1 Dynamic Connectivity

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1 15-850: Advanced Algorthms CMU, Sprng 2017 Lecture #3: Dynamc Graph Connectvty algorthms 01/30/17 Lecturer: Anupam Gupta Scrbe: Hu Han Chn, Jacob Imola Dynamc graph algorthms s the study of standard graph algorthmc problems n the setup where the graph changes over tme. For ths lecture, t s assume that the vertex set of the underlyng graph s fxed and the graph changes on the edge set wth update operatons Insert(u,v)and Delete(u,v), u and v beng vertces. Ths s known as the edge arrval model 1 Dynamc Connectvty Two connectvty queres that we would lke to support are Connected(u,v), whch ask f 2 vertces are n the same connected component and Connected(G), whch ask f the graph connected. Whle ths s a basc queston to ask and many results have been shown, t has not been completely resolved yet. Lets consder 2 nave approach to dynamc connectvty for 2 vertces. 1. Track updates to the graph as an adacency lst and run DFS at every query. Ths would take O(1) for update and O(m + n) for query. 2. Track the vertces parwse connectvty as a n n matrx. Check an entry n the matrx durng query. Ths would cost O(n 2 ) for update snce an edge can affect the connectvty between 2 O(n) szed component, and O(1) for query. Ths llustrates a trade-off between update and query. Below are some known results for dynamc connectvty. Authors Update Tme Query Tme Comments Frederckson [Fre85] O(m 1 2 ) O(1) Determnstc. Worst case Eppsten et al [EGIN97] O(n 1 2 ) O(1) Sparsfcaton Holm, de Lchtenberg, Throp O(log 2 n) O( log n log log n ) Amortzed. [HdLT98] Kapron, Kng, Mountoy O(log 5 n) O( log n log log n ) Randomzed. Worst case [KKM13] Let T u and T q be the update tme and query tme respectvely. The known lower bounds, due to [PD04], are ( ) Tq T u log log n T u ( ) Tu T q log log n T q 1

2 2 Frederckson s Algorthm Below we provde two mportant deas from Frederckson s algorthm [Fre85] that acheve O(m 2 3 ) update tme. The frst dea s pretendng that every vertex n G has maxmum degree 3. We can do ths by treatng any vertces v wth degree d > 3 as d vertces connected to each other by a cycle where each one s connected to a dstnct neghbor of v. The case for d = 4 s shown below. A B A B v 1 v 2 v v 3 v 4 C D C D Ths transformaton does not change the connectvty propertes of G. Thus, we wll assume for the rest of ths dscusson that G has maxmum degree 3. The second dea s mantanng a clustered spannng forest F of G. We wll cluster F n the followng way: Lemma 3.1. Gven any tree T = (V, E) and a parameter z, where V z, t s always possble to cluster V nto parttons V 1, V 2,..., V k such that each subgraph formed by V s connected and z V 3z. Proof. We wll use nducton on the sze of V. Our base case s all trees of sze at most 3z; then, nothng needs to be done. Assumng V > 3z, orent each edge (a, b) n T n the followng way: Remove (a, b) from the graph, and pont to a f the tree wth a n t s bgger than the tree wth b n t. Otherwse, pont to b. Snce T s now an orented tree, there must be a vertex v 0 whch s a snk. We wll analyze the case where v 0 has degree 3; the degree 1 and 2 cases are smpler. Suppose v 0 s connected to subtrees A, B, and C of szes S A, A B, and S C, as shown below: v 0 A B C Snce S A + S B + S C + 1 > 3z, not all three can be less than z. If all three are greater than z, then removng any edge connectng to v 0 wll produce two subtrees on whch we can apply the nductve hypothess. So, wthout loss of generalty, let S A < z and S B z. Snce the edge e that goes between B and v 0 ponts to v 0, we must have S A S C S B z. Thus, cuttng e wll produce two trees whch both have sze at least z, so they are able to be clustered by the nductve hypothess. The unon of these clusters s a way of clusterng T. Also, we can cluster T n O( E ) tme usng ths recursve strategy. 2

