Towards Compressing Web Graphs

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1 Towards Compressig Web Graphs Micah Adler Λ Uiversity of Massachusetts, Amherst Michael Mitzemacher y Harvard Uiversity Abstract We cosider the problem of compressig graphs of the lik structure of the World Wide Web. We provide efficiet algorithms for such compressio that are motivated by recetly proposed radom graph models for describig the Web. The algorithms are based o reducig the compressio problem to the problem of fidig a miimum spaig tree i a directed graph related to the origial lik graph. The performace of the algorithms o graphs geerated by the radom graph models suggests that by takig advatage of the lik structure of the Web, oe may achieve sigificatly better compressio tha atural Huffma-based schemes. We also provide hardess results demostratig limitatios o atural extesios of our approach. 1 Itroductio A sapshot of the World Wide Web ca be thought of as a graph, with Web pages represeted by odes ad hyperliks represeted by directed edges. This represetatio is used by a wide variety of Web algorithms, icludig algorithms for rakig pages based o their coectivity [12, 4] ad algorithms for fidig atural commuities of pages o a shared topic [14]. At least oe major search egie has desiged a tool called the coectivity server for storig the Web graph [3, 5]. Give this iterest, a atural questio to ask is how well the Web graph ad Web-like graphs ca be compressed. Such compressio would allow for more efficiet storage ad trasfer of Web graphs, ad may improve the performace of Web algorithms by allowig computatio to be performed i faster levels of computer memory hierarchies. Good compressio requires usig the structural properties of the Web graph, ad hece a importat first step is uderstadig this structure. Previous work gives us importat isights. It is clear that the Web graph appears to be sigificatly differet from the likely graphs resultig from traditioal radom graph models. I particular, there appear to be atural clusters of related pages with similar coectios. Hece, i [13, 15], a ew radom graph model was itroduced with these clusterig properties. The basis of this model is that pages ad liks eter the system dyamically, ad ew pages may lik to other pages by fidig oe or more referece pages ad copyig liks from these refereces. Λ Departmet of Computer Sciece, Uiversity of Massachusetts, Amherst, MA micah@cs.umass.edu. y Harvard Uiversity, Divisio of Egieerig ad Applied Scieces, 33 Oxford St., Cambridge, MA Supported i part by a Alfred P. Sloa Research Fellowship, NSF CAREER grat CCR , ad a equipmet grat from Compaq Computer Corporatio. michaelm@eecs.harvard.edu.

2 Recet studies of the Web graph suggest that the structure of the Web is actually more complex tha this radom graph model; see, for example [5]. However, as a first approximatio, this model captures importat high level behavior, ad it may be especially suitable for specific subdomais, such as all the pages withi a give uiversity. Hece, i this paper we focus o variatios of this graph model, although we also test our prototype o real data [10]. Our primary results are the followig: ffl We provide a compressio algorithm specifically desiged for graph structures with may shared liks. Uder appropriate assumptios, the ruig time of our algorithm is O( log ),where is the umber of odes i the graph. The algorithm requires fidig a directed miimum spaig tree o a graph associated with the origial graph. ffl We provide hardess results demostratig that several atural extesios of our algorithm are NP-Hard. ffl We demostrate the effectiveess of our approach o a testbed of radom graphs derived from the radom graph models that motivate our approach. Space limitatios require us to limit our expositio here. More details ca be foud i the complete exteded abstract [1]. 