Path Minima on Dynamic Weighted Trees Pooya Davoodi Aarhus University Aarhus University, November 17, 2010 Joint work with Gerth Stølting Brodal and S.

Slides:

Advertisements

Similar presentations

Turning Amortized to Worst-Case Data Structures: Techniques and Results Tsichlas Kostas.

Advertisements

Differential Forms for Target Tracking and Aggregate Queries in Distributed Networks Rik Sarkar Jie Gao Stony Brook University 1.

Every edge is in a red ellipse (the bags). The bags are connected in a tree. The bags an original vertex is part of are connected.

Gerth Stølting Brodal University of Aarhus Monday June 9, 2008, IT University of Copenhagen, Denmark International PhD School in Algorithms for Advanced.

Dynamic Graph Algorithms - I

1 Lecture 12 AVL Trees. 2 trees static dynamic game treessearch trees priority queues and heaps graphs binary search trees AVL trees 2-3 treestries Huffman.

Tight Bounds for Dynamic Convex Hull Queries (Again) Erik DemaineMihai Pătraşcu.

An Optimal Dynamic Interval Stabbing-Max Data Structure? Pankaj K. Agarwal, Lars Arge and Ke Yi Department of Computer Science Duke University.

Perfect Matching for Biconnected Cubic Graphs in O(nlog 2 n) Time Krzysztof Diks & Piotr Stańczyk.

1 Self-Adjusting Data Structures. 2 Lists [D.D. Sleator, R.E. Tarjan, Amortized Efficiency of List Update Rules, Proc. 16 th Annual ACM Symposium on Theory.

Unified Access Bound 1 [M. B ă doiu, R. Cole, E.D. Demaine, J. Iacono, A unified access bound on comparison- based dynamic dictionaries, Theoretical Computer.

Data Structures: Range Queries - Space Efficiency Pooya Davoodi Aarhus University PhD Defense July 4, 2011.

Time-Space Trade-Offs for 2D Range Minimum Queries

Dynamic Planar Range Maxima Queries (presented at ICALP 2011) Gerth Stølting Brodal Aarhus University Kostas Tsakalidis University of Primorska, October.

Update Persistent Data Structures (Version Control) v0v0 v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 Ephemeral query v0v0 v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 Partial persistence.

Update 1 Persistent Data Structures (Version Control) v0v0 v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 Ephemeral query v0v0 v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 Partial persistence.

Purely Functional Worst Case Constant Time Catenable Sorted Lists Gerth Stølting Brodal University of Aarhus Joint work with Christos Makris Kostas Tsichlas.

Balanced Graph Partitioning Konstantin Andreev Harald Räcke.

1 Finding Dominators in Flowgraphs Linear-Time Algorithm 1 and Experimental Study 2 Loukas Georgiadis 1 joint work with Robert E. Tarjan 2 joint work with.

Recent Development on Elimination Ordering Group 1.

1 On Dynamic Shortest Paths Problems Liam Roditty Uri Zwick Tel Aviv University ESA 2004.

Greedy Algorithms Reading Material: Chapter 8 (Except Section 8.5)

Fully Persistent B-Trees 23 rd Annual ACM-SIAM Symposium on Discrete Algorithms, Kyoto, Japan, January 18, 2012 Gerth Stølting Brodal Konstantinos Tsakalidis.

Comparison Based Dictionaries: Fault Tolerance versus I/O Efficiency Gerth Stølting Brodal Allan Grønlund Jørgensen Thomas Mølhave University of Aarhus.

1 Splay trees (Sleator, Tarjan 1983). 2 Motivation Assume you know the frequencies p 1, p 2, …. What is the best static tree ? You can find it in O(nlog(n))

Greedy Algorithms Like dynamic programming algorithms, greedy algorithms are usually designed to solve optimization problems Unlike dynamic programming.

Mergeable Trees1 Mergeable Trees Mergeable Trees Robert E. Tarjan Princeton University and HP Labs Joint work with Loukas Georgiadis, Haim Kaplan, Nira.

