These slides are from 2014 and contain a semi-serious error at one point in the

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1 Hello Internet! Hello Internet! These slides re rom 2014 nd ontin semi-serious error t one point in the These slides re rom 2014 nd ontin semi-serious error t one point in the presenttion. For more up-to-dte set o slides with orretion, plese visit presenttion. For more up-to-dte set o slides with orretion, plese visit Hope this helps! Hope this helps! -Keith -Keith

2 Euler Tour Trees

3 Outline or Tody Dynmi Connetivity on Trees Mintining onnetivity in hnging environment. Euler Tour Trees A dt struture or dynmi onnetivity. The Bottlenek Pth Prolem Putting everything together.

4 The Dynmi Connetivity Prolem

5 The Connetivity Prolem The grph onnetivity prolem is the ollowing: Given n undireted grph G, preproess the grph so tht queries o the orm re nodes u nd v onneted? Using Θ(m + n) preproessing, n preproess the grph to nswer queries in time O(1)

6 Dynmi Connetivity The dynmi onnetivity prolem is the ollowing: Mintin n undireted grph G so tht edges my e inserted n deleted nd onnetivity queries my e nswered eiiently. This is muh hrder prolem!

7 Dynmi Connetivity Best known dt struture: Edge insertions nd deletions tke verge time O(log 2 n) time. (Nottion: log (k) n is log log log n, k times. log k n is (log n) k. Connetivity queries tke time O(log 2 n / log log n). This is topi or lter in the qurter. The solution is not trivil. Tody, we'll look t restrited version o the prolem tht will serve uilding lok or the generl version.

8 Dynmi Connetivity in Forests Consider the ollowing speil-se o the dynmi onnetivity prolem: Mintin n undireted orest G so tht edges my e inserted n deleted nd onnetivity queries my e nswered eiiently. Eh deleted edge splits tree in two; eh dded edge joins two trees nd never loses yle.

9 Dynmi Connetivity in Forests Gol: Support these three opertions: link(u, v): Add in edge {u, v}. The ssumption is tht u nd v re in seprte trees. ut(u, v): Cut the edge {u, v}. The ssumption is tht the edge exists in the tree. is-onneted(u, v): Return whether u nd v re onneted. The dt struture we'll develop n perorm these opertions time O(log n) eh.

10 Euler Tours

11 Euler Tours In grph G, n Euler tour is pth through the grph tht visits every edge extly one. Mthemtilly ormultes the tre this igure without piking up your penil or redrwing ny lines puzzles.

12 Euler Tours on Trees In generl, trees do not hve Euler tours. d e d d d e Tehnique: reple eh edge {u, v} with two edges (u, v) nd (v, u). Resulting grph hs n Euler tour.

13 Euler Tours on Trees The dt struture we'll design tody is lled n Euler tour tree. High-level ide: Insted o storing the trees in the orest, store their Euler tours. Eh edge insertion or deletion trnsltes into set o mnipultions on the Euler tours o the trees in the orest. Cheking whether two nodes re onneted n e done y heking i they're in the sme Euler tour.

14 Properties o Euler Tours The sequene o nodes visited in n Euler tour o tree is losely onneted to the struture o the tree. d e g h i j k l g h i d d e d i j l j i h g g k g

15 Properties o Euler Tours The sequene o nodes visited in n Euler tour o tree is losely onneted to the struture o the tree. d e g h i j k l g h i d d e d i j l j i h g g k g

16 Properties o Euler Tours The sequene o nodes visited in n Euler tour o tree is losely onneted to the struture o the tree. d e g h i j k l g h i d d e d i j l j i h g g k g

17 Properties o Euler Tours The sequene o nodes visited in n Euler tour o tree is losely onneted to the struture o the tree. d e g h i j k l g h i d d e d i j l j i h g g k g

18 Properties o Euler Tours The sequene o nodes visited in n Euler tour o tree is losely onneted to the struture o the tree. Begin y direting ll edges towrd the the irst node in the tour. Clim: The sequenes o nodes visited etween the irst nd lst instne o node v gives n Euler tour o the sutree rooted t v.

