We study an agglomerative clustering problem motivated by interactive glyphs
in geo-visualization. Consider a set of disjoint square glyphs on an
interactive map. When the user zooms out, the glyphs grow in size relative to
the map, possibly with different speeds. When two glyphs intersect, we wish to
replace them by a new glyph that captures the information of the intersecting
glyphs.
  We present a fully dynamic kinetic data structure that maintains a set of $n$
disjoint growing squares. Our data structure uses $O(n (\log n \log\log n)^2)$
space, supports queries in worst case $O(\log^3 n)$ time, and updates in
$O(\log^7 n)$ amortized time. This leads to an $O(n\alpha(n)\log^7 n)$ time
algorithm to solve the agglomerative clustering problem. This is a significant
improvement over the current best $O(n^2)$ time algorithms.Comment: 14 pages, 7 figure

Castermans, Thom

Speckmann, Bettina

Staals, Frank

Verbeek, Kevin

English

arXiv

NARCIS 

Agglomerative clustering of growing squares

Pure OAI Repository

 Agglomerative clustering of growing squaresCitation for published version (APA):Castermans, T., Speckmann, B., Staals, F., & Verbeek, K. (2018). Agglomerative clustering of growing squares.8:1-8:6. Abstract from 34th European Workshop on Computational Geometry (EuroCG2018), Berlin, Germany.Document status and date:Published: 21/03/2018Document Version:Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)Please check the document version of this publication:• A submitted manuscript is the version of the article upon submission and before peer-review. There can beimportant differences between the submitted version and the official published version of record. Peopleinterested in the research are advised to contact the author for the final version of the publication, or visit theDOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and pagenumbers.Link to publicationGeneral rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            • Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying the publication in the public portal.If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, pleasefollow below link for the End User Agreement:www.tue.nl/taverneTake down policyIf you believe that this document breaches copyright please contact us at:openaccess@tue.nlproviding details and we will investigate your claim.Download date: 04. Oct. 2023Agglomerative Clustering of Growing Squares∗Thom Castermans1, Bettina Speckmann1, Frank Staals2, andKevin Verbeek11 TU Eindhoven[t.h.a.castermans|b.speckmann|k.a.b.verbeek]@tue.nl2 Utrecht Universityf.staals@uu.nl1 IntroductionWe study an agglomerative clustering problem motivated by interactive glyphs in geo-visualization. GlamMap [5] is a visual analytics tool for the eHumanities which allows theuser to interactively explore metadata of a book collection. Each book is depicted by asquare, color-coded by publication year, and placed on a map according to the location of itspublisher. Overlapping squares are recursively aggregated into a larger glyph until all glyphsare disjoint. As the user zooms out, the glyphs “grow” relative to the map to remain legible.As glyphs start to overlap, they are merged into larger glyphs to keep the map clear anduncluttered. To allow the user to filter and browse real world data sets at interactive speedwe hence need an efficient agglomerative clustering algorithm for growing squares (glyphs).Formal problem statement. Let P be a set of points in R2. Each point p ∈ P has apositive weight pw. Given a “time” parameter t, we interpret the points in P as squares.More specifically, let p(t) be the square centered at p with width tpw. For ease of expositionwe assume all point locations to be unique. Furthermore, we refer to P as a set of squaresrather than a set of center points of squares. Observe that initially, i.e. at t = 0, all squaresin P are disjoint. As t increases, the squares in P grow, and hence they may start to intersect.When two squares p(t) and q(t) intersect at time t, we remove both p and q and