A General Approach to Online Network Optimization Problems Seffi Naor Computer Science Dept. Technion Haifa, Israel Joint work: Noga Alon, Yossi Azar,

Algorithm Engineering, September 2013Data Structures, February-March 2010Data Structures, February-March 2006 Cache-Oblivious and Cache-Aware Algorithms,

The Encoding Complexity of Two Dimensional Range Minimum Data Structures MADALGO, October 2, 2013 (Work presented at ESA 2013) 1234  n

The Encoding Complexity of Two Dimensional Range Minimum Data Structures European Symposium on Algorithms, Inria, Sophia Antipolis, France, September 3,

Multiple-Source Shortest Paths in Planar Graphs Allowing Negative Lengths Philip Klein Brown University.

Strict Fibonacci Heaps Gerth Stølting Brodal Aarhus University George Lagogiannis Robert Endre Tarjan Agricultural University of Athens Princeton University.

Algorithms and Data Structures Gerth Stølting Brodal Computer Science Day, Department of Computer Science, Aarhus University, May 25, 2012.

2IL05 Data Structures Fall 2007 Lecture 13: Minimum Spanning Trees.

Spring 2015 Lecture 11: Minimum Spanning Trees

An Approximation Algorithm for Binary Searching in Trees Marco Molinaro Carnegie Mellon University joint work with Eduardo Laber (PUC-Rio)

Wolfgang Mulzer Institut f ür Informatik Data Structures on Event Graphs Bernard ChazelleWolfgang Mulzer FU Berlin Princeton University.

1 Splay trees (Sleator, Tarjan 1983). 2 Goal Support the same operations as previous search trees.

Succinct Dynamic Cardinal Trees with Constant Time Operations for Small Alphabet Pooya Davoodi Aarhus University May 24, 2011 S. Srinivasa Rao Seoul National.

Algorithms and Data Structures Gerth Stølting Brodal Computer Science Day, Department of Computer Science, Aarhus University, May 25, 2012.

I/O-Efficient Batched Union-Find and Its Applications to Terrain Analysis Pankaj K. Agarwal, Lars Arge, Ke Yi Duke University University of Aarhus.

Λίστα Εργασιών Data Structures for Tree Manipulation D. Harel and R.E. Tarjan. Fast Algorithms for finding nearest common ancestors. SIAM J. Computing,

Université de Nantes Dynamic Trees and Log-lists Irena Rusu Université de Nantes, LINA

Union-Find  Application in Kruskal’s Algorithm  Optimizing Union and Find Methods.

Gerth stølting Brodal 1 Closing a Classical Data Structure Problem Gerth Stølting Brodal : Strict Fibonacci Heaps joint work with George Lagogiannis Robert.

Equivalence Between Priority Queues and Sorting in External Memory

Splay Trees and the Interleave Bound Brendan Lucier March 15, 2005 Summary of “Dynamic Optimality -- Almost” by Demaine et. Al., 2004.

Minimum Bottleneck Spanning Trees (MBST)

Vasilis Syrgkanis Cornell University

Introduction Terrain Level set and Contour tree Problem Maintaining the contour tree of a terrain under the following operation: ChangeHeight(v, r) : Change.

Shortest Paths in Decremental, Distributed and Streaming Settings 1 Danupon Nanongkai KTH Royal Institute of Technology BIRS, Banff, March 2015.

Outline Standard 2-way minimum graph cut problem. Applications to problems in computer vision Classical algorithms from the theory literature A new algorithm.

1 CPSC 320: Intermediate Algorithm Design and Analysis July 14, 2014.

Some links between min-cuts, optimal spanning forests and watersheds Cédric Allène, Jean-Yves Audibert, Michel Couprie, Jean Cousty & Renaud Keriven ENPC.

Self-Adjusting Data Structures

Minimum Spanning Tree Chapter 13.6.

Improved Randomized Algorithms for Path Problems in Graphs

Persistent Data Structures (Version Control)

Searching in Trees Gerth Stølting Brodal Aarhus University

Minimum Spanning Tree.

