Block Crossings in Storyline Visualizations

Transcription

1 Journal of Graph Algorithms and Appliations vol. 21, no. 5, pp (2017) DOI: /jgaa Blok Crossings in Storylin Visualizations Thomas C. van Dijk 1 Martin Fink 2 Norbrt Fishr 1 Fabian Lipp 1 Ptr Markfldr 1 Alxandr Ravsky 3 Subhash Suri 2 Alxandr Wolff 1 1 Lhrstuhl für Informatik I, Univrsität Würzburg, Grmany 2 Univrsity of California, Santa Barbara, USA 3 Pidstryhah Institut for Applid Problms of Mhanis and Mathmatis, National Aadmy of Sins of Ukrain, Lviv, Ukrain Abstrat Storylin visualizations hlp visualiz nountrs of th haratrs in a story ovr tim. Eah haratr is rprsntd by an x-monoton urv that gos from lft to right visualizing progrssion of tim. A mting is rprsntd by having th haratrs that partiipat in th mting run los togthr for som tim. In ordr to kp th visual omplxity low, rathr than just minimizing pairwis rossings of urvs, w propos to ount blok rossings, that is, pairs of intrsting bundls of lins. In a blok rossing, two bloks of paralll lins intrst ah othr, whih is lss distrating than th sam numbr of individual rossings bing sprad ovr th drawing. In this papr, w show that minimizing th numbr of blok rossings is NPhard, vn if all mtings ar of siz 2. For this spial as, w prsnt a grdy huristi, whih w valuat xprimntally. W show that th gnral as is fixd-paramtr tratabl. Our main rsults is a onstant-fator approximation algorithm for mtings of boundd siz. Th algorithm is basd on (approximatly) solving a hyprdg dltion problm on hyprgraphs that may b of indpndnt intrst. Submittd: Dmbr 2016 Rviwd: Marh 2017 Aptd: August 2017 Artil typ: Rgular papr Rvisd: May 2017 Final: August 2017 Rviwd: Jun 2017 Publishd: Otobr 2017 Communiatd by: Y. Hu and M. Nöllnburg Rvisd: July 2017 A prliminary vrsion of this papr appard in th Prodings of th 24th Intrnational Symposium on Graph Drawing and Ntwork Visualization (GD 2016) [17]. addrsss: thomas.van.dijk@uni-wurzburg.d (Thomas C. van Dijk) fink@s.usb.du (Martin Fink) norbrt.fishr@stud-mail.uni-wurzburg.d (Norbrt Fishr) fabian.lipp@uniwurzburg.d (Fabian Lipp) ptr.markfldr@stud-mail.uni-wurzburg.d (Ptr Markfldr) alxandr.ravsky@uni-wurzburg.d (Alxandr Ravsky) suri@s.usb.du (Subhash Suri) orid.org/ x (Alxandr Wolff)

2 874 T. C. van Dijk t al. Blok Crossings in Storylin Visualizations 1 Introdution A storylin visualization is a onvnint abstration for visualizing th omplx narrativ of intrations among popl, objts, or onpts. Th motivation oms from th stting of a movi, novl, or play whr th narrativ dvlops as a squn of intronntd sns, ah involving a subst of haratrs. S Fig. 1 for an xampl. Th storylin abstration of haratrs and vnts ourring ovr tim an b usd as a mtaphor for visualizing othr situations, from physial vnts involving groups of popl mting in orporat organizations, politial ladrs managing global affairs, and groups of sholars ollaborating on rsarh to abstrat o-ourrns of topis suh as a global vnt bing ovrd on th front pags of multipl lading nws outlts, or diffrnt organizations turning thir attntion to a ommon aus. A storylin visualization maps a st of haratrs of a story to a st of urvs in th plan and a squn of mtings btwn th haratrs to rgions in th plan whr th orrsponding urvs om los to ah othr. Whil Minard s visualization of Napolon s Russian ampaign [11] an b sn as an arly (and xtrmly stark) form of storylin visualization (ombining tim and loation on a map), th urrnt form of storylin visualizations sms to hav bn invntd by Munro [14] (ompar Fig. 1), who usd it to visualiz, in a ompat way, whih substs of haratrs mt ovr th ours of a movi. Eah haratr is shown as an x-monoton urv. Mtings our at rtain tims from lft to right. A mting orrsponds to a point in tim whr th haratrs that mt ar nxt to ah othr with only small gaps btwn thm. Munro highlights mtings by undrlying thm with a gray shadd rgion, whil w us a vrtial lin for that purpos. Hn, a storylin visualization an b sn as a drawing of a hyprgraph whos vrtis ar rprsntd by th urvs and whos dgs om in at spifi points in tim. A natural objtiv for th quality of a storylin visualization is to minimiz unnssary rossings among th haratr lins. Th numbr of rossings alon, howvr, is a poor masur: two bloks of loally paralll lins rossing ah othr ar far lss distrating than an qual numbr of rossings randomly sattrd throughout th drawing. Thrfor, instad of pairwis rossings, w Figur 1: Storylin visualization for Jurassi Park by xkd [14] with svral blok rossings (on of whih w highlightd by a bold grn llips).

3 JGAA, 21(5) (2017) 875 fous on minimizing th numbr of blok rossings; ah blok rossing involvs two arbitrarily larg sts of loally paralll lins forming a rossbar, with no othr lin in th rossing ara; s Fig. 1 for an xampl. Prvious Work. Th InfoVis ommunity has quikly mbrad Munro s fftiv visualization thniqu. Kim t al. [8] usd storylins to visualiz gnalogial data; mtings orrspond to marriags and spial thniqus ar usd to indiat hild parnt rlationships. Tanahashi and Ma [15] omputd storylin visualizations automatially and showd how to adjust th gomtry of individual lins to improv th asthtis of thir visualizations. Muldr t al. [13] visualizd lustrd, dynami graphs as storylins, summarizing th bhavior of loal ntworks that surround usr-sltd foi. Only rntly, a mor thortial and prinipld study was initiatd by Kostitsyna t al. [10], who onsidrd th problm of minimizing pairwis rossings (that is, not blok rossings) in storylins. Thy provd that th problm is NP-hard in gnral, and showd that it is fixd-paramtr tratabl (FPT) with rspt to th (total) numbr of haratrs. Thir FPT algorithm runs in tim O(k! 2 k log k + k! 2 n) and uss spa O(k! 2 + n), whr k is th numbr of haratrs and n is th numbr of mtings. For th spial as of 2-haratr mtings without rptitions whr th mting graph whos dgs dsrib th mtings of pairs of haratrs is a tr, thy showd that Θ(k log k) rossings always suffi and ar somtims nssary. Vry rntly, Gronmann t al. [7] formulatd an intgr linar program that minimizs th total numbr of rossings for storylins with mtings of arbitrary ardinality. Thir approah an solv instans with haratrs and up to about 50 mtings optimally in a fw sonds. Du to th natur of intgr linar programming, howvr, thir approah boms unusabl for larg instans. Our work builds on th problm formulation of Kostitsyna t al. [10] but w onsidrably xtnd thir rsults by dsigning (approximation) algorithms for gnral mtings for a diffrnt optimization goal: w minimiz th numbr of blok rossings rathr than th numbr of pairwis lin rossings. Blok rossings wr introdud by Fink t al. [5] for visualizing mtro maps; in thir stting, blok rossings happn btwn mtro lins that run on top of an mbddd graph, th mtro ntwork. Problm Dfinition. Whil a blok rossing is visually asily rognizabl by a human radr, w start by arfully dfining th problm. In th ours of doing so, w will alrady gain furthr insight in what minimizing th numbr of blok rossings amounts to. A storylin S is a pair (C, M) whr C = {1,..., k} is a st of haratrs and M = [m 1, m 2,..., m n ] with m i C and m i 2 for i = 1, 2,..., n is a squn of mtings of at last two haratrs. W all any st g C of haratrs that has at last on mting, a group (i.., g = m i for som 1 i n). W dfin th group hyprgraph H = (C, Γ) as follows: its vrtis ar th haratrs and its hyprdgs ar th groups. This hyprgraph dos

