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**Dynamic Graph Algorithms - I**

Surender Baswana Department of CSE, IIT Kanpur.

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**About this talk animation**

Prerequisite: a course on data structure and algorithms Survey of the results: (not the main objective of the talk) Main contents: -- A novel data structure -- A fully dynamic algorithm for a fundamental graph problem AIM: To give an exposure to the beautiful area of dynamic algorithms animation

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**A (static) Graph Algorithm**

G=(V,E) Solution

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**A dynamic graph algorithm**

No assumption about the updates. For problems involving queries (connectivity, distance,…): initial graph G=(V,E) followed by a sequence q,u,u,q,u,u,u,q,u,q,q,u, … q: query u: insertion/deletion of edge Each query has to be answered in an online manner. For problems that aim to maintain some structure (matching, spanner, min-cut,…) u,u,u,u,u,u,u, … The structure has to be maintained in an online manner.

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**Dynamic graph algorithm**

Aim: Maintain a data structure which can answer each query efficiently (or maintain the structure), and process each update efficiently (much faster than the static algo) Types of dynamic graph algorithms: Incremental (only insertion of edges) Decremental (only deletion of edges) Fully dynamic (both insertion and deletion of edges)

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**A motivating example : Undirected Connectivity**

Static solution: v a b c d

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**A motivating example : Undirected Connectivity**

Incremental Algorithm: O(log* n) update time (Disjoint Set Union Algorithm [Tarjan 1975]) Decremental Algorithm: O(n) update time [Even and Shiloach, 1981] O(log n) update time [Thorup 1997]

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**A motivating example : Undirected Connectivity**

Fully Dynamic Algorithms: O( 𝒎 ) update time [Frederickson, 1982] O( 𝒏 ) update time [Eppstein, Galil, Italiano, Nissenzweig 1991] O(polylog n) expected update time, O(log n) query time [King and Henzinger 1995] 4. O(polylog n) update time, O(log n) query time [Holm, Litchenberg, Thorup 1998]

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Outline of the talk Dynamic graph algorithms for some important problems Data structure for dynamic trees Fully dynamic connectivity with polylog n update time Open problems

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**Examples of fully dynamic algorithms**

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**Fully dynamic algorithms for undirected graphs**

Connectivity 2-edge connectivity Bi-connectivity Bipartiteness Min. spanning tree O(polylog n) update time [Holm, Litchenberg, Thorup 1998]

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**Fully dynamic algorithms for undirected graphs**

Min-cut Best static algorithm: O(m polylog n) (Randomized) [Karger, 1996] Fully dynamic algorithm: O( 𝒏 ) update time [Thorup, 2001] Graph spanner Definition: A subgraph which is sparse and yet preserves all-pairs distances approximately. Best static algorithm: O(m) [Halperin, Zwick, 1996] Fully dynamic algorithm: O(polylog n) update time [Baswana, Khurana, and Sarkar, 2008]

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**Dynamic algorithms for directed graphs**

Maintaining BFS tree under deletion of edges : O(n) time per edge deletion [Even & Shiloach, 1981] Unbeaten till date. Used in many dynamic algorithms for directed graphs. Not as good bounds as undirected graphs

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**Dynamic algorithms for directed graphs**

Transitive Closure Incremental algorithm: O(n) update time [Italiano, 1986] Decremental algorithm: O(n) update time Randomized [Roditty and Zwick, 2002] Deterministic [Lacki, ] Fully Dynamic algorithm: O( 𝒏 𝟐 ) update time [Roditty, 2003] All-pairs Shortest paths Fully dynamic algorithms: Amortized O( 𝒏 𝟐 ) update time [Demetrescu and Italiano, 2003] Worst case O( 𝒏 𝟐.𝟕𝟓 ) update time [Thorup, 2005]

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**Data structure for dynamic trees**

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**Data structures Stacks Queues Binary heap Binary search tree ...**

Fibinaacci heap Too elementary Limited applications Too complex Hardly any application !

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**Power of Data structures: An inspirational example**

Maintain n records r(1),…,r(n) under the following operations Add(i,j,x) : Add x to each record from r(i),…,r(j). All-swap(i,j) : r(i)↔r(j), r(i+1) ↔ r(j-1), r(i+2) ↔ r(j-2). Report(i) : report record r(i). Report-min(i,j) : report the smallest record from r(i),…,r(j). Each operation in O(log n) worst case time.