19 Rerooting Tour The sutrees deined y rnges in Euler tours on trees depend on the root. In some ses, we will need to hnge the root o the tree. d e g h i j k l g h i d d e d i j l j i h g g k g

20 Rerooting Tour The sutrees deined y rnges in Euler tours on trees depend on the root. In some ses, we will need to hnge the root o the tree. d e g h i j k l g h i d d e d i j l j i h g g k g

21 Rerooting Tour The sutrees deined y rnges in Euler tours on trees depend on the root. In some ses, we will need to hnge the root o the tree. d e g h i j k l g h i d d e d i j l j i h g g k g

22 Rerooting Tour The sutrees deined y rnges in Euler tours on trees depend on the root. In some ses, we will need to hnge the root o the tree. d e g h i j k l h i d d e d i j l j i h g g k g g h

23 Rerooting Tour The sutrees deined y rnges in Euler tours on trees depend on the root. In some ses, we will need to hnge the root o the tree. d e g h i j k l h i d d e d i j l j i h g g k g g h

24 Rerooting Tour The sutrees deined y rnges in Euler tours on trees depend on the root. In some ses, we will need to hnge the root o the tree. d e g h i j k l h i d d e d i j l j i h g g k g g h

25 Rerooting Tour The sutrees deined y rnges in Euler tours on trees depend on the root. In some ses, we will need to hnge the root o the tree. d e g h i j k l h i d d e d i j l j i h g g k g g h

26 Rerooting Tour The sutrees deined y rnges in Euler tours on trees depend on the root. In some ses, we will need to hnge the root o the tree. d e g h i j k l d d e d i j l j i h g g k g g h i d

27 Rerooting Tour The sutrees deined y rnges in Euler tours on trees depend on the root. In some ses, we will need to hnge the root o the tree. d e g h i j k l d d e d i j l j i h g g k g g h i d

28 Rerooting Tour The sutrees deined y rnges in Euler tours on trees depend on the root. In some ses, we will need to hnge the root o the tree. d e g h i j k l d d e d i j l j i h g g k g g h i d

29 Rerooting Tour The sutrees deined y rnges in Euler tours on trees depend on the root. In some ses, we will need to hnge the root o the tree. d e g h i j k l d d e d i j l j i h g g k g g h i d

30 Rerooting Tour The sutrees deined y rnges in Euler tours on trees depend on the root. In some ses, we will need to hnge the root o the tree. d e g h i j k l g h i d d e d i j l j i h g g k g

31 Rerooting Tour Algorithm: Split the tour into three prts: S₁, R, nd S₂, where R onsists o the nodes etween the irst nd lst ourrene o the new root r. Delete the irst node in S₁. Contente R, S₂, S₁, {r}.

32 Euler Tours nd Dynmi Trees Given two trees T₁ nd T₂, where u T₁ nd v T₂, exeuting link(u, v) links the trees together y dding edge {u, v}. Wth wht hppens to the Euler tours: h i d e g j k d e g j k j i j g h g

33 Euler Tours nd Dynmi Trees Given two trees T₁ nd T₂, where u T₁ nd v T₂, exeuting link(u, v) links the trees together y dding edge {u, v}. Wth wht hppens to the Euler tours: h i d e g j k d e g j k j i j g h g

34 Euler Tours nd Dynmi Trees Given two trees T₁ nd T₂, where u T₁ nd v T₂, exeuting link(u, v) links the trees together y dding edge {u, v}. Wth wht hppens to the Euler tours: h i d e g j k d e g j k j i j g h g

35 Euler Tours nd Dynmi Trees Given two trees T₁ nd T₂, where u T₁ nd v T₂, exeuting link(u, v) links the trees together y dding edge {u, v}. Wth wht hppens to the Euler tours: h i d e g j k d e g j k j i j g h g

36 Euler Tours nd Dynmi Trees Given two trees T₁ nd T₂, where u T₁ nd v T₂, exeuting link(u, v) links the trees together y dding edge {u, v}. Wth wht hppens to the Euler tours: h i d e g j k e d g j k j i j g h g g