1.1 Previous ad Cocurret Work Sice developig this work, we have foud that similar ideas have bee used by the creators of the coectivity server, a tool for keepig compressed Web graph data [3]. They have foud the referece-based approach extremely effective [17], although they also use locality. Our approach, however, is sigificatly differet from theirs. For example, here we focus o optimal algorithms ad correspodig theoretical limitatios. The coectivity server project has focused o approximate algorithms that seem to be effective i practice, but without the same theoretical guaratees that we provide. We believe both works represet importat steps i uderstadig ad improvig compressio of Web graphs. The Web graph ca be thought of as a sparse bit array. Methods for compressig sparse bit arrays that make use of the probability distributios of the gaps betwee successive etries are discussed i [20]. Similar approaches are used by the creators of the coectivity server ad are useful for compressig local liks betwee pages o the same host [17]. Our referece approach, i cotrast, is desiged for liks betwee pages o distict hosts, ad thus is better suited to the simple Web models we examie here. Directed miimum spaig trees have bee used previously i other scearios to provide good compressio. Tate [18] uses such trees to obtai a reorderig of the bads of a multispectral image that allows for the optimal compressio. More recetly, a similar idea is alluded to i [6], i the cotext of compressig tables of data. There, the authors use oe colum to compress aother, ad metio that the problem ca be reduced to a miimum spaig tree problem, although i their case, edges are udirected. 1.2 Framework Whe we discuss compressig Web-like graphs, there are actually a variety of distict situatios we may wish to cosider:

3 1. Compressig the uderlyig graph for storage or trasmissio, up to isomorphism. This settig would be useful if we wat to store just the graph structure itself. 2. Compressig the uderlyig graph for storage or trasmissio, maitaiig a give orderig of the odes. For example, we might order the odes accordig to the sorted order of the URLs (so that the URLs ca be compressed by delta ecodig, as i [3]). 3. Compressig the uderlyig graph for use i its compressed form. That is, we desire a compressed form of the graph that still allows for efficiet computatio o the compressed form. Our primary focus i the paper is the secod settig; however, we will suggest coectios betwee the variatios as they arise. 1.3 A Web graph model We reiterate that i this paper, rather tha compress actual subgraphs of the Web, our focus is a recetly proposed Web graph model that captures certai aspects of Web graphs. The model, take primarily from [15], uses the followig basic outlie. The graph evolves over time by associated ode ad edge creatio ad deletio processes. The ituitio suggested from [15] is the followig: A ew page adds liks by pickig a existig page, ad copyig some liks from that page to itself. For example, a ew page v might examie the outedges from a page w ad lik to a subset of the pages that w liks to; we call this copyig outedges. This ituitio is based o the idea that a user decides what pages to lik a ew page to based o a page or pages that the user already likes. Give this framework, there are a variety of possible variatios, depedig o the specifics of the edge creatio ad deletio process as well as the copy process. We specify the model we use here. We begi with a iitial graph of 0 odes, with each havig d 0 outedges coected to odes chose uiformly at radom. There is o deletio process, oly a ode creatio process. Oe ew ode is created each time step to a total of odes. The creatio process is determied by probability distributios A; B; C; D; E; F. The distributio A provides a umber a, such that v is give a outedges with each edge poitig to a ode chose uiformly at radom from all odes existig at that time. Similarly, B provides a umber such that v copies outedges from b odes, agai chose uiformly at radom. The distributio C yields a probability; for each of the b odes w 1 ;:::;w b chose to copy from, a probability is idepedetly chose from C, ad each outedge from w i is copied idepedetly with the probability determied from C. Distributios D; E, adf are aalogous to A; B, adc respectively, except that they determie the iedges for a ew page. We call such graphs copy graphs. 2 A baselie Huffma-based scheme Experimets have demostrated that the idegrees ad outdegrees of Web pages follow a Zipfia distributio [2, 15, 5]. That is, the fractio of pages with idegree j is roughly proportioal to 1=j ff for some fixed costat ff, ad similarly the fractio of pages with outdegree j is roughly proportioal to 1=j fi for some fixed costat fi. Oe of the features of the copy graph model is that it yields graphs with such Zipfia distributios [15]. Give the large variace i degrees, it is atural to cosider Huffma-based compressio schemes. A simple such scheme goes through the odes i order ad lists the destiatio of