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**Balanced Binary Tree : a very powerful data structure**

: Additional information

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Dynamic Trees a j g b f e c d u w v u

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Dynamic Trees Aim : Maintain a forest of trees on n vertices under the following operation. Link(u,v) : Add an edge between u and v Cut(u,v) : Delete an edge between u and v Update() : Update information associated with nodes/edges Query() : Topological information associated with a tree, or a path

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**Data Structures for Dynamic Trees**

ST Tree [Sleator & Tarjan, 1983] Operations and queries on edges of paths ET tree [Henzinger and King, 1995] Operations and queries on nodes of a tree Top tree [Alstrup et al., TALG 2005] (generalization of Topology Tree [Frederickson, 1982]) Topological properties (diameter, center)

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**Dynamic Trees query and updates on trees**

j g b 5 3 2 17 Operations : Link(u,v) Cut(u,v) Update-weight-node(v,a): weight(v) a Add-weight-tree(v,x): add x to weight of each node of tree of v ReportMin(u): report min weight in the entire tree containing u 32 f 41 7 e c d -3 u 12 44 67 15 w v u

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**Dynamic Trees query and updates on trees**

ReportMin(v) = 12 a j g b 5 3 2 17 Operations : Link(u,v) Cut(u,v) Update-weight-node(v,a): weight(v) a Add-weight-tree(v,x): add x to weight of each node of tree of v ReportMin(u): report min weight in the entire tree containing u 32 f 41 7 e c d -3 u 12 44 67 15 w v u

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**Dynamic Trees query and updates on trees**

ReportMin(v) changes … a j g b 5 3 2 17 Operations : Link(u,v) Cut(u,v) Update-weight-node(v,a): weight(v) a Add-weight-tree(v,x): add x to weight of each node of tree of v ReportMin(u): report min weight in the entire tree containing u 32 f 41 7 e c d -3 u 12 44 67 1 w v u

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**Euler tour tree : A data structure for dynamic trees**

h d g a b c e f b-c-d-c-b-a- e-f-e-g-e-h-e -a-b How to transform a tree into a one dimensional data structure ? b d e c a f g h

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**Euler tour tree : A data structure for dynamic trees**

h d g a b c e : minimum value of all nodes in the subtree. f b-c-d-c-b-a- e-f-e-g-e-h-e -a-b f a f b c d e g h c h c g a a b d b e e e e b

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**Euler tour tree : A data structure for dynamic trees**

h d g a b c e f b-c-d-c-b-a- e-f-e-g-e-h-e -a-b

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**Euler tour tree : A data structure for dynamic trees**

h d g a b c e f b-c-d-c-b-a- e-f-e-g-e-h-e -a-b

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**Euler tour tree : A data structure for dynamic trees**

h d g T2 T1 a b c e f b-c-d-c-b-a a-b T1 e-f-e-g-e-h-e

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**Euler tour tree : A data structure for dynamic trees**

h d g T2 T1 a b c e f b-c-d-c-b-a a-b T1 e-f-e-g-e-h-e T1

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**Euler tour tree : A data structure for dynamic trees**

h d g T2 T1 a b c e f b-c-d-c-b-a a-b T1 e-f-e-g-e-h-e T1

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**Euler tour tree : A data structure for dynamic trees**

h d g T2 T1 a b c e f b-c-d-c-b-a a-b T1 e-f-e-g-e-h-e T1

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**Euler tour tree : A data structure for dynamic trees**

h d g T2 T1 a b c e Split(T,(e,a)) Merge(T1,T2,(u,v)) Change-origin(T,x) : change the origin of Euler tour to vertex x. f T1 T2 e-f-e-g-e-h-e b-c-d-c-b-a-b T1 T2

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**Fully dynamic algorithm for connectivity**

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**Fully dynamic randomized algorithm for connectivity with polylogarithmic update time**

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**Maintain ET tree for each tree in the spanning forest**

Fully dynamic randomized algorithm for connectivity with polylogarithmic update time Maintain ET tree for each tree in the spanning forest

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**Maintain ET tree for each tree in the spanning forest**