4 876 T. C. van Dijk t al. Blok Crossings in Storylin Visualizations 1 a b b+1 k Figur 2: Blok rossing (a, b, ) not inlud th tmporal aspt of th storylin it modls only th strutur of groups partiipating in th storylin. Eah group is rprsntd by a singl hyprdg, vn if it mts multipl tims. Th group hyprgraph an b built by sorting M and filtring th multiply-ourring mtings. This an b don in O(nk) tim by rprsnting th mtings as bitsts and using radix sort. Not that w do not nod th xat tims of th mtings: in a givn visualization, at any tim t, thr is a uniqu vrtial ordr π t of th haratrs. Without hanging π t by rossings, w an inras or dras vrtial gaps btwn lins. If a group g forms a ontiguous intrval in π t, thn w an bring th lins in g within a short vrtial distan δ group without any rossing, and also mak sur that all othr lins ar at a distan of δ sp > δ group ; s Fig. 12. Sin any group must b supportd at a tim just bfor its mting starts, omputing an output drawing onsists mainly of hanging th prmutation of haratrs ovr tim so that during a mting its group is supportd by th urrnt prmutation. W thrfor fous on hanging th prmutation by rossings ovr tim, and only hav to b onrnd about th ordr of mtings; th final drawing an b obtaind by a simpl post-prossing from this disrt st of prmutations. Suppos that w loally rnumbr th haratrs from top to bottom from 1 to k. Thn w an dfin, for a b <, a blok rossing (a, b, ) to b th xhang of th two onsutiv bloks a,..., b and b + 1,...,. This xhang maps th prmutation 1,..., a,..., b,...,,..., k to th prmutation 1,..., a 1, b + 1,...,, a,..., b, + 1,..., k ; s Fig. 2 for an illustration. Th sam dfinition was usd by Fink t al. [5]. A mting m fits a prmutation π (or a prmutation π supports a mting m) if th haratrs partiipating in m form an intrval in π. In othr words, thr is a prmutation of m that is part of π. If w apply a squn B of blok rossings to a prmutation π in th givn ordr, w dnot th rsulting prmutation by B(π). Problm 1 (Storylin Blok Crossing Minimization (SBCM)) Givn a storylin instan (C, M) with M = [m 1, m 2,..., m n ], find a solution onsisting of a start prmutation π 0 of C and a squn B = (B 1, B 2,..., B n ) of (possibly mpty) squns of blok rossings suh that th total numbr of blok rossings is minimizd and π i = B i (π i 1 ) supports m i for 1 i n.

5 JGAA, 21(5) (2017) 877 Not that in our problm dfinition no haratrs arriv aftr th start of th drawing or disappar bfor th nd; this is diffrnt in th xampl in Fig. 1. W also onsidr d-sbcm, a spial as of SBCM whr mtings involv groups of siz at most d, for an arbitrary onstant d. For xampl, 2-SBCM allows only 2-haratr mtings, a stting that was also studid by Kostitsyna t al. [10]. Our Rsults. W obsrv that a storylin has a rossing-fr visualization if and only if its group hyprgraph is an intrval hyprgraph, that is, if thr xists a prmutation of th haratr st suh that ah hyprdg orrsponds to a ontiguous blok of haratrs in this prmutation. This is quivalnt to th hyprgraph having path support [1]. A hyprgraph an b tstd for th intrval proprty in O(n 2 ) tim, whr n is th numbr of hyprdgs. W show that 2-SBCM is NP-hard (s St. 3) and that SBCM is fixd-paramtr tratabl with rspt to k (St. 4). On of our FPT algorithms an b modifid to handl pairwis rossings instad of blok rossings. Th modifid algorithm is fastr than th algorithm of Kostitsyna t al. [10] by a fator of k!. W arry out som xprimnts valuating this algorithm and othr approahs in a follow-up papr [18]. Th as of siz-2 mtings is of intrst; rall that Kim t al. [8] hav usd a storylin-lik visualization to tra gnalogial data. W invstigat strutural proprtis and prsnt a grdy algorithm for 2-SBCM that runs in O(k 3 n) tim for k haratrs. For k = 3, th grdy algorithm yilds optimal solutions. W xprimntally ompar grdy solutions to optimal solutions; s St. 5. On main rsult is a onstant-fator approximation algorithm for d-sbcm for th as that d is boundd and that mtings annot b rpatd; s St. 6. Our algorithm is basd on a solution for th following NP-omplt hyprgraph problm, whih may b of indpndnt intrst: givn a hyprgraph H, dlt th minimum numbr of hyprdgs so that th rmaindr is an intrval hyprgraph. For this problm, w dvlop a (d + 1)-approximation algorithm, whr d is th maximum ardinality of a hyprdg in H; s St. 7. Finally, w list som opn problms in St Prliminaris and Basi Obsrvations First, w onsidr th spial as whr vry mting onsists of two haratrs. For ths rstritd instans, any mting an b ralizd from any prmutation by a singl blok rossing. Obsrvation 1 Givn an instan of 2-SBCM, thr is a solution with at most on blok rossing bfor ah of th mtings. In partiular, thr is a solution with at most n blok rossings in total. Proof: Lt π b an arbitrary prmutation, and lt m = {, } M b th nxt mting. Lt i and j b th positions of th haratrs and, rsptivly, in th prmutation π, that is, π (i) = and π (j) =. Without loss of gnrality,