3 For each T n F we wll mantan a set of clusters C (T ) 1, C (T ) 2,..., C s (T ) T f T > z where each cluster has sze between z and 3z. If T < z, then ust mantan one cluster around the entre tree. Each C (T ) wll keep track of all edges wth at least one endpont n C (T ). Also, we wll keep track of an adacency lst A(T ) whch records whch clusters n T have edges between them. Snce the maxmum degree s at most 3, there are O(z) edges n total per cluster. The three operatons are outlned below: Insert(u,v): Let T u be the tree contanng u and T v the tree contanng v. If u = v, then we can add the edge to T u and be done. So, suppose u v. Let C (Tu) be the cluster contanng u and C (Tv) be the cluster contanng v. We wll merge C (Tu) and C (Tv). Mergng s smply a matter of lookng through the edges and vertces of C (Tu) and C (Tv), and takes O(z) tme. If the resultng cluster s too bg, we wll splt t n the way descrbed n the lemma whch takes O(z) tme. The total tme spent s O(z). Delete(u,v): Let T be the tree contanng (u, v). Let C (T ) be the cluster contanng u and C (T ) be the cluster contanng v. If, then skp to the next paragraph. Assumng =, we wll check f deletng ths edge makes C (T ) become dsconnected. Ths takes O(z) tme. If t does, then we need to do more work; we wll merge the two components of C (T ) to any neghborng clusters f they are too small. If the resultng clusters are too bg, we wll splt them n the way descrbed n the lemma. Fnally, we need to check f deletng (u, v) makes T become dsconnected. We wll do a graph search on A(T ) to( see f T s dsconnected, and f so, determne whch clusters should be splt ( off. Ths takes O m ) ) 2 z tme total. Query: Snce we have a spannng forest, queres are trval to answer. We can do t n O(1) tme. The runtme of Insert s O(z) and the runtme of Delete s O(z + (m/z) 2 ). Settng z = m 2 3 the best runtme for nserton and deleton of O(m 2 3 ). gves 3

4 3 Usng Randomzaton n Dynamc Connectvty Kapron-Kng-Mountoy n SODA 2013 [KKM13] showed a dynamc connectvty algorthm that uses randomzaton. It has O(polylogn) worst case update and O(polylogn) query wth 1-sded error, meanng Yes s always correct whle No has probablty 1 n error. c Smlar to most of the technques so far, KKM13 reles on mantanng a spannng forest usng dynamc trees. The man nnovaton s on dentfyng a canddate replacement edge on Delete. The hgh level deas of ther technque are as follows: 1. Consder a model where there are a seres of Insert followed by ust one Delete. Ths s needed to smplfy the analyss for ths lecture. 2. Each vertex,v, defne a O(log n) bt label called l(v) 3. Each edge, e = (u, v)) defne a label formed by the concatenaton of the labels of ts vertces l(e) = l(u)l(v) for some orderng of the vertces. 4. Let the bt XOR operator be. For a gven vertex V, defne sgn(v) = l(e), e are ncdent edges to x For a gven subset of vertces S, defne sgn(s) = v S sgn(v) Notce that sgn(s) s l(e), where e are edges whch has exactly 1 endpont n S. 5. The sgnature, sgn(v) operaton can be done on dynamc tree data structure, such as the Range operaton on lnk-cut trees, n O(log n) Suppose on Delete(u,v), edge e s beng remove. Let L be the subtree contanng, u and R be the subtree contanng v. Consder canddate replacement edges between L R n G\e. If there s no replacement edge, meanng G\e s dsconnected, then sgn(l) = sgn(r) = 0 If there s only 1 replacement edge, f, between L to R, then sgn(l) = sgn(r) = sgn(f) If there are many possble replacement edges then sgn(l) or sgn(r) does not help n dentfyng the replacement edge. The technque to fnd that replacement edge n the cutset, sze greater than 1, s to sample. Suppose O(log n) sets s mantan for each vertex. Sample an edge to the -th set wth probablty 2 whle dong Insert. For ths to work, one of these sets needs to have exactly one unque edge. Notce that f the cut set has C edges, then the probablty of the log C th set havng exactly 1 edge s: C 1 2 (1 1 2 )C 1 = C 1 2 log C (1 1 2 log C )C 1 = (1 1 C )C 1 > 1 e O(1) 4

