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Surender Baswana Department of CSE, IIT Kanpur. Surender Baswana Department of CSE, IIT Kanpur.

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About this talk 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

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A (static) Graph Algorithm Algorithm G=(V,E)Solution

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A dynamic graph algorithm 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,…) initial graph G=(V,E) followed by a sequence u,u,u,u,u,u,u, … The structure has to be maintained in an online manner. No assumption about the updates.

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Dynamic graph algorithm Aim: 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 u v a a a a b b b b b b c c c c c c d d d d Static solution:

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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

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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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Fully dynamic algorithms for undirected graphs 1. Connectivity 2.2-edge connectivity 3. Bi-connectivity 4. Bipartiteness 5.Min. spanning tree O(polylog n) update time [Holm, Litchenberg, Thorup 1998]

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Fully dynamic algorithms for undirected graphs

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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

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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: 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 ef a u c gj d b w v u

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Aim 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 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 a u c g ef j d b w v u

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Dynamic Trees query and updates on trees ReportMin(v) = 12 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 a u c g ef j d b w v u

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Dynamic Trees query and updates on trees ReportMin(v) changes … 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 a u c g ef j d b w v u

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Euler tour tree : A data structure for dynamic trees ab c e f g h d b-c-d-c-b-a-e-f-e-g-e-h-e-a-b bd be e c c a f e g h a e b How to transform a tree into a one dimensional data structure ?

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Euler tour tree : A data structure for dynamic trees : minimum value of all nodes in the subtree. ab c e f g h d b-c-d-c-b-a-e-f-e-g-e-h-e-a-b bd be e c c a f e g h a e b a f b c d e g h

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Euler tour tree : A data structure for dynamic trees ab c e f g h d 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 ab c e f g h d 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 ab c e f g h d a-b b-c-d-c-b-a e-f-e-g-e-h-e T1 T2 T1

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Euler tour tree : A data structure for dynamic trees ab c e f g h d a-b b-c-d-c-b-a T1 T2 T1 e-f-e-g-e-h-e

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Euler tour tree : A data structure for dynamic trees ab c e f g h d a-b b-c-d-c-b-a e-f-e-g-e-h-e T1 T2 T1

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Euler tour tree : A data structure for dynamic trees ab c e f g h d a-b b-c-d-c-b-a e-f-e-g-e-h-e T1 T2 T1

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Euler tour tree : A data structure for dynamic trees Split(T,(e,a)) Merge(T 1,T 2,(u,v)) Change-origin(T,x) : change the origin of Euler tour to vertex x. ab c e f g h d b-c-d-c-b-a-b e-f-e-g-e-h-e T1 T2 T1 T2

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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

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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 log n

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12 A 2-level algorithm

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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 T 1 and T 2. Let µ be the number of non-tree edges incident on T 2. Replacement edge can be found in time O(µ log n) time. Random sampling

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The role of random sampling Uniform random sampling with replacement

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Handling the deletion of a tree edge T2 T1 T T2 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 ?

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Handling the deletion of a tree edge 2-Level approach A partition of E into two levels : (E 1, E 2 ) In the beginning, E 1 = E and E 1 = Ø F 1 : spanning forest of E 1 F 2 : spanning forest of E, F 1 is subset of F 2 Level 1 Level 2

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Handling the deletion of a tree edge Level 1 Level 2

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Handling the deletion of a tree edge Level 1 Level 2 Trivial algorithm at level 2 Random sampling at level 1

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

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Bounding Bounding µ 2 (number of non-tree edges at level 2) Level 1 Level 2 T2T1

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Analysis

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Transforming to fully dynamic environment

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Fully dynamic randomized algorithm for connectivity with polylogarithmic update time

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A Hierarchical algorithm log n c 2

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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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