replacethem by a new point z = κp + (1 − κ)q, with κ = pw/(pw + qw), of weight zw = pw + qw.Our goal is to compute the complete sequence of events where squares intersect and merge.Results. We present a fully dynamic data structure that can maintain a set P of n disjointgrowing squares. Our data structure can report the first time two squares in P intersect,and supports updates (inserting or deleting a square) in O(log7 n) amortized time. Queriesasking whether a query square q currently intersects a square p in P take O(log3 n) time,and the space usage is O(n(log n log log n)2). Using this data structure we can compute theagglomerative clustering for n squares in O(nα(n) log7 n) time. Here, α is the extremelyslowly growing inverse Ackermann function. To the best of our knowledge, this is the firstfully dynamic clustering algorithm which beats the straightforward O(n2 log n) time bound.This abstract focuses on the update and query times for our data structure. Omitted proofsand detailed bounds on space usage, as well as related discussions on the relation betweencanonical subsets in dominance queries, can be found in the full version [4].Related Work. Funke, Krumpe, and Storandt [6] introduced so-called “ball tournaments”.Their input is a set of balls in Rd with an associated set of priorities. The balls growlinearly and whenever two balls touch, the lower priority ball is eliminated. The goal is to∗ The Netherlands Organisation for Scientific Research (NWO) is supporting T. Castermans (projectnumber 314.99.117), B. Speckmann (project number 639.023.208), F. Staals (project number 612.001.651),and K. Verbeek (project number 639.021.541).34th European Workshop on Computational Geometry, Berlin, Germany, March 21–23, 2018.This is an extended abstract of a presentation given at EuroCG’18. It has been made public for the benefit of thecommunity and should be considered a preprint rather than a formally reviewed paper. Thus, this work is expectedto appear eventually in more final form at a conference with formal proceedings and/or in a journal.T. Castermans, B. Speckmann, F. Staals and K. Verbeek 8:3◮ Lemma 2.3. Let t∗ be the time that a square p of a point p ∈ D(q) touches q. We have(i) rq(t∗)y = ℓp(t∗)y, and ℓp(t∗) is the point with minimumy-coordinate among the points in L−(q)(t∗) at time t∗ if p ∈ D−(q), and(ii) rq(t∗)x = ℓp(t∗)x, and ℓp(t∗) is the point with minimumx-coordinate among the points in L+(q)(t∗) at time t∗ if p ∈ D+(q).3 A Kinetic Data Structure for Growing SquaresWe describe a data structure that can detect intersections between all pairs of squares p,qin P such that p ∈ D+(q). We build an analogous data structure for p ∈ D−(q), and thenuse four copies of these data structures, one for each quadrant, to detect the first intersectionamong all pairs of squares.3.1 The Data StructureOur data structure consists of two three-layered trees T L and T R, and a set of certificateslinking nodes from T L and T R. These trees essentially form two 3D range trees on thecenters of the squares in P , taking the third coordinate pγ of each point to be their rank inthe order (from left to right) along the line γ. The third layer of T L doubles as a kinetictournament tracking the bottom left vertices of squares. Similarly, T R tracks the top rightvertices of the squares.The Layered Trees. The tree T L is a 3D-range tree storing the center points in P . Eachlayer is implemented by a bb[α] tree [8], and each node µ corresponds to a canonical subsetPµ of points stored in the leaves of the subtree rooted at µ. The points are ordered onx-coordinate first, then on y-coordinate, and finally on γ-coordinate. Let Lµ denote the setof bottom left vertices of squares corresponding to the set Pµ, for some node µ.Consider the associated structure XLv of some secondary node v. We consider XLv asa kinetic tournament on the x-coordinates of the points Lv [1]. More specifically, everyinternal node w ∈ XLv corresponds to a set of points Pw consecutive along the line γ. Sincethe γ-coordinates of a point p