierarchies Including the History of Treemap Research at the University of Maryland Retrieved February 23 2010 Cox Amanda Fairfield Hannah February 25 2007 The health of the car van SUV and truck market The New York Times Retrieved March 12 2010 Carter Shan Cox Amanda February 14 2011 Obama s 2012 Budget Proposal How 3 7 Trillion is Spent The New York Times Retrieved February 15 2011 External links edit nbsp Wikimedia Commons has media related to Treemaps Treemap Art Project produced exhibit for the National Academies in Washington DC Discovering Business Intelligence Using Treemap Visualizations Ben Shneiderman April 11 2006 Comprehensive survey and bibliography of Tree Visualization techniques Vliegen Roel van Wijk Jarke J van der Linden Erik Jan September October 2006 Visualizing Business Data with Generalized Treemaps PDF IEEE Transactions on Visualization and Computer Graphics 12 5 789 796 doi 10 1109 TVCG 2006 200 PMID 17080801 S2CID 18891326 Archived from the original PDF on 24 July 2011 History of Treemaps by Ben Shneiderman Hypermedia exploration with interactive dynamic maps Paper by Zizi and Beaudouin Lafon introducing the squarified treemap layout algorithm named improved treemap layout at the time Indiana University description Live interactive treemap based on crowd sourced discounted deals from Flytail Group Treemap sample in English from The Hive Group Several treemap examples made with Macrofocus TreeMap Visualizations using dynamic treemaps and online treemapping software by drasticdata Retrieved from https en wikipedia org w index php title Treemapping amp oldid 1179145602, wikipedia, wiki, book, books, library,

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