6 878 T. C. van Dijk t al. Blok Crossings in Storylin Visualizations π 0 π 1 π Figur 3: Optimal solution for S from th proof of Lmma 1. assum i < j. If π dos not support m, w an raliz it using th blok rossings (i, i, j 1), that is, moving th lin of dirtly abov that of. Obsrvation 1 shows that thr is a solution with at most on blok rossing bfor any mting. This raiss th qustion whthr thr is also an optimal solution that fulfills this ondition. Th answr is ngativ. Lmma 1 Thr is an instan S of 2-SBCM and a start prmutation π 0 suh that thr is no optimal solution (π 0, B) of S that starts with π 0 and uss at most on blok rossing bfor th first and btwn ah pair of onsutiv mtings. Proof: W prov th laim by ontradition. Considr th instan S = (C, M) with C = {1, 2, 3, 4, 5, 6, 7, 8} and M = [{6, 3}, {7, 2}, {1, 5}, {5, 6}, {6, 3}, {3, 4}, {4, 8}, {8, 7}]. Lt π 0 = 1, 2, 3, 4, 5, 6, 7, 8 b th start prmutation. Thr is a solution with only two blok rossings, namly (π 0, B) with B = [(2, 4, 7), (4, 5, 8)]; s Fig. 3. Lt π 1 b th prmutation aftr th first blok rossing of B on π 0, and π 2 th prmutation aftr both blok rossings. Th prmutation π 2 supports all mtings in M. Th first mting {6, 3} in M fits nithr π 0 nor π 1, that is, both blok rossings must our bfor th first mting. Now assum that thr is anothr solution (π 0, B ) with B 2 that has at most on blok rossing bfor ah mting. Starting from π 0 thr ar xatly nin fasibl blok rossings that allow th first mting. Thy yild th following prmutations (whr th bars indiat th rossing bloks): 1, 2, 4, 5, 6, 3, 7, 8 4, 5, 1, 2, 3, 6, 7, 8 1, 2, 3, 6, 4, 5, 7, 8 1, 2, 5, 6, 3, 4, 7, 8 1, 4, 5, 2, 3, 6, 7, 8 1, 2, 3, 6, 7, 4, 5, 8 1, 2, 6, 3, 4, 5, 7, 8 1, 2, 4, 5, 3, 6, 7, 8 1, 2, 3, 6, 7, 8, 4, 5 Non of ths prmutations supports th sond mting {7, 2}. So w nd th sond blok rossing bfor this mting. This sond blok rossing nds to prpar all of th rmaining mtings baus othrwis B > 2. Ths mtings an only b supportd by th prmutation σ = 1, 5, 6, 3, 4, 8, 7, 2 or

7 JGAA, 21(5) (2017) 879 m π 0 π 1 π Figur 4: Optimal solution for S from th proof of Thorm 1. its rvrs prmutation σ R. It rmains to show that non of th prmutations yildd by th fasibl first blok rossing an b transformd to σ or σ R by on additional blok rossing. All prmutations ontaining 3, 6 as a subsqun ar infasibl baus thr is only on blok rossing that swaps two nighboring haratrs and it dos not produ σ. For prmutations starting with 1, 2, thr is only on possibl blok rossing to bring haratr 2 to th nd of th prmutation whil haratr 1 stays at th first position, whih also dos not yild σ. Similarly, w an show that thr is also no blok rossing aftr any of th fasibl blok rossings for th first stp that lads to σ R. Now w modify this instan suh that w do not nd to prsrib th start prmutation. Thorm 1 Thr is an instan S of 2-SBCM suh that thr is no optimal solution (π 0, B) of S that uss at most on blok rossing btwn ah pair of onsutiv mtings. Proof: W prov th laim by onstruting an xampl. Considr th instan S = (C, M) dpitd in Fig. 4. Th instan has no solution with lss than two blok rossings, but thr is a solution with two blok rossings, namly (π 0, B) with start prmutation π 0 = 1, 2, 3, 4, 5, 6, 7, 8 and blok rossings B = [(2, 4, 7), (4, 5, 8)]; s Fig. 4. Lt π 1 b th prmutation aftr th first blok rossing of B on π 0, and lt π 2 = 1, 5, 6, 3, 4, 8, 7, 2 b th prmutation aftr th sond blok rossing in B. Th prmutation π 2 supports all mtings in th right half of M. Th first ourrn of mting {6, 3} in M (right aftr th third dashd lin in Fig. 4) fits nithr π 0 nor π 1, that is, both blok rossings our bfor that mting. Now w argu that thr is no solution with two blok rossings that ar sparatd by at last on mting. Lt m b th third ourrn of mting {1, 2} (right bfor th first dashd lin in Fig. 4). At som point in tim bfor m, ithr prmutation π 0 or π0 R must our, othrwis w would nd at last two blok rossings to raliz th mtings bfor m and at last on blok rossing aftr m to raliz th last svn mtings (baus th prmutation at m is not π 2 or π2 R ). W distinguish thr ass for th two blok rossings.

8 880 T. C. van Dijk t al. Blok Crossings in Storylin Visualizations a) Both rossings our bfor m: This is not possibl baus th rmaining mtings annot b ralizd with haratrs 1 and 2 bing adjant. b) Both rossings our aftr m: Thn th start prmutation nds to b π 0 or π R 0, whih is xatly th as handld in Lmma 1. ) Th first rossing is bfor m, th sond on aftr m. Sin w must raliz th last svn mtings with only on blok rossing, th final prmutation is π 2 or π R 2. Assum that it is π 2. Thn going bakwards from π 2, w hav to raliz m by on blok rossing. Thr ar xatly nin suh blok rossings yilding th following prmutations: 5, 6, 3, 4, 8, 7, 2, 1 6, 3, 4, 8, 7, 2, 1, 5 3, 4, 8, 7, 2, 1, 5, 6 4, 8, 7, 2, 1, 5, 6, 3 8, 7, 2, 1, 5, 6, 3, 4 7, 2, 1, 5, 6, 3, 4, 8 2, 1, 5, 6, 3, 4, 8, 7 1, 2, 5, 6, 3, 4, 8, 7 5, 6, 3, 4, 8, 7, 1, 2 It is asy to hk that non of thm ontains as a subintrval what w all a triplt; thr onsutiv numbrs in thir natural (or invrtd) ordr (for instan, 3, 4, 5 or 5, 4, 3 ). Rall that at som point in tim bfor m, prmutation π 0 or π0 R must our. Now not that any prmutation rahabl from π 0 or π0 R by on blok rossing ontains a triplt (sin w hav ight haratrs). Thus w obtain a ontradition. Th as whn th final prmutation is π2 R an b tratd similarly. Dtting Crossing-Fr Storylins. If a storylin admits a rossing-fr visualization, thn th vrtial prmutation of th haratr lins rmains th sam ovr tim, and all mtings involv groups that form ontiguous substs in that prmutation. (Th visualization an b obtaind by plaing haratrs along a vrtial lin in th orrt prmutation and for ah mting bringing its lins togthr for th duration of th mting and thn sparating thm apart again.) In othr words, a singl prmutation supports ah group of H = (C, Γ). This holds if and only if H is an intrval hyprgraph. Rall that this is th as if thr xists a prmutation π = v 1,..., v k of C suh that ah hyprdg Γ orrsponds to a ontiguous blok of haratrs in this prmutation. Not that a graph is an intrval hyprgraph if and only if all of th graph s onntd omponnts ar paths. Hn, a graph bing an intrval hyprgraph is not th sam proprty as bing an intrval graph. An intrval hyprgraph an b visualizd by plaing all of its vrtis on a lin, and drawing ah hyprdg as an intrval that inluds all vrtis of and no vrtx of V \. Chking whthr a k-vrtx hyprgraph is an intrval hyprgraph taks O(k 2 ) tim [16]. Rall that w an build H in O(nk log n) tim. Thorm 2 Givn th group hyprgraph H of an instan of SBCM with k haratrs, w an hk in O(k 2 ) tim whthr a rossing-fr solution xists. If this is th as, a prmutation ralizing a rossing-fr solution an also b found within th sam tim bound.