and its bottom left vertex ℓp are equal, this means w alsocorresponds to a set of consecutive bottom left vertices Lw. Node w stores the vertex ℓp inLw with minimum x-coordinate, and will maintain certificates that guarantee this [1].The tree T R has the same structure as T L: it is a three-layered range tree on the centerpoints in P . The difference is that a ternary structure XRv , for some secondary node v, formsa kinetic tournament maintaining the maximum x-coordinate of the points in Rv, where Rvare the top right vertices of the squares (with center points) in Pv. Hence, every ternarynode z ∈ XRv stores the vertex rq with maximum x-coordinate among Rv. Let XL and X Rdenote the set of all kinetic tournament nodes in T L and T R, respectively.Linking the Trees. Next, we describe how to add linking certificates between the kinetictournament nodes in the trees T L and T R that guarantee the squares are disjoint. Morespecifically, we describe the certificates, between nodes w ∈ X L and z ∈ X R, that guaranteethat the squares p and q are disjoint, for all pairs q ∈ P and p ∈ D+(q).Consider a point q. There are O(log2 n) nodes in the secondary trees of T L, whosecanonical subsets together represent exactly D(q). For each of these nodes v we can thenfind O(log n) nodes in XLv representing the points in L+(q). So, in total q is interested in aset QL(q) of O(log3 n) kinetic tournament nodes. It now follows from Lemma 2.3 that if wewere to add certificates certifying that rq is left of the point stored at the nodes in QL(q) wecan detect when q intersects with a square of a point in D+(q). However, as there mayEuroCG’18T. Castermans, B. Speckmann, F. Staals and K. Verbeek 8:5◮ Lemma 3.3. Let p and q, with p ∈ D+(q), be the first pair of squares to intersect, atsome time t∗. There is a pair of nodes w, z that have a linking certificate that fails at time t∗.From Lemma 3.3 it follows that we can now detect the first intersection between a pair ofsquares p,q, with p ∈ D+(q). We define an analogous data structure for when p ∈ D−(q).Following Lemma 2.3, the kinetic tournaments will maintain the vertices with minimum andmaximum y-coordinate for this case. We then again link up the kinetic tournament nodes inthe two trees appropriately.Space Usage. Our trees T L and T R are range trees in R3, and thus use O(n log2 n)space. However, it is easy to see that this is dominated by the space required to storethe certificates. For all O(n log2 n) kinetic tournament nodes we store at most O(log3 n)certificates (Lemma 3.1), and thus the total space used by our data structure is O(n log5 n).In the full version [4], we show that the number of certificates that we maintain (and thusthe space used by our data structure) is actually only O(n(log n log log n)2).3.2 Inserting or Deleting a SquareAt an insertion or deletion of a square p we proceed in three steps. (1) We update TL andT R, restoring range tree properties, and ensure that the ternary data structures are correctkinetic tournaments. (2) For each kinetic tournament node in X L affected by the update,we query T R to find a new set of linking certificates. We update X R analogously. (3) Weupdate the global event queue.◮ Lemma 3.4. Inserting or deleting a square in T L takes O(log3 n) amortized time.Clearly we can update T R in O(log3 n) amortized time as well. Next, we update the linkingcertificates. We say that a kinetic tournament node w in T L is affected by an update if (i)the update added or removed a leaf node in the subtree rooted at w, (ii) node w was involvedin a tree rotation, or (iii) w occurs in a newly built associated tree XLv (for some node v).Let X Li denote the set of nodes affected by update i (XRi of TR is defined analogously). Foreach node w ∈ X Li , we query TR to find the set of O(log3 n) nodes whose canonical subsetsrepresent QR(w). For each node z in this set, we test if we have to add a linking certificatebetween w and z. As we show next, this takes constant time for each node z, and thusO(∑i |XLi | log3 n) time in total, for