Lecture 19-Problem Solving 4 Incremental Method

CS 583 Analysis of Algorithms

Closing a Classical Data Structure Problem Gerth Stølting Brodal

Strict Fibonacci Heaps

Splay trees (Sleator, Tarjan 1983)

More Trees B.Ramamurthy 4/6/2019 BR.

Dynamic Graph Algorithms

Minimum Spanning Trees (MSTs)

Lecture 10 Graph Algorithms

Presentation transcript:

Path Minima on Dynamic Weighted Trees Pooya Davoodi Aarhus University Aarhus University, November 17, 2010 Joint work with Gerth Stølting Brodal and S. Srinivasa Rao

Path Minima Problem Definition Forest of unrooted trees Operations: make-tree, path-minima, weight-update, link, cut f b c e a g d i make-tree(i) link(g,b,2) path-minima(d,f) cut(e,g) (g,b) weight-update(b,c,1) 1 path-minima: bottleneck edge query (beq) h 10 2 Applications: Network Flows, Minimum Spanning Trees, Transportation Problem, Network Optimization Algorithms

Computational Models 3

Outline 4 Path Minima Problem make-tree, beq, update, link, cut Dynamic Trees of Sleator and Tarjan (STOC’81) Dynamic Trees is Optimal Patrascu and Demaine (STOC’04) Lower Bounds The Problem is Open New Reductions

Dynamic Trees (Link-Cut Trees) Sleator and Tarjan (STOC’81) Arbitrary roots with operation evert (more operations: parent, root, LCA) Vertex-disjoint path decomposition Each path represented by a biased search tree or a splay tree Operations in O(log n) Model: Semigroup by J. Erickson, C. Osborn 5

Dynamic Trees is Optimal Fully Dynamic Connectivity 6 u v

Lower Bounds Connectivity 7 uv r w Patrascu and Demaine (STOC’04) (Cell Probe)

Lower Bounds Incremental Connectivity 8 Kaplan et. al. (STOC'02)

Lower Bounds 1D-RMQ Just a Path with no link & cut Brodal et. al.(SWAT'96) reduction from Insert-Delete-FindMin in (Comparison) Alstrup et. al.(FOCS'98): reduction from Priority Search Trees (Cell Probe) Patrascu and Demaine (SODA'04): reduction from Dynamic Partial Sums (Semigroup) 9

Path Minima Open Problems 10 (RAM model) Conjecture of Patrascu and Thorup (STOC’06) (Comparison and RAM models)

Variants OperationsPreprocessingPath MinimaUpdatelink & cutComments beq, update & link no results beq & link no results Semigroup & Comparisons -RAM, Kaplan et al. (ESA’08) beq Semigroup & Comparison, Chazelle (FOCS’84) Alon & Shieber (TecRep’87) Pettie (FOCS’02) RAM, Kaplan et al. (ESA’08) beq & update Comparison – New RAM - New beq, leaf-link & leaf-cut Semigroup – New RAM, Kaplan et al. (ESA’08) 11

Static Trees with Dynamic Weights 12 Transformation: add O(m) edges make it rooted Path Minima on

Static Trees with Dynamic Weights 13 Path Minima on u v cont.

Leaf-Link-Cut Trees with Static Weights 14 make it rooted Topological Partitioning Recursion link: Split & Update cut: Global Rebuilding Path Minima on

Path Minima Open Problems 15 (RAM model) Conjecture of Patrascu and Thorup (STOC’06) (Comparison and RAM models)

Variants OperationsPreprocessingPath MinimaUpdatelink & cutComments beq, update & link no results beq & link no results Semigroup & Comparisons -RAM, Kaplan et al. (ESA’08) beq Semigroup & Comparison, Chazelle (FOCS’84) Alon & Shieber (TecRep’87) Pettie (FOCS’02) RAM, Kaplan et al. (ESA’08) beq & update Comparison – New RAM - New beq, leaf-link & leaf-cut Semigroup – New RAM, Kaplan et al. (ESA’08) 16

17 THANK YOU