9 JGAA, 21(5) (2017) 881 For 2-SBCM w only nd to hk (in O(k) tim) whthr H is a olltion of vrtx-disjoint paths; this is dominatd by th tim (O(n)) for building H, as long as k = O(n). 3 NP-Compltnss of SBCM In this stion w prov that SBCM is NP-omplt. This is known for th problm Blok Crossing Minimization (BCM), introdud by Fink t al. [5]. But SBCM is not simply a gnralization of BCM baus in SBCM w an hoos an arbitrary start prmutation. Thrfor, th ida of our hardnss proof is to for a rtain start prmutation by adding som haratrs and mtings. W rdu from Sorting by Transpositions (SBT), whih has also bn usd to show th hardnss of BCM [5]. In SBT, on is givn a prmutation π and an intgr k, and th task is to did whthr thr is a squn of transpositions (whih ar quivalnt to blok rossings) of lngth at most k that transforms π to th idntity. SBT was rntly shown NP-hard by Bultau t al. [2]. W show hardnss for 2-SBCM, whih implis that SBCM is NP-hard, too. It is asy to s that th dision vrsion of SBCM is in NP: Obviously, th maximum numbr of blok rossings ndd for any numbr of haratrs and mtings is boundd by a polynomial in k and n. Thrfor also th siz of a possibl rtifiat is boundd by a polynomial. To tst th fasibility of a solution ffiintly, w simply tst whthr th prmutations btwn th blok rossings support th mtings in th right ordr from lft to right. W will us th following obvious fat. Obsrvation 2 If prmutation π nds blok rossings to b sortd, any prmutation ontaining π as a subsqun nds at last blok rossings to b sortd. Thorm 3 2-SBCM is NP-omplt. Proof: It rmains to show th NP-hardnss. W rdu from SBT. Givn an instan of SBT, that is, a prmutation π of {1,..., k}, w show how to onstrut a orrsponding instan of 2-SBCM. W xtnd th st of haratrs {1, 2,..., k} to C = {1,..., k, 1, 2,..., 2k }. Corrspondingly, w xtnd th prmutation π = π(1), π(2),..., π(k) to π = 1,..., 2k, π(1),..., π(k) and ι to ι = 1, 2,..., 2k, 1, 2,..., k. Lt M π and M ι b th squns of mtings of all nighboring pairs in π and ι, rsptivly. Lt M 1 and M 2 b th onatnations of k + 1 opis of M π and M ι, rsptivly. This yilds th instan S = (C, M) of 2-SBCM, whr M is th onatnation of M 1 and M 2 ; s Fig. 5. W show that th numbr of blok rossings ndd for th 2-SBCM instan S quals th numbr of transpositions to solv instan π of SBT, that is, to transform π to th idntity ι = 1, 2,..., k. Not that π an b sortd by at most k blok rossings. So k is an uppr bound for an optimal solution of instan π of SBT.

10 882 T. C. van Dijk t al. Blok Crossings in Storylin Visualizations M 1 M π π π 2k π 1 π k k 1 k 1 B 2k 1 k ι ι ι Figur 5: Solution for th 2-SBCM instan S orrsponding to a solution B of instan π of SBT. Th box B rprsnts th blok rossings. First, lt B b a shortst squn of blok rossings to sort π. Thn, (π, B) is a fasibl solution for S. Th start prmutation π supports all mtings in M 1 without any blok rossing. Using B, th lins ar sortd to ι, and this prmutation supports all mtings in M 2 without any furthr blok rossings; s Fig. 5. Hn, th numbr of blok rossings in any solution of π is an uppr bound for th minimum numbr of blok rossings ndd for S. For th othr dirtion, lt (π, B ) b an optimal solution for S. In th input of SBCM, w do not fix a rtain ordr of th haratrs. So any solution of 2-SBCM givs ris to a symmtri solution that is obtaind by rvrsing th ordr of th haratrs. In th following, without loss of gnrality, w assum that π (rathr than th rvrs prmutation π R ) ours in M, that is, thr is a tim t in th visualization suh that th vrtial ordr of th haratrs at t is dsribd by π. Nxt, w show that th start prmutation π ours somwhr in M 1 and that ι ours somwhr in M 2. If thr is a squn M π of mtings btwn whih thr is no blok rossing, th prmutation at this position an only b th start prmutation π or its rvrs. For a ontradition, assum that π dos not our during M 1 in th layout indud by (π, B ). Thn thr is no suh squn without any blok rossing in it. As this squn is rpatd k + 1 tims, th solution would nd at last k + 1 blok rossings. This ontradits our uppr bound, whih is k. Analogously, w an show that th prmutation ι or its rvrs ours in M 2. W now want to show that th unrvrsd vrsion of ι ours in M 2. For a ontradition, assum th opposit. W forgt about th lins 1,..., k and only onsidr th squn π = 1,..., 2k in π whih is rvrsd to ι R = 2k,..., 1 in ι R. Eriksson t al. [4] showd that w nd (l + 1)/2 blok rossings to rvrs a prmutation of l lmnts. This implis that w nd k + 1 blok rossings to transform π to ι R. As π and ι R ontain ths squns as subsquns, Obsrvation 2 implis that th transformation from π to ι R also nds at last k + 1 blok rossings. As th optimal solution uss at most k blok rossings, w know that w annot rah ι R and thus th squn of prmutations ontains π and ι.

11 JGAA, 21(5) (2017) 883 Th squn of blok rossings that transforms π to ι yilds a squn B of blok rossings of th sam lngth that transforms π to ι. Thrfor th lngth of a solution for S is an uppr bound for th lngth of an optimal solution of th orrsponding SBT instan π. Thus, th two lngths ar qual. Hardnss Without Rptitions. With arbitrarily larg mtings, w an slightly modify our hardnss proof, and show that minimizing th numbr of blok rossings is also hard without rpating th sam mting many tims. Th ida to hang our rdud SBCM instan, is to rpla th rpatd squn of 2-haratr mtings so that in ah rptition th group siz is inrasd by on for all mtings; s Fig. 6. Du to th ovrlapping strutur of th groups in a singl squn, thy an only b all supportd at th sam tim if also th 2-haratr mtings that thy rplad ar supportd. Th only thing that w hav to b arful about is that whn th groups gt largr than 3k/2, that is, half of th numbr of haratrs, thr is a growing st of haratrs in π (or ι ) that ar ontaind in xatly th sam groups, and thir rlativ ordr dos not mattr for th mtings; s Fig. 6(a). W will avoid this situation in th following way (shown in Fig. 6(b)). Sin w hav k + 1 squns of rpatd mtings at th bginning as wll as at th nd of th timlin, and w kp inrasing th group sizs, w hav groups of 2k + 3 haratrs in th nd. W rpla 1,..., 2k by a nw squn 1,..., 10k of haratrs without hanging th strutur furthr. Thn, w an inras th group siz up to 2k + 3 whil in th nd still lss than half of all haratrs ar involvd in ah group. Sin th growing mtings ompltly simulat th dsird 2-haratr mtings, th rst of th rdution and its proof stay th sam, and w gt th following rsult. Thorm 4 SBCM is NP-hard vn if mtings ar not rpatd. Not that in this rdution (diffrnt from Thorm 3) w us mtings whos siz dpnds on th input. So for this variant without rpatd mtings w do not show hardnss for d-sbcm for any fixd d. (a) Instan in whih th groups ar too larg for th numbr of haratrs. Th ara highlightd in gray shows th haratrs involvd in all mtings of a rtain siz. Thrfor, th rlativ ordr of ths haratrs dos not mattr anymor. (b) By adding haratrs (and th orrsponding mtings), w avoid haratrs bing involvd in vry mting. Figur 6: W simulat rpatd 2-haratr mtings by using groups of inrasing siz.