all nodes w (analogously for X Ri ).We have to add a link between a node z ∈ QR(w) and w if and only if we also havew ∈ QL(z). We test this as follows. Let v be the node whose associated tree XLv contains w,and let u be the node in T L whose associated tree contains v. We have that w ∈ QL(z) ifand only if u ∈ C(T L, [mzx, ∞)), v ∈ C(Tu, [mzy, ∞)), and w ∈ C(XLv , [mzγ , ∞)). We can testeach of these conditions in constant time:◮ Observation 3.5. Let q be a query point in R1, let w be a node in a binary search tree T ,and let xp = min Pp of the parent p of w in T , or xp = −∞ if no such node exists. We havethat w ∈ C(T, [q, ∞)) if and only if q ≤ min Pw and q > xp.Finally, we delete all certificates involving no longer existing nodes from our global eventqueue, and replace them by all newly created certificates. This takes O(log n) time percertificate. We charge the cost of deleting a certificate to when it gets created. Since everynode w affected creates at most O(log3 n) new certificates, all that remains is to boundthe total number of affected nodes. Here we can use basically the same argument as whenbounding the update time.◮ Lemma 3.6. Inserting a disjoint square into P , or deleting a square from P takes O(log7 n)amortized time.EuroCG’188:6 Agglomerative Clustering of Growing Squares3.3 Running the SimulationAll that remains is to analyze the number of events processed, and the time to do so. Sinceeach failure of a linking certificate produces an intersection, and thus an update, the numberof such events is at most the number of updates. To bound the number of events created bythe tournament trees we use an argument similar to that of Agarwal et al. [1].◮ Theorem 3.7. We can maintain a set P of n disjoint growing squares in a fully dynamicdata structure such that we can detect the first time that a square q intersects with a squarep, with p ∈ D+(q). Our data structure uses O(n(log n log log n)2) space, supports updatesin O(log7 n) amortized time, and queries in O(log3 n) time. For a sequence of m operations,the structure processes a total of O(mα(n) log3 n) events in a total of O(mα(n) log7 n) time.To simulate the process of growing the squares in P , we now maintain eight copies of thedata structure from Theorem 3.7: two data structures for each quadrant (one for D+, theother for D−). Using these data structures we obtain the following agglomerative glyphclustering solution.◮ Theorem 3.8. Given a set of n initial square glyphs P , we can compute an agglomerativeclustering of the squares in P in O(nα(n) log7 n) time using O(n(log n log log n)2) space.References1 P. K. Agarwal, H. Kaplan, and M. Sharir. Kinetic and Dynamic Data Structures for ClosestPair and All Nearest Neighbors. ACM Transactions on Algorithms, 5(1):4:1–4:37, 2008.2 H.-K. Ahn, S. W. Bae, J. Choi, M. Korman, W. Mulzer, E. Oh, J.-W. Park, A. van Renssen,and A. Vigneron. Faster Algorithms for Growing Prioritized Disks and Rectangles. InProceedings of the 28th International Symposium on Algorithms and Computation, pages3:1–3:13, 2017.3 G. Alexandron, H. Kaplan, and M. Sharir. Kinetic and dynamic data structures for convexhulls and upper envelopes. Computational Geometry: Theory and Applications, 36(2):144–1158, 2007.4 T. Castermans, B. Speckmann, F. Staals, and K. Verbeek. Agglomerative clustering ofgrowing squares. ArXiv e-prints, 2017. arXiv:1706.10195.5 T. Castermans, B. Speckmann, K. Verbeek, M. A. Westenberg, A. Betti, and H. van denBerg. GlamMap: Geovisualization for e-Humanities. In Proceedings of the 1st Workshopon Visualization for the Digital Humanities, 2016.6 S. Funke, F. Krumpe, and S. Storandt. Crushing Disks Efficiently. In Proceedings of the27th International Workshop on Combinatorial Algorithms, pages 43–54, 2016.7 S. Funke and S. Storandt. Parametrized Runtimes for Ball Tournaments. In Proceedingsof the 33rd European Workshop on Computational Geometry, pages 221–224, 2017.8 J. Nievergelt and E. M. Reingold. Binary Search Trees of Bounded Balance. SIAM Journalon Computing, 2(1):33–43, 1973.