37 Euler Tours nd Dynmi Trees Given two trees T₁ nd T₂, where u T₁ nd v T₂, exeuting link(u, v) links the trees together y dding edge {u, v}. Wth wht hppens to the Euler tours: h i d e g j k e d g j k j i j g h g g

38 Euler Tours nd Dynmi Trees Given two trees T₁ nd T₂, where u T₁ nd v T₂, exeuting link(u, v) links the trees together y dding edge {u, v}. Wth wht hppens to the Euler tours: h i d e g j k e d g j k j i j g h g g

39 Euler Tours nd Dynmi Trees Given two trees T₁ nd T₂, where u T₁ nd v T₂, exeuting link(u, v) links the trees together y dding edge {u, v}. To link T₁ nd T₂ y dding {u, v}: Let E₁ nd E₂ e Euler tours o T₁ nd T₂, respetively. Rotte E₁ to root the tour t u. Rotte E₂ to root the tour t v. Contente E₁, E₂, {u}.

40 Euler Tours nd Dynmi Trees Given tree T, exeuting ut(u, v) uts the edge {u, v} rom the tree (ssuming it exists). Wth wht hppens to the Euler tour o T: h i d e g j k e d g j k j i j g h g g

41 Euler Tours nd Dynmi Trees Given tree T, exeuting ut(u, v) uts the edge {u, v} rom the tree (ssuming it exists). Wth wht hppens to the Euler tour o T: h i d e g j k e d g j k j i j g h g g

42 Euler Tours nd Dynmi Trees Given tree T, exeuting ut(u, v) uts the edge {u, v} rom the tree (ssuming it exists). Wth wht hppens to the Euler tour o T: h i d e g j k d e g j k j i j g h g g

43 Euler Tours nd Dynmi Trees Given tree T, exeuting ut(u, v) uts the edge {u, v} rom the tree (ssuming it exists). Wth wht hppens to the Euler tour o T: h i d e g j k d e g j k j i j g h g g

44 Euler Tours nd Dynmi Trees Given tree T, exeuting ut(u, v) uts the edge {u, v} rom the tree (ssuming it exists). Wth wht hppens to the Euler tour o T: h i d e g j k d g j k j i j g h g g e g

45 Euler Tours nd Dynmi Trees Given tree T, exeuting ut(u, v) uts the edge {u, v} rom the tree (ssuming it exists). Wth wht hppens to the Euler tour o T: h i d e g j k d g h g g e g j k j i j

46 Euler Tours nd Dynmi Trees Given tree T, exeuting ut(u, v) uts the edge {u, v} rom the tree (ssuming it exists). To ut T into T₁ nd T₂ y utting {u, v}: Let E e n Euler tour or T. Rotte u to the ront o E. Split E into E₁, V, E₂, where V is the spn etween the irst nd lst ourrene o v. T₁ hs the Euler tour ormed y ontenting E₁ nd E₂, deleting the extr u t the join point. T₂ hs Euler tour V.

47 The Story So Fr Gol: Implement link, ut, nd is-onneted s eiiently s possile. By representing trees vi their Euler tours, n implement link nd ut so tht only O(1) joins nd splits re neessry per opertion. Questions to nswer: How do we eiiently implement these joins nd splits? One we hve the tours, how do we nswer onnetivity queries?

48 Implementing the Struture The opertions we hve seen require us to e le to eiiently do the ollowing: Identiy the irst nd lst opy o node in sequene. Split sequene t those positions. Contente sequenes. Add new opy o node to sequene. Delete duplite opy o node rom sequene. How do we do this eiiently?

49 An Initil Ide: Linked Lists d e d e d e d e

50 An Initil Ide: Linked Lists Eh split or ontente tkes time O(1). The irst nd lst opy o node n e identiied in time O(1). A new opy o node n e ppended to the end o the sequene in time O(1). A redundnt opy o node n e deleted in time O(1). Everything sounds gret! Question: How do you test or onnetivity?