5 1 Hence, wth hgh probablty of 1 scheme works. polylogn, there wll be some set wth exactly 1 edge, thus the The frst Delete on the tree s depends only on the state of the tree when delete occurs. However, subsequent Deletes would have dependences on other Delete. Thus the calculatons n the above analyss cannot use ndepedence the same way. Refer to [KKM13] for the technque to get around ths lmtaton n the analyss. 4 Amortzed analyss for O(polylogn) determnstc bound We would present a hgh level dea of [HdLT98] to a show dynamc connectvty algorthm wth O(log 2 n) amortzed bound. Consder storng the graph G as log n subgraph, each wth a mnmum spannng forest. For each edge, have a level that s an nteger n [0, log n]. Ths would be the potental functon used for the amortzed analyss. An edge s nserted at level 0 and moves up 1 level everythng t s scanned durng a delete operaton. Let G be the subgraph of G consstng of the edges of level and F be ts spannng forest. Each of the forests, F, s mantaned by a dynamc tree such as Lnk-Cut Trees where the amortzed cost for each of these tree operatons s O(log n). 2 nvarants would be mantaned throughout the executon of the algorthm. 1. I1: Every connected component of F has at most 2 vertces. 2. I2: F log n F log n 1 F 0. Also ths means f u, v G are connected, then u, v are connected n F On Insert(u,v), add the edge to G 0 and update the assocated data structures n O(log n). On Delete(u,v)=e, f edge, e s at level l, look at the tree, T whch contans e. Splt T nto sub tree L, R such that L R and rase the edges of L to level l + 1. Ths mantans I1 as L T 2 n 1 2 l 2 by nducton. Now start scannng the non-tree edges ncdent to L at level l. If the edge s wthn L, rase t to level l + 1. If t goes to R, then we have a replacement edge, and add t to F l, F l 1,, F 0 to mantan I2. Suppose that there no replacement edges at level l, then rerun the process at level l 1. Each edge charged O(log n) tmes thus t cost O(log 2 n) per Update. On Connected(u,v), check f both u, v are n the same component by I2. Ths can be done n O(log n) tme n the dynamc tree. 5 Key Ideas for the Lecture 1. Clusterng and tree separators as seen n Fredrckson. 2. Amortzed analyss by Holm et al. 3. Bt trcks such XOR to detect cuts and use of samplng as seen n KKM13. 5

6 References [EGIN97] Davd Eppsten, Zv Gall, Guseppe F. Italano, and Amnon Nssenzweg. Sparsfcaton;a technque for speedng up dynamc graph algorthms. J. ACM, 44(5): , September [Fre85] Greg N. Frederckson. Data structures for on-lne updatng of mnmum spannng trees, wth applcatons. SIAM Journal on Computng, 14(4): , , 2 [HdLT98] Jacob Holm, Krstan de Lchtenberg, and Mkkel Thorup. Poly-logarthmc determnstc fully-dynamc algorthms for connectvty, mnmum spannng tree, 2-edge, and bconnectvty. In Proceedngs of the Thrteth Annual ACM Symposum on Theory of Computng, STOC 98, pages 79 89, New York, NY, USA, ACM. 1, 4 [KKM13] Bruce M. Kapron, Valere Kng, and Ben Mountoy. Dynamc graph connectvty n polylogarthmc worst case tme. In Proceedngs of the Twenty-fourth Annual ACM- SIAM Symposum on Dscrete Algorthms, SODA 13, pages , Phladelpha, PA, USA, Socety for Industral and Appled Mathematcs. 1, 3, 3 [PD04] Mha Pǎtraşcu and Erk D. Demane. Lower bounds for dynamc connectvty. In Proceedngs of the Thrty-sxth Annual ACM Symposum on Theory of Computng, STOC 04, pages , New York, NY, USA, ACM. 1 6

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