Adjacency-Preserving Spatial Treemaps
Total Page：16
File Type：pdf, Size：1020Kb
Adjacency-Preserving Spatial Treemaps Kevin Buchin1, David Eppstein2, Maarten Loffler¨ 2, Martin Nollenburg¨ 3, and Rodrigo I. Silveira4 1 Dept. of Mathematics and Computer Science, TU Eindhoven. 2 Dept. of Computer Science, University of California, Irvine. 3 Institute of Theoretical Informatics, Karlsruhe Institute of Technology. 4 Dept. de Matematica` Aplicada II, Universitat Politecnica` de Catalunya. Abstract. Rectangular layouts, subdivisions of an outer rectangle into smaller rectangles, have many applications in visualizing spatial information, for instance in rectangular cartograms in which the rectangles represent geographic or political regions. A spatial treemap is a rectangular layout with a hierarchical structure: the outer rectangle is subdivided into rectangles that are in turn subdivided into smaller rectangles. We describe algorithms for transforming a rectangular layout that does not have this hierarchical structure, together with a clustering of the rectangles of the layout, into a spatial treemap that respects the clustering and also respects to the extent possible the adjacencies of the input layout. 1 Introduction Spatial treemaps are an effective technique to visualize two-dimensional hierarchical information. They display hierarchical data by using nested rectangles in a space-filling layout. Each rectangle represents a geometric or geographic region, which in turn can be subdivided recursively into smaller regions. On lower levels of the recursion, rectangles can also be subdivided based on non-spatial attributes. Typically, at the lowest level some attribute of interest of the region is summarized by using properties like area or color. Treemaps were originally proposed to represent one-dimensional information in two dimensions [14]. How- ever, they are well suited to represent spatial—two-dimensional—data because the containment metaphor of the nested rectangles has a natural geographic meaning, and two-dimensional data makes an efficient use of space [18]. Spatial treemaps are closely related to rectangular cartograms [13]: distorted maps where each region is represented by a rectangle whose area corresponds to a numerical attribute such as population. Rectangular cartograms can be seen as spatial treemaps with only one level; multi-level spatial treemaps in which every rectangle corresponds to a region are also known as rectangular hierarchical cartograms [15, 16]. Spatial treemaps and rectangular cartograms have in common that it is essential to preserve the recognizability of arXiv:1105.0398v1 [cs.CG] 2 May 2011 the regions shown [17]. Most previous work on spatial treemaps reflects this by focusing on the preservation of distances between the rectangular regions and their geographic counterparts (that is, they minimize the displacement of the regions). However, often small displacement does not imply recognizability (swapping the position of two small neighboring countries can result in small displacement, but a big loss of recog- nizability). In the case of cartograms, most emphasis has been put on preserving adjacencies between the geographic regions. It has also been shown that while preserving the topology it is possible to keep the displacement error small [3, 17]. In this paper we are interested in constructing high-quality spatial treemaps by prioritizing the preservation of topology, following a principle already used for rectangular cartograms. Previous work on treemaps has recognized that preserving neighborhood relationships and relative positions between the regions were important criteria [7,11,18], but we are not aware of treemap algorithms that put the emphasis on preserving topology. The importance of preserving adjacencies in spatial treemaps can be appreciated by viewing a concrete example. Figure 1, from [15], shows a spatial treemap of property transactions in London between 2000 and 2008, with two levels formed by the boroughs and wards of London and colors representing average prices. To see whether housing prices of neighboring wards are correlated, it is important to preserve adjacencies: 2 Buchin et al. Bounds Green White Hart Lane Northumberland Park Hatch End Harrow Weald Stanmore Park Canons Cockfosters Alexandra Endlebury Chingford Green Hatch Lane Hale End and Highams Park Monkhams Underhill High Barnet East Barnet Highlands Chase Turkey Street Enfield Lock Valley Edgware Fairlop Chadwell Bridge Hainault Noel Park Tottenham Hale Headstone North Wealdstone Marlborough Belmont Woodside Pinner Larkswood Chapel End Wood Street Edgware Totteridge Oakleigh Brunswick Park Southbury Enfield Highway Higham Hill Grange Town Roding Clayhall Fullwell Aldborough Fortis Green Church End Bruce Grove Queensbury Southgate Hornsey