We study an agglomerative clustering problem motivated by interactive glyphs in geo-visualization. Consider a set of disjoint square glyphs on an interactive map. When the user zooms out, the glyphs grow in size relative to the map, possibly with different speeds. When two glyphs intersect, we wish to replace them by a new glyph that captures the information of the intersecting glyphs. We present a fully dynamic kinetic data structure that maintains a set of n disjoint growing squares. Our data structure uses O(n(log n log log n)2) space, supports queries in worst case O(log3 n) time, and updates in O(log7 n) amortized time. This leads to an O(n α(n) log7 n) time algorithm (where α is the inverse Ackermann function) to solve the agglomerative clustering problem, which is a significant improvement over the straightforward O(n2 log n) time algorithm.</p

Castermans, THA Thom

Speckmann, B Bettina

Verbeek, KAB Kevin

Repository TU/e

\u3cp\u3eWe study an agglomerative clustering problem motivated by interactive glyphs in geo-visualization. Consider a set of disjoint square glyphs on an interactive map. When the user zooms out, the glyphs grow in size relative to the map, possibly with different speeds. When two glyphs intersect, we wish to replace them by a new glyph that captures the information of the intersecting glyphs. We present a fully dynamic kinetic data structure that maintains a set of n disjoint growing squares. Our data structure uses O(n(log n log log n)\u3csup\u3e2\u3c/sup\u3e) space, supports queries in worst case O(log\u3csup\u3e3\u3c/sup\u3e n) time, and updates in O(log\u3csup\u3e7 \u3c/sup\u3en) amortized time. This leads to an O(n α(n) log\u3csup\u3e7 \u3c/sup\u3en) time algorithm (where α is the inverse Ackermann function) to solve the agglomerative clustering problem, which is a significant improvement over the straightforward O(n\u3csup\u3e2\u3c/sup\u3e log n) time algorithm.\u3c/p\u3