4 each outedge directed from that ode. Each page is assiged a Huffma codeword based o its idegree. To separate the outedges of each ode we utilize a special stop symbol. May simple variatios are possible. The compressio scheme could also be based o the edges directed ito each ode, whichever is better. I the case where we oly eed to store a isomorphism of the graph, we might avoid the stop symbol. Istead, we ca sed a implicit or explicit represetatio of the outdegree distributio, sort the odes by outdegree, ad list the outedges for each ode as before without the stop symbol. This approach achieves sigificat compressio with little complexity; it could aturally be used i the framework of the coectivity server [3], or i ay system that wated efficiet computatio o the compressed form of the graph. Thus we shall use it as a baselie for the compariso of algorithms that compress the Web graph. Note, however, that the Huffmabased scheme igores the atural clusterig structure iduced i copy graphs. 3 The FIND-REFERENCE algorithm Our basic algorithm is based o the followig isight: sice copyig liks is a basic operatio i our graph model, we ca attempt to fid odes that share several commo outedges, correspodig to cases where oe ode might have copied the liks of aother. Oce a appropriate eighbor is idetified, the differece, or delta, betwee the outedges of the two odes ca be idetified. Whe ode i is compressed i this way usig ode j, we say that ode j is a referece for ode i. For example, let us cosider a specific scheme (we describe possible improvemets to this scheme i Sectio 3.1). If ode j is labeled as a referece of ode i, we ca iclude a 0/1 bit vector deotig which outedges of ode j are also outedges of ode i. Other outedges of i ca the be separately idetified, say usig dlog e bits i a ode graph. We must also idetify i, which is aother dlog e bits. Let N (i) ad N (j) represet the set of outedges for ode i ad ode j respectively. The cost of compressig ode i usig ode j as a referece with this scheme is the cost(i; j) =out-deg(j) +dlog e (jn (i) N (j)j +1): Give a descriptio of a graph i this kid of compressed format, cosider how we would determie where a lik from ode i ecoded usig ode j as a referece actually poits. If the correspodig lik from ode j is ecoded usig aother ode k as a referece, the we would eed to determie where the correspodig lik from ode k poits. Evetually, we must reach a lik that is ecoded without usig a referece ode. I order to satisfy this requiremet, we shall ot allow ay cycles amog refereces. For example, we shall ot allow i to be compressed usig j as a referece, j to be compressed usig k as a referece, ad k to be compressed usig i as a referece. A itermediate structure that FIND-REFERENCE uses is the affiity graph G S for the give Web graph G W. Specifically, the odes of G S are the same as the odes of G W.Weset w(i; j), the weight of the directed edge from ode i to ode j, to be the cost of compressig ode i usig ode j as a referece. We add to the affiity graph a root ode r to which every other ode has a directed edge ad from which there are o directed edges. The weight of the edge from i to r is the cost of compressig i without usig ay other ode as a referece. We assume that ode i has a directed edge to ode j if ad oly if w(i; j) <w(i; r).

5 Give a Web graph, the algorithm FIND-REFERENCE first computes the correspodig affiity graph for the give cost fuctio, ad the fids a optimal set of refereces uder the restrictios that (a) each ode has at most oe referece, ad (b) there are o cycles amog refereces. The problem of fidig the globally best mappig from odes to refereces (or to the dummy ode) is equivalet to fidig the miimum weight directed spaig tree with root r o the affiity graph. Thus, a high level descriptio of the compressio algorithm is as follows: Algorithm FIND-REFERENCE ffl Give a Web graph G W, compute the correspodig affiity graph G S. ffl Compute a miimum directed spaig tree D rooted at r for the graph G S. ffl Compress the graph G W, where ode i uses ode j as a referece if ad oly if ode i poits to ode j i D. Theorem 1 ForaWebgraphG W, let be the umber of odes i G W, ad let t G W (i) be the idegree P of ode i. Algorithm FIND-REFERENCE ca be realized to ru i time O ( log + i=1 (t GW (i)2 )). Proof: The affiity