12 884 T. C. van Dijk t al. Blok Crossings in Storylin Visualizations 4 Exat Algorithms W prsnt two xat algorithms for SBCM. Conptually, both build up a squn of blok rossings whil kping trak of how many mtings hav alrady bn aomplishd. Th first uss polynomial spa; th sond improvs th runtim at th ost of xponntial spa. Th lattr gnralizs to improv a rsult by Kostitsyna t al. [10] about minimizing pairwis rossings (rathr than blok rossings). W start with a data strutur to kp trak of prmutations, blok rossings and mtings. It is initializd with a givn prmutation and has two oprations. Th Chk opration rturns whthr a givn mting fits th urrnt prmutation. Th BlokMov opration prforms a givn blok rossing on th prmutation and thn rturns whthr th most-rntly Chkd mting now fits. Lmma 2 A squn of arbitrarily intrlavd BlokMov and Chk oprations an b prformd in O(β + µ) tim, whr β is th numbr of alls to BlokMov and µ is sum of ardinalitis of th mtings givn to Chk. Th spa usag is O(k), whr k is th numbr of haratrs. Proof: Rprsnt th prmutation as a doubly-linkd list. 1 First onsidr a 2- mting. It taks onstant tim to hk whthr it fits: hk th prvious/nxt pointrs. Sin a blok rossing hangs at most 6 adjanis, BlokMov an updat th linkd list in onstant tim. (This rquirs giving out pointrs into th linkd-list rprsntation, sin w nd to find th haratrs at th dgs of th blok in onstant tim.) Not that all possibl blok rossings an b numratd in onstant tim ah. Now w look at a mting of ardinality m. Intrprt th linkd list as a path and onsidr th subgraph indud by th nods in th mting. If th mting fits th prmutation, this subgraph is onntd and, bing a path, has m 1 dgs; if th mting dos not fit, this subgraph has mor omponnts and thrfor fwr dgs. Th Chk opration on a mting of siz m an b prformd in O(m) tim by ounting at vry nod in th mting whthr zro, on or two of its nighbors ar also in th mting. For th amortizd runtim ovr a squn of oprations, rmmbr this ount: BlokMov an updat it in onstant tim, sin again at most six adjanis hang. In trms of spa, thr is only th doubly linkd list and th ount. Now w provid an output-snsitiv algorithm for SBCM, th runtim of whih dpnds on th numbr of blok rossings rquird by th optimum. Thorm 5 An instan S = (C, M) of SBCM an b solvd in O(k! ( k3 k 6 ) β (β + µ)) tim and O(βk) working spa if a solution with β blok rossings xists, whr µ = m M m. 1 W assum that th mtings givn to Chk ar rprsntd by rfrns to th nods in this list; if nssary, this rprsntation an b onstrutd ffiintly in prprossing.

13 JGAA, 21(5) (2017) 885 Proof: Considr a branhing algorithm that starts from a prmutation of th haratrs and kps trying all possibl blok rossings. Th instan of SBCM is solvd by finding th shortst squn of blok rossings that supports th mtings. A blok rossing an b rprsntd by indis (a, b, ) with 1 a b < k. Th numbr of possibl blok rossings is givn by adding th numbr of blok rossings with a = b and th numbr of blok rossings with a b; hn, thr ar ( ( k 2) + k ) 3 = k 3 k 6 distint blok rossings on a prmutation of lngth k. W an numrat ths in onstant tim ah by numrating all appropriat tripls (a, b, ). W us dpth-first itrativ-dpning sarh [9] from all possibl start prmutations, applying blok rossings, until w find a squn of prmutations that fulfills all mtings. Whil sarhing, th algorithm kps trak of how many mtings fit th urrnt squn of prmutations using th data strutur from Lmma 2. Corrtnss follows from th itrativ dpning of th sarh, sin w want an (unwightd) shortst squn of blok rossings. Th runtim and spa bounds follow from th standard analysis of itrativ-dpning sarh, obsrving that a nod uss O(k) spa and it taks O(β + µ) tim in total to valuat a path from root to laf. Not that µ is O(kn): thr ar n mtings and ah onsists of at most k haratrs. At th ost of xponntial spa, w an improv th runtim and gt rid of th dpndn on β, showing th problm to b fixd paramtr linar for k. Th following algorithm an asily b adaptd to optimiz pairwis rossings rathr than blok rossings. In that as w improv upon th algorithm of Kostitsyna t al. [10] by a fator of roughly k!, in trms of both running tim and spa onsumption: rall that thir FPT algorithm runs in O(k! 2 k log k + k! 2 n) tim and uss O(k! 2 + n) spa. Thorm 6 An instan of SBCM an b solvd in O(k! k 3 n) tim and O(k! k n) spa. Proof: Th algorithm is basd on th following qustion: how many blok rossings ar rquird for th first l mtings givn that w nd on prmutation π? Formally, lt f(π, l) b th optimal numbr of blok rossings in a solution to th givn instan whn rstritd to th first l mtings and to hav π as its final prmutation. Not that by dfinition th solution for th atual instan is givn by min π f(π, n), whr th minimum rangs ovr all possibl prmutations. As a bas as, f(π, 0) = 0 for all π, sin th mpty st of mtings is supportd by any prmutation. Lt π and π b prmutations that ar on blok rossing apart and lt 0 l l. If th mtings {m l+1,..., m l } all fit π, thn w hav f(π, l ) f(π, l) + 1: if w an support th first l mtings and nd on π, thn with on additional blok rossing w an support th first l mtings and nd on π. W now modl th abov obsrvation in a graph. Lt G b an unwightd dirtd graph on nods (π, l), whr π is a prmutation of haratrs and