 Agglomerative clustering of growing squaresCitation for published version (APA):Castermans, T., Speckmann, B., Staals, F., & Verbeek, K. (2018). Agglomerative clustering of growing squares.8:1-8:6. Abstract from 34th European Workshop on Computational Geometry (EuroCG2018), Berlin, Germany.Document status and date:Published: 21/03/2018Document Version:Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)Please check the document version of this publication:• A submitted manuscript is the version of the article upon submission and before peer-review. There can beimportant differences between the submitted version and the official published version of record. Peopleinterested in the research are advised to contact the author for the final version of the publication, or visit theDOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and pagenumbers.Link to publicationGeneral rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            • Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying the publication in the public portal.If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, pleasefollow below link for the End User Agreement:www.tue.nl/taverneTake down policyIf you believe that this document breaches copyright please contact us at:openaccess@tue.nlproviding details and we will investigate your claim.Download date: 05. Oct. 2023Agglomerative Clustering of Growing Squares∗Thom Castermans1, Bettina Speckmann1, Frank Staals2, andKevin Verbeek11 TU Eindhoven[t.h.a.castermans|b.speckmann|k.a.b.verbeek]@tue.nl2 Utrecht Universityf.staals@uu.nl1 IntroductionWe study an agglomerative clustering problem motivated by interactive glyphs in geo-visualization. GlamMap [5] is a visual analytics tool for the eHumanities which allows theuser to interactively explore metadata of a book collection. Each book is depicted by asquare, color-coded by publication year, and placed on a map according to the location of itspublisher. Overlapping squares are recursively aggregated into a larger glyph until all glyphsare disjoint. As the user zooms out, the glyphs “grow” relative to the map to remain legible.As glyphs start to overlap, they are merged into larger glyphs to keep the map clear anduncluttered. To allow the user to filter and browse real world data sets at interactive speedwe hence need an efficient agglomerative clustering algorithm for growing squares (glyphs).Formal problem statement. Let P be a set of points in R2. Each point p ∈ P has apositive weight pw. Given a “time” parameter t, we interpret the points in P as squares.More specifically, let p(t) be the square centered at p with width tpw. For ease of expositionwe assume all point locations to be unique. Furthermore, we refer to P as a set of squaresrather than a set of center points of squares. Observe that initially, i.e. at t = 0, all squaresin P are disjoint. As t increases, the squares in P grow, and hence they may start to intersect.When two squares p(t) and q(t) intersect at time t, we remove both p and q and replacethem by a new point z = κp + (1 − κ)q, with κ = pw/(pw + qw), of weight zw = pw + qw.Our goal is to compute the complete sequence of events where squares intersect and merge.Results. We present a fully dynamic data structure that can maintain a set P of n disjointgrowing squares. Our data structure can report the first time two squares in P intersect,and supports updates (inserting or deleting a square) in O(log7 n) amortized time. Queriesasking whether a query square q currently intersects a square p in P take O(log3 n) time,and the space usage is O(n(log n log log n)2). Using this data structure we can compute theagglomerative clustering for n squares in O(nα(n) log7 n) time. Here, α is the extremelyslowly growing inverse Ackermann function. To the best of our knowledge, this is the firstfully dynamic clustering algorithm which beats the straightforward O(n2 log n) time bound.This abstract focuses on the update and query times for our data structure. Omitted proofsand detailed bounds on space usage, as well as related discussions on the relation betweencanonical subsets in dominance queries, can be found in the full version [4].Related Work. Funke, Krumpe, and Storandt [6] introduced so-called “ball tournaments”.Their input is a set of balls in Rd with an associated set of priorities. The balls growlinearly and whenever two balls touch, the lower priority ball is eliminated. The goal is to∗ The Netherlands Organisation for Scientific Research (NWO) is supporting T. Castermans (projectnumber 314.99.117), B. Speckmann (project number 639.023.208), F. Staals (project number 612.001.651),and K. Verbeek (project number 639.021.541).34th European Workshop on Computational Geometry, Berlin, Germany, March 21–23, 2018.This is an extended abstract of a presentation given at EuroCG’18. It has been made public for the benefit of thecommunity and should be considered a preprint rather than a formally reviewed paper. Thus, this work is expectedto appear eventually in more final form at a conference with formal proceedings and/or in a