Dynamic Graph Algorithms - I Surender Baswana Department of CSE, IIT Kanpur.

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

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

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.

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)

A motivating example : Undirected Connectivity Static solution: v a b c d

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]

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]

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

Examples of fully dynamic algorithms

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]

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]

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

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, 2011 ] 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]

Data structure for dynamic trees

Data structures Stacks Queues Binary heap Binary search tree ... Fibinaacci heap Too elementary Limited applications Too complex Hardly any application !

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.

Balanced Binary Tree : a very powerful data structure : Additional information

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

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

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)

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

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

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

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

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

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

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

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

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

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

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

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

Fully dynamic algorithm for connectivity

Fully dynamic randomized algorithm for connectivity with polylogarithmic update time

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

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

Fully dynamic randomized algorithm for connectivity with polylogarithmic update time A Hierarchical algorithm 1 2 3 2 log n

Fully dynamic randomized algorithm for connectivity with O( 𝒎 ) update time A 2-level algorithm 2 1

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

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

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

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

Handling the deletion of a tree edge Level 2 Level 1

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

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

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

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 )

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)

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

Fully dynamic randomized algorithm for connectivity with polylogarithmic update time A Hierarchical algorithm 2 log n 2 2 c 1

Open Problems

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 ?