graph G S ca be computed from the origial graph G W by usig a matrix multiplicatio. Whe G W is a Web graph, we expect it to be sparse, ad so we describe the algorithm i terms of a sparse matrix multiplicatio. Let M represet the adjacecy matrix of G W. Itiseasytoverifythat(MM T ) ij is the umber of odes P that both i ad j have outedges to. The matrix (MM T ) ca be computed i time O ( i=1 t G W (i)2 ), assumig that we compute a list of the o-zero etries of (MM T ). We also P compute a array R, where R[j] = out-deg(j). This requires time O( + m), wherem = i=1 t G W (i) is the umber of edges i G W.Give(MM T ) ij ad R[j], cost(i; j) ca be computed i costat time. Note that there will ever be a edge from i to j i G S uless odes i ad j i G W have a outgoig edge to at least oe shared eighbor. Thus, to compute the set of edges i G S, we oly eed to compute cost(i; r) for every i, ad the for every edge from i to j such that (MM T ) ij > 0, compute cost(i; j), ad compare it to cost(i; r). The set of edges i G S is f(i; j) s.t. cost(i; j) < cost(i; r)g ad the set of edges from every other P vertex to r. Thisalso gives us the weight of each edge i G S. Sice there ca be at most i=1 t G P W (i)2 ozero etries i (MM T ), the total time required to compute the graph G S is O ( + i=1 t G W (i)2 ). Computig a miimum directed spaig tree with root r i a directed graph is geerally referred to i the literature as a brachig with root r. 1 For iformatio o brachigs, see for example [7, 9, 11, 19]. Miimum spaig trees i directed graphs with x odes ad y edges ca be foud determiistically i time O(x log x+y) [9]. A simpler algorithm that rus i time O(y log x) is suitable for the case of sparse graphs [19, 7], which P will geerally be the case i our cotext. Sice the total umber of edges i G S is at most i=1 t G W (i)2 +, P the total time required to compute the miimum directed spaig tree i G S is O ( log + i=1 t G W (i)2 ). All that remais is to perform the compressio usig the computed directed tree to specify a referece for each ode. To do this, we compute for each ode i with referece ode j a liked list of outedges that i ad j have i commo. This set of lists ca be computed i time 1 Brachigs geerally refer to the (equivalet) maximum weight problem. They are sometimes also referred to as arboresceces.

6 O ( P i=1 t G W (i)2 ). With the list of edges that i ad j have i commo, the compressed versio of ode i ca be computed i time O(out-deg(i)). Thus, the etire algorithm rus i time O ( log + P i=1 t G W (i)2 ). 2 Note that the performace of this algorithm is particularly good whe G W is sparse, as we expect of P Web graphs. For example, if the distributio of idegrees i G W is Zipfia with ff>3,the i=1 t G W (i)2 = O(). 3.1 Additioal improvemets ad related problems I practice, after we have foud the refereces via the directed miimum spaig tree, there are various improvemets that ca be implemeted. For example, we may wish to fid additioal refereces for greater compressio. This ca be doe by strippig edges from the origial graph hadled by the first refereces, re-calculatig the cost fuctio accordigly, ad re-ruig the algorithm. This algorithm is ot optimal, sice i some cases, better compressio is possible if we choose the refereces of the first stage keepig i mid that we have further stages comig. I geeral, however, fidig multiple refereces i a optimal maer is NP-hard, as we show i Sectio 4. Oce we have foud the best refereces, we may agai use a Huffma ecodig to hadle the edges ot covered by refereces. Note that by doig this, we ivalidate the cost fuctio we used to determie the refereces, so that the set of refereces may ot be optimal. However, util we choose the refereces, we caot determie the cost of edges ot covered by refereces, so it seems difficult to take this ito accout properly i the cost fuctio. Other possible improvemets iclude usig differet compressed represetatios. We have suggested usig a bit vector to deote which liks a ode has copied from its referece. These bit vectors ca be Huffma or ru-legth ecoded. Although there are a variety of possible ehacemets that may slightly improve compressio, we believe the mai cocept of usig similar pages for compressio provides the bulk of the beefit. Although our algorithm is desiged for storig Web graphs, we believe these techiques ca also apply whe we wish to compute with the compressed form. The potetial problem