journal.T. Castermans, B. Speckmann, F. Staals and K. Verbeek 8:3◮ Lemma 2.3. Let t∗ be the time that a square p of a point p ∈ D(q) touches q. We have(i) rq(t∗)y = ℓp(t∗)y, and ℓp(t∗) is the point with minimumy-coordinate among the points in L−(q)(t∗) at time t∗ if p ∈ D−(q), and(ii) rq(t∗)x = ℓp(t∗)x, and ℓp(t∗) is the point with minimumx-coordinate among the points in L+(q)(t∗) at time t∗ if p ∈ D+(q).3 A Kinetic Data Structure for Growing SquaresWe describe a data structure that can detect intersections between all pairs of squares p,qin P such that p ∈ D+(q). We build an analogous data structure for p ∈ D−(q), and thenuse four copies of these data structures, one for each quadrant, to detect the first intersectionamong all pairs of squares.3.1 The Data StructureOur data structure consists of two three-layered trees T L and T R, and a set of certificateslinking nodes from T L and T R. These trees essentially form two 3D range trees on thecenters of the squares in P , taking the third coordinate pγ of each point to be their rank inthe order (from left to right) along the line γ. The third layer of T L doubles as a kinetictournament tracking the bottom left vertices of squares. Similarly, T R tracks the top rightvertices of the squares.The Layered Trees. The tree T L is a 3D-range tree storing the center points in P . Eachlayer is implemented by a bb[α] tree [8], and each node µ corresponds to a canonical subsetPµ of points stored in the leaves of the subtree rooted at µ. The points are ordered onx-coordinate first, then on y-coordinate, and finally on γ-coordinate. Let Lµ denote the setof bottom left vertices of squares corresponding to the set Pµ, for some node µ.Consider the associated structure XLv of some secondary node v. We consider XLv asa kinetic tournament on the x-coordinates of the points Lv [1]. More specifically, everyinternal node w ∈ XLv corresponds to a set of points Pw consecutive along the line γ. Sincethe γ-coordinates of a point p and its bottom left vertex ℓp are equal, this means w alsocorresponds to a set of consecutive bottom left vertices Lw. Node w stores the vertex ℓp inLw with minimum x-coordinate, and will maintain certificates that guarantee this [1].The tree T R has the same structure as T L: it is a three-layered range tree on the centerpoints in P . The difference is that a ternary structure XRv , for some secondary node v, formsa kinetic tournament maintaining the maximum x-coordinate of the points in Rv, where Rvare the top right vertices of the squares (with center points) in Pv. Hence, every ternarynode z ∈ XRv stores the vertex rq with maximum x-coordinate among Rv. Let XL and X Rdenote the set of all kinetic tournament nodes in T L and T R, respectively.Linking the Trees. Next, we describe how to add linking certificates between the kinetictournament nodes in the trees T L and T R that guarantee the squares are disjoint. Morespecifically, we describe the certificates, between nodes w ∈ X L and z ∈ X R, that guaranteethat the squares p and q are disjoint, for all pairs q ∈ P and p ∈ D+(q).Consider a point q. There are O(log2 n) nodes in the secondary trees of T L, whosecanonical subsets together represent exactly D(q). For each of these nodes v we can thenfind O(log n) nodes in XLv representing the points in L+(q). So, in total q is interested in aset QL(q) of O(log3 n) kinetic tournament nodes. It now follows from Lemma 2.3 that if wewere to add certificates certifying that rq is left of the point stored at the nodes in QL(q) wecan detect when q intersects with a square of a point in D+(q). However, as there mayEuroCG’18T. Castermans, B. Speckmann, F. Staals and K. Verbeek 8:5◮ Lemma 3.3. Let p and q, with p ∈ D+(q), be the first pair of squares to intersect, atsome time t∗. There is a pair of nodes w, z that have a linking certificate that fails at time t∗.From Lemma 3.3 it follows that we can now detect the first intersection between a pair ofsquares p,q, with p ∈ D+(q). We define an analogous data structure for when p ∈ D−(q).Following Lemma 2.3, the kinetic tournaments will maintain the vertices with minimum andmaximum y-coordinate for this case. We then again link up the kinetic tournament nodes inthe two trees appropriately.Space Usage. Our trees T L and T R are range trees in R3, and thus use O(n log2 n)space. However, it is easy to see that this is dominated by the space required to storethe certificates. For all O(n log2 n) kinetic tournament nodes we store at most O(log3 n)certificates (Lemma 3.1), and thus the total space used by our data structure is O(n log5 n).In the full version [4], we show that the number of certificates that we maintain (and thusthe space used by our data structure) is actually only O(n(log n log log n)2).3.2 Inserting or Deleting a SquareAt an insertion or deletion of a square p we proceed in three steps. (1) We update TL andT R, restoring range tree properties, and ensure that the ternary data structures are correctkinetic tournaments. (2) For each kinetic tournament node in X L affected by the update,we query T R to find a new set of linking certificates. We update X R analogously. (3) Weupdate the global event queue.