14 886 T. C. van Dijk t al. Blok Crossings in Storylin Visualizations 0 l n. Call a nod start nod if l = 0. Thr is an ar from (π, l) to (π, l ) if and only if π and π ar on blok rossing apart, l l, and th mtings {m l+1,..., m l } fit π. Not that w allow l = l sin w may nd to allow blok rossings that do not immdiatly ahiv an additional mting (f. Proposition 1), so G is not ayli. By onstrution w hav that f(π, l) quals th minimum graph distan from a start nod to th nod (π, l). Call a path from a start nod that ralizs this distan optimal. In G, onsidr any lngth-3 path [(π 1, l 1 ), (π 2, l 2 ), (π 3, l 3 )] with strit inquality l 3 > l 2. If mting l fits π 2, thn [(π 1, l 1 ), (π 2, l 2 + 1), (π 3, l 3 )] is also a path. Rpating this transformation shows that for all π, th nod (π, n) has an optimal path in whih vry ar maximally inrass l. Lt G b th graph whr w drop all ars from G that do not maximally inras l. Not that G still ontains a path that orrsponds to th global optimum. Th graph G has O(k! n) nods. Eah nod has outdgr O(k 3 ), sin any blok rossing ontributs at most on out-ar to a nod. Thn a bradth-first sarh from all start nods to any nod (π, n) ahivs th laimd tim and spa bounds, assuming w an numrat th outgoing ars of a nod in onstant tim ah. For a givn nod (π, l) w an numrat all possibl blok rossings in onstant tim ah, as bfor. To gnrat its outgoing ars in G, w also nd to know th maximum l suh that all mtings l + 1 up to l fit π, whr π is th prmutation rsulting from th blok rossing. Not that l only dpnds on l and π ; in partiular, it dos not dpnd on π. W an thrfor promput a tabl M(π, l) that givs this valu. Computing M(π, l) for a givn π and all l taks a total of O(kn) tim: first omput for vry m i whthr it fits π, thn omput th implid forward pointrs using a linar san. So using O(k! kn) prprossing tim and O(k! n) spa, w hav an ffiint implmntation of th bradth-first sarh. Th thorm follows. 5 A Grdy Huristi In this stion w dvlop an O(kn)-tim grdy algorithm to quikly draw good storylin visualizations for 2-SBCM. Givn an instan S = (C, M), w rsrv a list B = [ ] that th algorithm will us to stor th blok rossings. Th algorithm starts with an arbitrary prmutation π 0 of th haratrs. In vry stp th algorithm rmovs all mtings from th bginning of M that ar supportd by th urrnt prmutation π i. Subsquntly, th algorithm piks a blok rossing b suh that th rsulting prmutation π i+1 = b(π i ) supports th maximum numbr of mtings from th bginning of M, and b is appndd to th list B. This pross rpats until M is mpty. Th algorithm rturns th solution (π 0, B).

15 JGAA, 21(5) (2017) (a) Grdy solution (b) Optimal solution Figur 7: Th grdy algorithm is not optimal. Rall that, from any partiular prmutation, thr ar O(k 3 ) blok rossings. To find th appropriat blok rossings, th algorithm ould simply hk all of thm. Howvr, most of thos will rsult in prmutations that do not vn support th nxt mting, whih annot b th grdy hoi dsribd abov. Hn, our algorithm numrats only th rlvant blok rossings in th following sns: blok rossings yilding a prmutation that supports th upoming mting. Lt {, } b th nxt mting in M. If x and y ar th positions of and in th urrnt prmutation (i.., π i (x) = and π i (y) = ; without loss of gnrality, assum x < y), th rlvant blok rossings ar: {(z, x, y 1): 1 z x} {(x, z, y): x z < y} {(x + 1, y 1, z): y z k}. W s that th numbr of rlvant blok rossings in ah stp is k + 1. Lt n i b th maximum numbr of mtings at th bginning of M w an support by on of ths blok rossings. W us th data strutur in Lmma 2 and hk for ah rlvant blok rossing how many mtings an b don with this prmutation. Hn, w an idntify a blok rossing ahiving th maximum numbr in O(kn i ) tim sin w hav to hk k + 1 options, ah lasting up to n i mtings ah. Th numbrs of mtings n i in ah itration of th algorithm sum up to n and thrfor th algorithm runs in O(kn) total tim. In th dsription abov, this grdy algorithm starts with an arbitrary prmutation. Instad, w ould start with a prmutation that supports th maximum numbr of mtings bfor th first blok rossing nds to b don. In othr words, w ould to find a maximal prfix M of M suh that (C, M ) an b rprsntd without any blok rossings. W an find M in O(kn) tim: start with an mpty graph on th haratrs and sussivly add dgs for th mtings. Aftr ah addition w hk whthr th graph is still a olltion of paths, whih an b don in O(k) tim. Aftr this pross trminats, w onstrut a prmutation that supports all mtings in M, whih is asy givn th olltion of paths. S Fig. 7 for an xampl that uss th huristi start prmutation. Whil this is a snsibl huristi, w do not prov that it rdus th total numbr of blok rossings. Indd, w xprimntally obsrv that, whil this huristi for th start prmutation is gnrally good, it is not always th bst; this is disussd in th xprimntal valuation latr in this stion. Th grdy algorithm atually yilds optimal solutions for spial ass of 2-SBCM. For th following lmma, w assum that no two subsqunt mtings

16 888 T. C. van Dijk t al. Blok Crossings in Storylin Visualizations in th input ar th sam. W all an instan normal if this is th as. An instan an b normalizd by simply dropping th rpatd mtings. This dos not afft th optimal numbr of blok rossings or th bhavior of th grdy algorithm, but not that it dos lowr n. Lmma 3 A normal instan of 2-SBCM with k = 3 an b solvd using at most n/2 1 blok rossings. Proof: Not that thr ar only thr possibl mtings, namly {1, 2}, {1, 3}, and {2, 3}. Any prmutation supports prisly two of ths and not th third, and is quivalnt in this sns to its rvrs. For xampl, th prmutation 1, 2, 3 and its rvrs both support th mtings {1, 2} and {2, 3}, but not {1, 3}. Lt π and π b distint prmutations. Cas distintion shows that it is always possibl with a singl blok rossing to gt from π to ithr π or to th rvrs of π. For th analysis, w partition th squn of mtings into pohs as follows. W start from th first mting and kp going until th third distint mting ours: ths mtings form th first poh. That is, an poh altrnats btwn two diffrnt mtings. Rpating this pross partitions th ntir squn of mtings into pohs, possibly with a singl rmaining mting as final poh. A solution an hoos th start prmutation π 0 that supports th first poh. Aftr that it an always gt to a prmutation that supports th ntir nxt poh with on blok rossing. (Not that th grdy algorithm uss xatly this stratgy.) In th worst as all pohs hav lngth 2, and w nd n/2 1 blok rossings. Sin th grdy algorithm an hoos an arbitrary start prmutation it dos not nd to prform a blok rossing for th first squn of mtings baus it an build a start prmutation using th sam stratgy as dsribd abov. This lavs us with at most n/2 1 blok rossings. Thorm 7 For k = 3, th grdy algorithm produs optimal solutions. Proof: W look at th pohs from Lmma 3 again. Th grdy algorithm produs on blok rossing fwr than th numbr of pohs. Considr any poh xpt th last on and inlud th mting aftr it. By onstrution, this is th third distint mting and thrfor ths mtings togthr annot fit a singl prmutation. Thn in any solution to th problm, a blok rossing must our aftr at last on of th mtings in th poh. This holds for all pohs xpt th last on and sin thy ar disjoint, th numbr of pohs rdud by on is a lowr bound for th optimum numbr of blok rossings. Th rsult of th grdy algorithm ralizs this bound. W not hr that th grdy algorithm dos not snsibly gnraliz to SBCM with arbitrary mtings. Considr th stp that finds a blok rossing that supports th maximum numbr of subsqunt mtings. With arbitrary mtings (as opposd to only ardinality-2 mtings), it may b th as that this maximum is zro that is, thr may not xist any singl blok rossing that supports th nxt mting. In this as our algorithm has no grdy way mak progrss.