is that fidig the iedges or outedges of a ode may require goig through multiple refereces i the directed miimum spaig tree, which may take more computatio time tha desired. To boud the umber of refereces to pass through i our sigle-referece settig it is sufficiet to boud the depth of the directed miimum spaig tree we fid o the affiity graph. Ufortuately, fidig the optimal directed miimum spaig tree of bouded depth is NPhard; for example, if we allow depth at most two, the the problem of fidig the optimal directed miimum spaig tree is equivalet to the facility locatio problem. (This coectio to the facility locatio problem was a major poit i the work o compressig tables of data metioed earlier [6].) I practice, we expect that usig the FIND-REFERENCE algorithm to iitially fid a directed tree ad the choppig the tree to maitai a depth boud (by chagig some odes to be compressed without a referece ad thus likig them to the root r) will be suitable. 4 Hardess results Sice we ca fid the optimal compressio give a appropriate cost metric whe we allow a sigle referece ode usig brachig algorithms, a atural questio to ask is whether we ca

7 similarly achieve optimality whe we allow more tha oe referece ode. We show hardess results related to this questio. We focus o the case where up to two odes ca be used as refereces, but everythig described is easily geeralized to ay umber of referece odes. For up to two referece odes, the affiity graph becomes the followig kid of structure: Defiitio 1 A 2-supergraph is a directed hypergraph where each hyperedge is directed from a sigle ode to two other odes. These two other odes ca be the same, but must be differet from the source ode. Give a Web graph, we shall cosider the correspodig weighted 2-supergraph, where w ijk, the weight of the hyperedge from i to j ad k, represets the cost of ecodig i usig both j ad k as refereces. The weight w ijj represets the cost of ecodig ode i usig oly ode j as a referece; i the correspodig hyperedge the ode i poits to j twice. Note that w ijk varies depedig o the overlap betwee the set of edges of the Web graph that odes i ad j have i commo ad the set of edges that odes i ad k have i commo. We call the resultig 2-supergraph a affiity 2-supergraph. Give a Web graph, computig the affiity 2-supergraph for a give lik compressio scheme ca easily be doe i polyomial time. Usig the affiity 2-supergraph to compute the best compressio usig up to two referece odes is equivalet to the followig geeralizatio of fidig optimal brachigs: Defiitio 2 Give a 2-supergraph G ad a desigated root ode r,a 2-brachig is a subset S of the hyperedges of G such that each ode except the desigated root has exactly oe outgoig hyperedge i S, ad r has o outgoig hyperedges i S. I additio, the hypergraph formed by the set of hyperedges i S has o directed cycles. The optimum 2-brachig is the 2-brachig that miimizes the total weight of the edges i S. Ufortuately, i geeral fidig the optimal 2-brachig is ot oly NP-Hard, it is hard to approximate. We demostrate a approximatio preservig polyomial time reductio from the problem of fidig the optimal directed Steier tree to the problem of fidig the optimal 2-brachig. It is kow that if P 6= NP, the o polyomial time algorithm ca fid a log -approximatio to the directed Steier tree problem [8]. Theorem 2 Ay polyomial algorithm that provides a k-approximatio for the 2-brachig problem also provides a k-approximatio for the directed Steier tree problem. Proof: Please see the exteded abstract [1]. The iapproximability result demostrates the hardess of the geeral problem of fidig a optimal 2-brachig. This result does ot however directly imply that it is eve NP-Hard to fid the best compressio usig at most two refereces, sice the graphs that we reduce the directed Steier tree problem to may ot correspod to actual affiity graphs that arise as a result of a Web graph. We also provide a more direct reductio showig that it is i fact NP-Hard to fid the best compressio of a Web graph based o usig up to two referece odes. I fact, eve if we igore the additioal difficulty imposed by takig ito accout the asymmetry of the affiity graph, the problem remais NP-Hard. I particular, we demostrate that the problem of fidig the assigmet of referece odes that maximizes the total umber of edges i the Web graph that are represeted by a correspodig edge i a referece ode is NP-Hard.