51 d e g d e g d e g d e g

52 The Story So Fr Euler tours give simple, lexile enoding o tree strutures. Using douly-linked lists, ontention nd splits tke time O(1) eh, ut testing onnetivity tkes time Θ(n) in the worst-se. Cn we do etter?

53 Using Blned Trees Clim: It is possile to represent sequenes o elements lned inry trees. These re not inry serh trees. We're using the shpe o red/lk tree to ensure lne. e d g d e g

54 Using Blned Trees Oservtion 1: Cn still store pointers to the irst nd lst ourrene o eh tree node. e d g d e g

55 Using Blned Trees Oservtion 2: I nodes store pointers to their prents, n nswer is-onneted(u, v) in time O(log n) y seeing i u nd v re in the sme tree. e d g d e g

56 Using Blned Trees Oservtion 3: Red/lk trees n e split nd joined in time O(log n) eh. e d g d e g

57 Using Blned Trees Oservtion 3: Red/lk trees n e split nd joined in time O(log n) eh. d e g d e g

58 Euler Tour Trees The dt struture: Represent eh tree s n Euler tour. Store those sequenes s lned inry trees. Eh node in the originl orest stores pointer to its irst nd lst ourrene. Eh node in the lned trees stores pointer to its prent. link, ut, nd is-onneted queries tke time only O(log n) eh.

59 Augmented Euler Tour Trees Euler tour trees re lyered top red/lk trees. We n thereore ugment Euler tour trees to store dditionl inormtion out eh tree. Exmples: Keep trk o the minimum-weight or mximumweight eh in eh tree. Keep trk o the numer o nodes in eh tree. We'll use this oth lter tody nd in ew weeks when we tlk out dynmi grph onnetivity.

60 Time-Out or Announements!

61 CS Csul Dinner Csul dinner or women studying omputer siene tomorrow night on the Gtes Fith Floor, 6PM 8PM. Everyone is welome!

62 OH Updte This week only: I'll e overing the Thursdy oie hours nd the TAs will over tody's OH right ter lss. Thursdy OH rooms TBA. We will proly e hnging Thursdy OH times in uture weeks; detils lter.

63 Your Questions

64 Could you reommend some (other) wesome CS lsses (preerly in Theory, ut doesn't hve to e)? My My Reommendtions CS143 CS143 (Compilers) (Compilers) CS343 CS343 (Dynmi (Dynmi Anlysis) Anlysis) CS154 CS154 (Automt (Automt nd nd Complexity Complexity Theory) Theory) CS181 CS181 (Computers, (Computers, Ethis, Ethis, nd nd Puli Puli Poliy) Poliy) CS261 CS261 // CS361B CS361B (Advned (Advned Algorithms) Algorithms) CS224W CS224W (Soil (Soil Network Network Anlysis) Anlysis)

65 Wht perentge o student sumitted the irst prolem set individully versus in pirs? I'm I'm going going to to hold hold o o on on nnouning this this until until Wednesdy when when we we get get n n estimte or or Prolem Set Set Two. Two. I I expet there there will will e e n n inrese.

66 I've never written Mkeile eore. Where n I ind the resoures to lern how to mke one or Assignment 2? Chek out out the the CS107 resoures pge: pge:

67 Wht's the est wy to impress everyone t oktil prty? Blning ork ork nd nd spoon spoon on on toothpik, then then setting setting the the toothpik toothpik on on ire. ire. Either Either tht tht or or wlking wlking through through sheet sheet o o pper. pper. Everyone Everyone will will ind ind you you impressive i i you you tlk tlk out out the the history history o o ood ood nd nd nutrition. nutrition. Bonus Bonus points points or or throwing throwing out out the the words words nixtmliztion nd nd orngery. Or Or just just e e nie nie nd nd good good listener. listener.

68 Bk to CS166!

69 Applition: Bottlenek Edge Queries

70 Bottlenek Pths Let G e n undireted grph where eh edge hs n ssoited, positive, relvlued pity. The ottlenek edge on pth is the edge on the pth with minimum pity. Chllenge: Find the mximumottlenek pth etween nodes u nd v.