17 JGAA, 21(5) (2017) 889 Tim (in s.) n Figur 8: Runtim of th xat algorithm of Thorm 5 on random instans with k = 4( ), 5(+), 6( ), 7( ). Eah data point is th avrag of 50 random instans. Exprimntal Evaluation. In this stion, w rport on som prliminary xprimntal rsults for 2-SBCM. W hav gnratd random instans as follows. Givn n and k, w pik n pairs of haratrs as mtings, uniformly at random using rjtion sampling to nsur that onsutiv mtings ar diffrnt. (This mans that th gnratd instans ar normal in th sns of Lmma 3.) First, w onsidr th xat algorithm of Thorm 5. As xptd, its runtim dpnds havily on k (Fig. 8). Prhaps somwhat unxptdly, w obsrv xponntial runtim in n. This is atually a proprty of our random instans, in whih β tnds to inras linarly with n. Not that this xprimnt dos not invalidat th algorithm sin in pratial appliations w may b intrstd mainly in instans for whih β is indd small. W hav also gnratd instans with small solutions as follows. Pik k, n and β, thn sampl a uniformly-random start prmutation and β uniformlyrandom blok rossings. Considr th squn of prmutations rsulting from ths blok rossings. W randomly sampl n mtings by piking, for ah on indpndntly, on of th prmutations at random and thn two adjant haratrs from this prmutation. An instan onstrutd in this way might hav a solution with fwr than β blok rossings, but by onstrution th optimum is at most β. On ths instans, th runtim of th algorithm of Thorm 5 sals as xptd. Sin th onstrution of ths instans is somwhat ompliatd and arbitrarily, w now rturn to th mor natural first typ of random instans. Th xat algorithm is fasibl only for rathr small instans, so w now shift our fous to th grdy algorithm. Rall that it starts with an arbitrary

18 890 T. C. van Dijk t al. Blok Crossings in Storylin Visualizations Frquny Blok movs Figur 9: Histogram of th numbr of blok rossings usd by th grdy algorithm for all k! diffrnt start prmutations, on a singl random instan with n = 100 and k = 8. Th bst grdy solution uss 45 blok rossings; th avrag is 51.2, with standard dviation Frquny Frquny Exss blok movs Exss blok movs Figur 10: Lft: histogram of HuristiGrdy minus BstGrdy, 200 instans with with k = 7 and n = 100. Right: histogram of RandomGrdy minus HuristiGrdy, 1000 instans with k = 30 and n = 200. prmutation and prods grdily. Th histogram in Fig. 9 shows th numbr of blok rossings usd by th grdy algorithm for ah of th k! possibl start prmutations, for a singl fixd instan. This bll urv is typial for instans gnratd by our random modl, and is shown for illustration. W s, qualitativly, that thr indd xist rar start prmutations that do notiably bttr than almost all othrs (hr with 45 blok rossings). Indd, for th rportd instan, a random start prmutation dos 7.2 blok rossings wors in xptation than th bst possibl start prmutation, whil th numbr of blok rossings for diffrnt start prmutations for this instan has standard dviation of 2.4. W all th bst possibl rsult of th grdy algorithm ovr all start prmutations BstGrdy, whih w alulat by brut for. This is dtrministi

19 JGAA, 21(5) (2017) Frquny Frquny Exss blok movs Exss blok movs Figur 11: Lft: Huristi-grdy minus OPT with k = 5 and n = 12, 1000 runs. Right: th sam, xpt k = 6 and 100 runs. givn th instan. Lt RandomGrdy b th rsult of th grdy algorithm starting with a prmutation hosn uniformly at random, and lt Huristi- Grdy b th rsult of starting with th huristi prmutation that w hav dsribd abov. Th histogram in Fig. 10 (lft) shows how many mor blok rossings HuristiGrdy uss than BstGrdy on random instans. This distribution is havist nar zro (maning th huristi is oftn los to optimal), but thr ar instans whr th diffrn is larg. On avrag th huristi is 4.1 blok rossings wors than bst possibl start prmutation on ths instans, whih is an improvmnt ovr random start prmutations; s Fig. 10, right. Not that w do not know how to omput BstGrdy ffiintly, but this xprimnt dmonstrats that thr is room for improvmnt within th grdy algorithm by piking a good start prmutation. Lastly, w ompar th grdy algorithm to th optimum, whih (baus of runtim) w only do for small k and n. On 1000 random instans with k = 5 and n = 12, HuristiGrdy gav an optimal solution 56% of th tim. It ndd somtims on (38%), two (5%), or thr (1%) blok rossings mor, but nvr mor than that. With k = 6, w gt similar numbrs; s Fig. 11. This is a promising bhavior, but larly annot b xtrapolatd vrbatim to largr instans. Basd on ths xprimnts, w rommnd HuristiGrdy as an ffiint, rasonabl huristi for th 2-SBCM problm whn it ariss. W hav not run xprimnts to visualiz ral storylins (movis, books, t tra) sin th rstrition to ardinality-2 mtings is too svr. Th gnalogial visualizations of Kim t al. [8] do fit this rstrition, but rquir furthr visualization work on top of solving th SBCM instan, whih w hav not prformd. Instad, w hav fousd ths xprimnts on som ombinatorial proprtis of random instans.

20 892 T. C. van Dijk t al. Blok Crossings in Storylin Visualizations Figur 12: Mting {, 1, 2, 3 } 6 Approximation Algorithm W now dvlop a onstant-fator approximation algorithm for d-sbcm whr d is a onstant. W initially assum that ah group mting ours xatly on, but latr show how to xtnd our rsults to th stting whr th sam group an mt a boundd numbr of tims. Ovrviw. Our approximation algorithm has th following thr main stps. 1. Rdu th input group hyprgraph H = (C, Γ) to an intrval hyprgraph H f = (C, Γ \ Γ p ) by dlting a subst Γ p Γ of th dgs of H. 2. Choos a prmutation π 0 of th haratrs that supports all groups of this intrval hyprgraph H f. Thus, π 0 is th ordr of haratrs at th bginning of th timlin. 3. Inrmntally rat support for ah dltd mting of Γ p in ordr of inrasing tim, as follows. W arbitrarily fix on haratr of th mting and mov th othr haratrs of th mting nxt to it. Aftr th mting w mov th haratrs bak to thir original positions; s Fig. 12. Stp 2 is straightforward: Stion 2 shows how to find a prmutation supporting all th groups for an intrval hyprgraph. Th main thnial parts of th algorithm ar Stp 1 and an analysis to harg at most a onstant numbr of blok rossings in Stp 3 to a blok rossing in th optimal visualization. Stp 1 rquirs solving a hyprgraph problm; this is thnially th most hallnging part, and onsums th ntir Stion 7. Bounds and Analysis. W all th dgs in Γ p paid dgs, and th dgs in Γ f = Γ \ Γ p fr dgs. Intuitivly, a fr dg an b ralizd without a blok rossing baus H f is an intrval hyprgraph, whil ah dg Γ p must b hargd to blok rossings of th optimal drawing. W initializ th drawing by plaing th haratrs in th vrtial ordr π 0, whih supports all th groups in Γ f. Now w onsidr th paid dgs in lft-to-right ordr. Suppos that th nxt mting involvs a group g Γ p. W hav g d and fix on haratr of g. To bring th rmaining haratrs of g in this haratr s viinity, w nd at most (d 1) blok rossings, on pr lin. Whn th mting is ovr, w