◮ Lemma 3.4. Inserting or deleting a square in T L takes O(log3 n) amortized time.Clearly we can update T R in O(log3 n) amortized time as well. Next, we update the linkingcertificates. We say that a kinetic tournament node w in T L is affected by an update if (i)the update added or removed a leaf node in the subtree rooted at w, (ii) node w was involvedin a tree rotation, or (iii) w occurs in a newly built associated tree XLv (for some node v).Let X Li denote the set of nodes affected by update i (XRi of TR is defined analogously). Foreach node w ∈ X Li , we query TR to find the set of O(log3 n) nodes whose canonical subsetsrepresent QR(w). For each node z in this set, we test if we have to add a linking certificatebetween w and z. As we show next, this takes constant time for each node z, and thusO(∑i |XLi | log3 n) time in total, for all nodes w (analogously for X Ri ).We have to add a link between a node z ∈ QR(w) and w if and only if we also havew ∈ QL(z). We test this as follows. Let v be the node whose associated tree XLv contains w,and let u be the node in T L whose associated tree contains v. We have that w ∈ QL(z) ifand only if u ∈ C(T L, [mzx, ∞)), v ∈ C(Tu, [mzy, ∞)), and w ∈ C(XLv , [mzγ , ∞)). We can testeach of these conditions in constant time:◮ Observation 3.5. Let q be a query point in R1, let w be a node in a binary search tree T ,and let xp = min Pp of the parent p of w in T , or xp = −∞ if no such node exists. We havethat w ∈ C(T, [q, ∞)) if and only if q ≤ min Pw and q > xp.Finally, we delete all certificates involving no longer existing nodes from our global eventqueue, and replace them by all newly created certificates. This takes O(log n) time percertificate. We charge the cost of deleting a certificate to when it gets created. Since everynode w affected creates at most O(log3 n) new certificates, all that remains is to boundthe total number of affected nodes. Here we can use basically the same argument as whenbounding the update time.◮ Lemma 3.6. Inserting a disjoint square into P , or deleting a square from P takes O(log7 n)amortized time.EuroCG’188:6 Agglomerative Clustering of Growing Squares3.3 Running the SimulationAll that remains is to analyze the number of events processed, and the time to do so. Sinceeach failure of a linking certificate produces an intersection, and thus an update, the numberof such events is at most the number of updates. To bound the number of events created bythe tournament trees we use an argument similar to that of Agarwal et al. [1].◮ Theorem 3.7. We can maintain a set P of n disjoint growing squares in a fully dynamicdata structure such that we can detect the first time that a square q intersects with a squarep, with p ∈ D+(q). Our data structure uses O(n(log n log log n)2) space, supports updatesin O(log7 n) amortized time, and queries in O(log3 n) time. For a sequence of m operations,the structure processes a total of O(mα(n) log3 n) events in a total of O(mα(n) log7 n) time.To simulate the process of growing the squares in P , we now maintain eight copies of thedata structure from Theorem 3.7: two data structures for each quadrant (one for D+, theother for D−). Using these data structures we obtain the following agglomerative glyphclustering solution.◮ Theorem 3.8. Given a set of n initial square glyphs P , we can compute an agglomerativeclustering of the squares in P in O(nα(n) log7 n) time using O(n(log n log log n)2) space.References1 P. K. Agarwal, H. Kaplan, and M. Sharir. Kinetic and Dynamic Data Structures for ClosestPair and All Nearest Neighbors. ACM Transactions on Algorithms, 5(1):4:1–4:37, 2008.2 H.-K. Ahn, S. W. Bae, J. Choi, M. Korman, W. Mulzer, E. Oh, J.-W. Park, A. van Renssen,and A. Vigneron. Faster Algorithms for Growing Prioritized Disks and Rectangles. InProceedings of the 28th International Symposium on Algorithms and Computation, pages3:1–3:13, 2017.3 G. Alexandron, H. Kaplan, and M. Sharir. Kinetic and dynamic data structures for convexhulls and upper envelopes. Computational Geometry: Theory and Applications, 36(2):144–1158, 2007.4 T. Castermans, B. Speckmann, F. Staals, and K. Verbeek. Agglomerative clustering ofgrowing squares. ArXiv e-prints, 2017. arXiv:1706.10195.5 T. Castermans, B. Speckmann, K. Verbeek, M. A. Westenberg, A. Betti, and H. van denBerg. GlamMap: Geovisualization for e-Humanities. In Proceedings of the 1st Workshopon Visualization for the Digital Humanities, 2016.6 S. Funke, F. Krumpe, and S. Storandt. Crushing Disks Efficiently. In Proceedings of the27th International Workshop on Combinatorial Algorithms, pages 43–54, 2016.7 S. Funke and S. Storandt. Parametrized Runtimes for Ball Tournaments. In Proceedingsof the 33rd European Workshop on Computational Geometry, pages 221–224, 2017.8 J. Nievergelt and E. M. Reingold. Binary Search Trees of Bounded Balance. SIAM Journalon Computing, 2(1):33–43, 1973.

Agglomerative Clustering of Growing Squares

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