8 Theorem 3 The problem of fidig a ecodig for a graph G W, with each ode ecoded usig up to two referece odes, that maximizes the total umber of edges that are ecoded usig a referece ode is NP-Hard. Proof: Please see the exteded abstract [1]. 5 Experimets We preset the results from a prelimiary prototype ruig o artificial radom copy graphs ad oe subset of a sapshot of the Web graph. We emphasize that these experimets are meat as a prelimiary proof of cocept. I particular, the prototype does ot output a compressed file, but rather the compressed size of the file. Moreover, whe usig Huffma codig, the compressed size does ot iclude the size of ay associated Huffma tables. We chose ot to iclude this as the size of the Huffma table depeds o whether oe compresses it further; this makes the Huffma scheme look better tha it would be i practice. We first describe the graphs we tested. For the radom copy graphs, our tests all had odes. (Smaller test graphs had similar performace.) Each graph bega with 1024 seed odes with three outedges, where the ed of the outedge was chose uiformly at radom from all odes. Whe ew odes were added, they were give oly outedges. The outedges were determied by copyig edges from some umber of odes ad by geeratig edges with edpoits chose uiformly at radom from all preset odes. We show the parameters for the copy graphs tested i Table 1. The field # radom copies deotes the umber of odes whose outedges were copied. A rage such as [1; 2] deotes that a iteger value was chose uiformly over that rage. Each edge was copied with a fixed probability, listed as the copy probability. The field # radom edges gives the umber of edges that were geerated with radom destiatios; agai, a rage deotes that a iteger value was chose uiformly over that rage. We ote that for the large graphs G 3 ad G 4, we were forced by memory cosideratios to limit the affiity graph to allow edges betwee odes i ad j oly if their outedges share at least three destiatios. This ca oly hurt our compressio efforts. Further testig suggests that the differece is miimal if two shared destiatios ca be hadled. We also tested our compressio scheme o real Web data from from the TREC-8 (Text REtrieval Coferece 8) Web track [10]. Our data set was the WT2g data set which was chose as a small subset of the Web for testig iformatio by the TREC retrieval coferece. This data set is larger tha our radom sets; hece agai to costruct the affiity graph o the TREC database we oly created edges betwee pages with at least three shared liks. Table 2 presets the compressio results i terms of total bits required divided by edges i the graph. For the radom graphs, we have take the average of te trials, where a differet radom graph is produced for each trial. We ote that there is little deviatio betwee the rus. For the ucompressed size i bits per edge, we use the uderestimate log 2 (#odes). As see i graph G 1, whe the amout of copyig is low, ad thus the average degree is very small, the referece algorithm aloe does slightly worse tha the Huffma algorithm, although usig a Huffma code i cojuctio with the referece algorithm leads to better performace. Whe the amout of copyig is larger, as for G 2, G 3,adG 4, our FIND- REFERENCE algorithm greatly outperforms Huffma codig. We expect repeated passes might allow eve greater compressio. The Huffma algorithm compresses the outedges for each edge, so the code words are based o the idegree. For the FIND-REFERENCE algo-

9 rithm, we test both the straightforward algorithm as well as oe which first determies the refereces ad the uses Huffma codig o the remaiig outedges. Our results are actually best for the TREC database, demostratig that our approach should be effective o real Web data as well. Our belief is that our good results o the TREC data set arise because liks appear to have sigificat locality. Ideed, the strog effect of locality has bee corroborated by the upublished related work o the coectivity server [17]. They use the fact that most Web liks are betwee pages o the same server to fid good refereces quickly. This lack of ackowledgmet of locality i curret Web models is problematic, although it is beig addressed [16]. Note that our approach ca be used to take advatage of locality; for example, our referece-based algorithm could be ru o a host-byhost basis, usig for example oly the outliks o a give host, to improve speed. Name G 1 G 2 G 3 G 4 TREC Nodes # Radom Copies 1 1 [1,2] [0,4] NA Copy prob NA # Radom Edges 1 1 [1,2] [1,2] NA Table 1: Parameters of the test graphs. Name G 1 G 2 G 3 G 4 TREC Ucompressed Huffma FIND-REFERENCE F.Ref. + Huff Table 2: Results from the test graphs; bits per edge. 6 Future Work We have iitiated study ito how to compress Web graphs usig the copy graph model, a radom graph family with properties similar to Web graphs. Usig this structure, we have desiged a compressio algorithm based o fidig similarity amog the liks of the pages ad tested it o simple copy graphs. We have also show that various geeralizatios of this idea lead to NP-Hard problems. There are several directios remaiig to pursue, icludig refiig algorithms for dealig with the locality of pages i the same domai. Although desigig such a system ca be doe via experimetatio, developig a appropriate model that allows us to uderstad the tradeoffs would be a iterestig problem. Aother iterestig issue is whether these compressio algorithms ca be used to test proposed radom graph models. A possible techique to test how accurately a proposed radom graph model captures the structure of real Web graphs would be to ru our compressio algorithm (or ay other compressio algorithm) o both kids of graphs, ad compare the compressio obtaied. Refereces [1] M. Adler ad M. Mitzemacher. Towards compressig Web graphs. U. of Mass. CMPSCI Techical Report Available at ftp://ftp.cs.umass.edu/pub/techrept/techreport/2000.