Headstone South Pinner South Greenhill West Green WalthamHoe StreetForest William Morris HarrowWest Harrow Muswell Hill Edmonton Green Haringey Seven Kings Woodhouse RedbridgeBarkingside Mill Hill Bush Hill Park Cranbrook Kenton West Southgate Green Winchmore Hill Newbury Hale Leytonstone Crouch End Harringay Tottenham Green Forest Enfield High Street Leyton Rayners Lane Roxeth Harrow on the Hill Snaresbrook Jubilee Ponders End Goodmayes Kenton East Roxbourne Palmers Green Haselbury Markhouse Grove Green Clementswood Coppetts Bowes Lower Edmonton Stroud Green St Ann's Seven Sisters Wanstead Loxford Highgate Lea Bridge Cathall Upper Edmonton Valentines Cann Hall Mayfield Burnt Oak Eastcote and East Ruislip Finchley Church End Northwood Northwood Hills Cavendish West Finchley Harefield Brownswood Springfield Colindale Hampstead Town Leabridge Chadwell Heath Havering Park Tollington Highbury West Highbury East Whalebone Valence Eastbrook Garden Suburb Fortune Green Frognal and Fitzjohns New River Mawneys Heaton Gooshays Harold Wood Uxbridge North Hendon Ickenham West Ruislip Hillrise Highgate Manor Kentish Town Lordship Cazenove Hackney Downs King's Park Barnet Becontree Parsloes Heath Swiss Cottage Gospel Oak Junction Mildmay Brooklands Emerson Park Cranham Uxbridge South Hillingdon East Childs Hill Pettits Squirrel's Heath South Ruislip Golders Green Stoke Newington Central Hackney Central Charville East Finchley Chatham Canonbury Abbey Finsbury Park West Hendon Haverstock Cantelowes West Hampstead Mayesbrook Goresbrook CamdenCamden Town with Primrose Hill Alibon Hillingdon St Pancras and Somers Town Islington Hackney King's Cross Clissold Barking St George's Holloway South Hornchurch St Peter's Queensbridge Havering West Drayton Barnhill Kenton Dalston Wick Brunel Fryent Welsh Harp Hylands Kilburn Belsize St Mary's Bloomsbury Holborn and Covent Garden Longbridge Dollis Hill Regent's Park Queensbury St Andrew's Upminster Botwell Northwick Park Yeading Caledonian Barnsbury Clerkenwell Haggerston Rainham and Wennington Bunhill Hoxton Victoria Gascoigne Romford Town Elm Park Village Thames Eastbury De Beauvoir Mapesbury St Charles River Heathrow Villages Barnhill Hacton Yiewsley Pinkwell Preston Colville Pembridge Campden Townfield Notting Barns Queen's Park Norland East Ham North Golborne Green Street East Manor Park Dudden Hill Brondesbury Park Maida Vale Cripplegate East Ham South Forest Gate North Plaistow South Little Ilford Sudbury Tokyngton Queen's Gate Brompton Regent's Park Plaistow North Forest Gate South Northolt Mandeville North Greenford Greenford Green Harrow Road Perivale Stratford and New Town Abbey Road Marylebone High Street Wembley CentralBrentWillesden Green Westbourne Stonebridge Kilburn Royal Docks Holland KensingtonAbingdon Canning Town North Canning Town South Wall End Courtfield Hans Town West Ham Custom House Beckton City of London Boleyn East Ham Central Little Venice Green Street West Alperton Kensal Green Queens Park West End Newham Harlesden Northolt West End Greenford Broadway Hanger Hill East Acton Royal Hospital St James's Bethnal Green North WestminsterBayswater Weavers Stanley Church Street Charlton Woolwich Riverside Thamesmead Moorings College Park and Old Oak Earl's Court Redcliffe Cremorne Bow East Dormers Wells Mile End and Globe Town East India and Lansbury Avonmore and Brook Green Bromley-by-Bow Lady Margaret Addison Warwick Vincent Square Spitalfields and Banglatown Hyde Park Wormholt and White City Peninsula Shepherd's Bush Green Woolwich Common Bryanston and Dorset Square Bethnal Green South Ealing Broadway Acton Central Cleveland Thamesfield St Mary's Park Queenstown Blackwall and Cubitt Town Askew Knightsbridge and Belgravia Southall Broadway Fulham Reach Fulham Broadway Churchill Tachbrook North End Lancaster Gate Glyndon Abbey Wood Ealing Bow West Blackheath Westcombe Mile End East Tower Hamlets Kidbrooke with Hornfair Hobbayne Elthorne Hammersmith Greenwich Palace Riverside St Katharine's and Wapping Eltham West Shooters Hill Greenwich West Clapham Town Bishop's Prince's Vassall Ravenscourt Park South Acton Ealing Common Town Shaftesbury Northcote Shadwell Southall Green Oval Latchmere West Putney Whitechapel Middle Park and Sutcliffe Coldharbour and New Eltham Eltham North Hammersmith Broadway St Dunstan's and Stepney Green Millwall Eltham South Walpole Munster Parsons Green and Walham Limehouse Norwood Green Stockwell Plumstead Southfield Sands End Larkhall Herne Hill Northfield Clapham Common Coldharbour East Putney Fairfield Balham East Walworth Heston West Coombe Hill Coombe Vale Thurlow Park Cathedrals North End Osterley and Spring Grove Chiswick Homefields Tudor Tulse Hill Brentford Canbury Ferndale Riverside Erith Rotherhithe Belvedere Lesnes Abbey Heston East Thornton Thamesmead East Cranford Lambeth