21 JGAA, 21(5) (2017) 893 again us up to (d 1) blok rossings to rvrt th lins bak to thir original position prsribd by π 0 ; s Fig. 12. W do this for ah paid hyprdg, giving ris to at most 2(d 1) Γ p blok rossings. W now prov that this bound is within a onstant fator of optimal. W first stablish a lowr bound on th optimal numbr of blok rossings assuming a fixd start prmutation. Lmma 4 Lt π b a prmutation of th haratrs, lt Γ f b th st of groups supportd by π, and lt Γ p = Γ \ Γ f. Any storylin visualization with start prmutation π nds at last 2 Γ p /(3d 2 ) blok rossings. Proof: Lt g Γ p. Sin g is not supportd by π, th optimal drawing dos not ontain th haratrs of g as a ontiguous blok initially. Howvr, in ordr to support this mting, ths haratrs must vntually bom ontiguous bfor th mting starts. Th ordr hangs only through (blok) rossings; w bound th numbr of groups that an bom supportd aftr ah blok rossing. Aftr a blok rossing, at most thr pairs of lins that wr not nighbors bfor an bom nighbors in th prmutation: aftr th bloks C 1, C 2 C ross, thr is on position in th prmutation whr a lin of C 1 is nxt to a lin of C 2, and two positions with a lin of C 1 (C 2, rsptivly) and a lin of C \ (C 1 C 2 ). Any group that was not supportd, but is supportd aftr th blok rossing, must ontain on of ths pairs. W an dsrib ah suh group in th nw prmutation by spifying th nw pair and th numbrs d 1 and d 2 of haratrs of th group abov and blow th nw pair in th prmutation. Sin th group siz is at most d, w hav d 1 + d 2 d. For any d 1 1, thr ar d d 1 possibl hois for d 2. Togthr with d 1, d 2 1 (sin th nw pair is inludd), w gt at most d 1 d (d d 1=1 1) = d 1 d d 1=1 1 = d(d 1)/2 d 2 /2 possibl groups for ah nw pair. Thus, th total numbr of nwly supportd groups aftr a blok rossing is at most 3d 2 /2, whih shows that th optimal numbr of blok rossings is at last 2 Γ p /(3d 2 ), omplting th proof. W now bound th loss of optimality ausd by not knowing th initial prmutation usd by th optimal solution. Th ky ida hr is to us a onstantfator approximation for th problm Intrval Hyprgraph Edg Dltion: dlt th minimum numbr of hyprdgs from H so that H boms an intrval hyprgraph. W prov th following thorm in Stion 7. Thorm 8 W an find a (d + 1)-approximation for Intrval Hyprgraph Edg Dltion on group hyprgraphs with m hyprdgs of rank d in O(m 2 ) tim. With th hlp of Thorm 8, w an show th main rsult of this papr. Thorm 9 d-sbcm admits a (3d 2 (d 2 1))-approximation algorithm that runs in O(kn) tim.

22 894 T. C. van Dijk t al. Blok Crossings in Storylin Visualizations Proof: Lt Γ p b th st of paid dgs in our algorithm, and lt Γ OPT b th st of paid dgs in th optimal solution, that is, th st of dgs that ar not supportd by th initial prmutation of th optimal solution. By Thorm 8, w hav Γ p (d + 1) Γ OPT, sin vn Γ OPT annot b smallr than an optimum solution for th intrval hyprgraph dg dltion problm. Lt ALG and OPT b th numbrs of blok rossings for our algorithm and th optimal solution, rsptivly. Our algorithm nsurs that ALG 2(d 1) Γ p 2(d 1)(d + 1) Γ OPT. On th othr hand, by Lmma 4, w hav OPT 2 Γ OPT /(3d 2 ), whih givs Γ OPT 3d 2 /2 OPT. Combining th prvious inqualitis, w gt ALG 3d 2 (d 2 1) OPT as dsird. Now w analyz th tim omplxity. W hav to onsidr th prmutation (of lngth k) of haratrs bfor and aftr ah of th n mtings, as wll as aftr ah of th O(n) blok rossings. This rsults in O(kn) tim for th last part of th algorithm, but this is dominatd by th tim (O(n 2 )) ndd for finding Γ p and for dtrmining th start prmutation. W an improv th running tim to O(kn) by a slight modifiation: using th approximation algorithm for Intrval Hyprgraph Edg Dltion is only nssary for spars instans. If H has suffiintly many dgs, any start prmutation will yild a good approximation. Sin no mting involvs mor than d haratrs, no start prmutation an support mor than dk mtings. If n 2dk, thn vn th optimal solution must thrfor rmov at last half of th dgs. Hn, taking an arbitrary start prmutation yilds an approximation fator of at most 2 < d + 1. W now hang th algorithm to us an arbitrary start prmutation if n 2dk and only us th approximation for Intrval Hyprgraph Edg Dltion othrwis. In partiular, w us th approximation only if thr ar n O(k) dgs. Hn, for spars instans w hav O(n 2 ) O(kn), and for dns instans, w skip th ostly O(n 2 )-tim stp mntiond abov. This yilds th dsird running tim of O(kn), whih is worst-as optimal sin th output omplxity is of th sam ordr. Rmark. W assumd that ah group mts only on, but w an xtnd th rsult if ah group an mt α tims, for onstant α. Our algorithm thn yilds a (α 3d 2 (d 2 1))-fator approximation; ah rptition of a mting may triggr a onstant numbr of blok rossings not prsnt in th optimal solution. Improvmnts for 2-SBCM. By using spifi struturs for 2-haratr mtings w an improv approximation fator and runtim; not that th gnral algorithm yilds a 36-approximation. For 2-haratr mtings th group hyprgraph is a graph; an intrval hyprgraph in this stting is a olltion of vrtx-disjoint paths. Our algorithm for Intrval Hyprgraph Edg Dltion for d = 2 yilds a 3-approximation. W dvlop a bttr approximation using th following obsrvation. Considr a haratr in th olltion of paths supportd in th bginning of som solution. If has two nighbors 1 and 2 in its path, but s first mting is with