71 Bottlenek Pths

72 Bottlenek Pths

73 Bottlenek Edge Queries The ottlenek edge query prolem is the ollowing: Given n undireted grph G with edge pities, preproess the grph so tht the ottlenek edge etween ny pir o nodes u nd v n e ound eiiently. Applitions in network routing, shipping, nd mximum low lgorithms. Like RMQ, ould solve y preomputing the ottlenek pths etween ll pirs o nodes. Cn we do etter?

74 Bottlenek Pths

75 Bottlenek Pths This This is is mximum mximum spnning spnning tree tree o o the the undireted undireted grph. grph. It's It's like like minimum 1 minimum spnning spnning tree, tree, ut ut with with the the mximum mximum possile possile weight. weight. 7 5 A mximum mximum spnning spnning tree tree in in grph grph n n e e ound ound in in time time O(m O(m + n 5 log log n). n). Detils Detils next next week! week!

76 Bottlenek Pths Clim: Clim: For For ny ny pir pir o o nodes, nodes, there there is is mximum- mximumottlenek ottlenek pth pth etween etween those those nodes nodes using using only only the the edges edges in in the the mximum mximum spnning spnning tree. tree

77 Bottlenek Pths Any Any edge edge not not in in the the tree tree n n e e repled repled y y the the edges edges in in the the yle yle it it loses loses without without lowering lowering the the ottlenek. ottlenek. Repeting Repeting this this proess proess turns turns ny ny pth pth into into pth pth using using only only tree tree edges. edges

78 Bottleneks nd Trees Beuse o the previous oservtion, we only relly need to ous on this prolem s pplied to trees. Good News: There is n elegnt reursive oservtion out this prolem on trees. 3 d 6 e g 17 h Any Any nodes nodes on on opposite opposite sides sides o o this this edge edge hve hve ottlenek ottlenek ost ost i j

79 Crtesin Trees o Trees Given tree T, the Crtesin tree o tree T is deined reursively: The root o the tree is the minimum-pity edge in T; ll tht edge {u, v}. The let sutree is the Crtesin tree o the u sutree nd the right sutree is the Crtesin tree o the v sutree. The Crtesin tree o one-node tree is tht node itsel d e e {,e} {,} {,} {d,} d {,d}

80 Crtesin Trees o Trees Crtesin trees o trees re useul or the ollowing reson: In tree T, the ottlenek etween u nd v is the lowest ommon nestor o u in v in the Crtesin tree o T d e e {,e} {,} {,} {d,} d {,d}

81 RMQ nd LCA A B D G C E F A B C B A D E D F D A G A Solving Solving RMQ RMQ on on the the Euler Euler tour tour o o tree tree gives gives solution solution to to LCA LCA on on tht tht tree. tree. LCA LCA hs hs n n O(n), O(n), O(1) O(1) solution! solution!

82 The Solution As preproessing: Compute mximum spnning tree T* in time O(m + n log n). Construt the Crtesin tree o T* in time O(n log n) (detils in seond). Construt n LCA dt struture or tht Crtesin tree in time O(n). Totl preproessing: O(m + n log n). To mke query or the ottlenek edge etween u nd v: Compute the LCA o u nd v. Totl time: O(1).

83 Summry Trees n e represented vi Euler tours. Given the Euler tours o two trees, those trees n e linked y doing O(1) ontentions nd splits. Given the Euler tour o tree, n ut tht tree y doing O(1) ontentions nd splits. Representing the Euler tours o vrious trees s lned trees gives O(log n) link, ut nd is-onneted opertions. The ottlenek edge query prolem n e solved in time O(m + n log n), O(1) using pproprite dt strutures.

84 Next Time Amortized Anlysis Cn we trde worst-se eiieny or glol eiieny? The Bnker's Method Putting redits on dt strutures. The Potentil Method Deining the potentil energy o dt struture. Amortized Eiient Dt Strutures A smpling o mortized dt strutures.