Fully dynamic randomized algorithm for connectivity with polylogarithmic update time Maintain ET tree for each tree in the spanning forest

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**Fully dynamic randomized algorithm for connectivity with polylogarithmic update time**

A Hierarchical algorithm 1 2 3 2 log n

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**Fully dynamic randomized algorithm for connectivity with O( 𝒎 ) update time**

A 2-level algorithm 2 1

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**Decremental O( 𝒎 ) update time algorithm for connectivity**

Key tools in addition to ET tree data structure: Trivial algorithm (for handling deletion of a tree edge) : Let (u,v) be a tree edge in the spanning forest. Let its deletion creates trees T1 and T2. Let µ be the number of non-tree edges incident on T2. Replacement edge can be found in time O(µ log n) time. Random sampling

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**The role of random sampling**

Exercise: If none of 2k log n balls is blue, then with probability 1−𝑛 −2 , the fraction of blue balls is less than 1/k. Uniform random sampling with replacement

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**Handling the deletion of a tree edge**

How to augment ET-tree to sample an edge ? Few samplings needed if the fraction of blue edges is large What if fraction of blue edges is small ? T2 T1 T

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**Handling the deletion of a tree edge 2-Level approach**

A partition of E into two levels : (E1, E2) In the beginning, E1 = E and E1 = Ø F1 : spanning forest of E1 F2 : spanning forest of E, F1 is subset of F2 Level 1 Level 2

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**Handling the deletion of a tree edge**

Level 2 Level 1

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**Handling the deletion of a tree edge**

Trivial algorithm at level 2 Level 1 Level 2 Random sampling at level 1

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**Algorithm for handling deletion of a tree edge**

O(µ2) time If (e ϵ F2 \ F1) scan non-tree edges at level 2 to find replacement edge. Else Let T be the tree to which e belongs; (T1,T2) Split(T,e); Repeat k log n times { (u,v) Sample-edge(T2); If (u,v) is a cut-edge { add (u,v) to F1; Merge(T1,T2, (u,v)); return; } Scan all non-tree edges incident on T2; If less than 1/k fraction are cut-edges move all edges of cut(T1,T2) to Level 2 and add one of them to F2. Else add an edge of cut(T1,T2) to F1 O(k logn) time O(µ1(T2)) time 1−𝑛 −2 probability

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**Bounding µ2 (number of non-tree edges at level 2)**

Upon splitting T into T1 and T2, how many edges are passed to level 2 ? ≤ 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑛𝑜𝑛_𝑡𝑟𝑒𝑒 𝑒𝑑𝑔𝑒𝑠 𝑖𝑛𝑐𝑖𝑑𝑒𝑛𝑡 𝑜𝑛 𝑇2 𝑘 charge 𝑑𝑒𝑔(𝑣) 𝑘 to each 𝑣 ϵ T2 Level 2 Level 1 T1 T2

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**Analysis A vertex v is processed only O(log n) times.**

Whenever v is processed The processing cost of v is O(deg(v)) The number of edges that move to level 2 is less than 𝑑𝑒𝑔(𝑣) 𝑘 Hence µ2 , the number of edges at level 2 is O( 𝑚 𝑘 log n) Processing cost per update : 𝑚 𝑘 log n k O(m) [1−𝑛 −2 ] = O( 𝒎 )

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**Transforming to fully dynamic environment**

Add every newly inserted edge to level 2. Periodically rebuild the data structure after every ( 𝒎 ) insertions. Expected amortized time per update : O( 𝒎 𝒍𝒐𝒈 𝒏)

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**Fully dynamic randomized algorithm for connectivity with polylogarithmic update time**

A partition of E into 2log n levels : (E1, E2, …) In the beginning, E1 = E and Ei = Ø for all i>1 F1 : spanning forest of E1 Fi : spanning forest of 𝑗=1 𝑖 𝐸𝑗 Fi-1 is subset of Fi for all i>1

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**Fully dynamic randomized algorithm for connectivity with polylogarithmic update time**

A Hierarchical algorithm 2 log n 2 2 c 1

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Open Problems

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**Open problems Amortized cost versus worst case bounds**

Specific problems : Min-cut, s-t min cut, max-flow, … Specific graph family : Planar graphs Better lower bounds ?

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