10 [2] W. Aiello, F. Chug, ad L. Lu. A radom graph model for massive graphs. I Proceedigs of the 32d Aual ACM Symposium o Theory of Computig, pages , [3] K. Bharat, A. Broder, M. Heziger, P. Kumar, ad S. Vekatasubramaia. The Coectivity Server: fast access to likage iformatio o the Web. I Proceedigs of the 7th World Wide Web Coferece, Available at [4] S. Bri ad L. Page. The aatomy of a large-scale hypertextual Web search egie. I Proceedigs of the 7th World Wide Web Coferece, Available at [5] A. Broder, R. Kumar, F. Maghoul, P. Raghava, S. Rajagopala, R. Stata, A. Tomkis, ad J. Wieer. Graph structure i the Web: experimets ad models. I Proceedigs of the 9th World Wide Web Coferece, Available at [6] A. L. Buchsbaum, D. F. Caldwell, K. W. Church, G. S. Fowler, ad S. Muthukrisha. Egieerig the compressio of massive tables: a experimetal approach. I Proceedigs of 11th Aual ACM-SIAM Symposium o Discrete Algorithms, pages , [7] P. M. Camerii, L. Fratta, ad F. Maffioli. A ote o fidig optimum brachigs. Networks, 9: , [8] M. Charikar, C. Chekuri, T. Cheug, Z. Dai, A. Goel, S. Guha, ad M. Li. Approximatio algorithms for directed Steier problems. I Proceedigs of the Nith Aual ACM-SIAM Symposium o Discrete Algorithms, pages , Sa Fracisco, Califoria, Jauary [9] H. N. Gabow, Z. Galil, T. Specer, ad R. E. Tarja. Efficiet algorithms for fidig miimum spaig trees i udirected ad directed graphs. Combiatorica 6(2): , [10] D. Hawkig, E. Voorhees, N. Craswell, ad P. Bailey. Overview of the TREC- 8 Web Track. I The 8th Text Retrieval Coferece, Available at proceedigs.html. [11] R. M. Karp. A simple derivatio of Edmods algorithm for optimum brachigs. Networks, 1: , [12] J. Kleiberg. Authoritative sources i a hyperliked eviromet. I Proceedigs of 9th Aual ACM-SIAM Symposium o Discrete Algorithms, pages , [13] J. Kleiberg, R. Kumar, P. Raghava, S. Rajagopala, ad A. Tomkis. The Web as a graph: Measuremets, Models, ad Methods. I Proceedigs of the Iteratioal Coferece o Combiatorics ad Computig, [14] R. Kumar, P. Raghava, S. Rajagopala, ad A. Tomkis. Trawlig the Web for emergig cybercommuities. I Proceedigs of the 8th World Wide Web Coferece, [15] R. Kumar, P. Raghava, S. Rajagopala, ad A. Tomkis. Extractig large scale kowledge bases from the Web. I Proceedigs of the 25th VLDB Coferece, [16] M. Mitzemacher ad J. Wieer. Improved graph models for the Web. Mauscript i preparatio. [17] R. Stata ad J. Wieer. Persoal commuicatio, o the Compaq Systems Research Ceter coectivity server. [18] S. R. Tate. Bad Orderig i Lossless Compressio of Multispectral Images. IEEE Trasactios o Computers, Vol. 46, No. 4, 1997, pp [19] R. E. Tarja. Fidig Optimum Brachigs. Networks, 7:25-35, [20] I. Witte, A. Moffat, T. Bell. Maagig Gigabytes: 2d Editio. Morga Kaufma, Sa Fracisco, 1999.

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