Approximation Strategies for Generalized Binary Search in Weighted - PowerPoint PPT Presentation

Introduction Preliminaries Building a QPTAS O ( p log n ) -approximation algorithm Conclusion and Perspective Approximation Strategies for Generalized Binary Search in Weighted Trees Dariusz Dereniowski 1 , Adrian Kosowski 2 , Przemys ł aw Uzna´ nski 3 , Mengchuan Zou 2 [1]Gda´ nsk University of Technology, Poland [2]Inria Paris and IRIF, France [3]ETH Zürich, Switzerland ANR DESCARTES, Poitier Oct. 4th, 2017 ANR DESCARTES, Poitier Oct. 4th, 2017 1 / ANR DESCARTES 2017 57

Introduction Preliminaries Building a QPTAS O ( p log n ) -approximation algorithm Conclusion and Perspective Content Introduction 1 2 Preliminaries Building a QPTAS 3 p O ( log n ) -approximation algorithm 4 Conclusion and Perspective 5 ANR DESCARTES, Poitier Oct. 4th, 2017 2 / ANR DESCARTES 2017 57

Introduction Preliminaries Building a QPTAS O ( p log n ) -approximation algorithm Conclusion and Perspective Introduction General Configuration of Searching Problem A set of data organized in some structure ANR DESCARTES, Poitier Oct. 4th, 2017 3 / ANR DESCARTES 2017 57

Introduction Preliminaries Building a QPTAS O ( p log n ) -approximation algorithm Conclusion and Perspective Introduction General Configuration of Searching Problem A set of data organized in some structure An oracle replies to queries on the data ANR DESCARTES, Poitier Oct. 4th, 2017 4 / ANR DESCARTES 2017 57

Introduction Preliminaries Building a QPTAS O ( p log n ) -approximation algorithm Conclusion and Perspective Introduction General Configuration of Searching Problem A set of data organized in some structure An oracle replies to queries on the data The oracle returns a subset of the data set which contains the target element ANR DESCARTES, Poitier Oct. 4th, 2017 5 / ANR DESCARTES 2017 57

Introduction Preliminaries Building a QPTAS O ( p log n ) -approximation algorithm Conclusion and Perspective Generalized Binary Search in Trees Binary Search For an ordered array (or totally ordered set) ANR DESCARTES, Poitier Oct. 4th, 2017 6 / ANR DESCARTES 2017 57

Introduction Preliminaries Building a QPTAS O ( p log n ) -approximation algorithm Conclusion and Perspective Generalized Binary Search in Trees Binary Search For an ordered array (or totally ordered set) Our problem : Searching in Trees Data organized into a tree Target node x is known to the oracle, but not to the search algorithm The oracle returns the subtree in which the target lies ANR DESCARTES, Poitier Oct. 4th, 2017 11 / ANR DESCARTES 2017 57

Introduction Preliminaries Building a QPTAS O ( p log n ) -approximation algorithm Conclusion and Perspective Query Model for Trees – Query : a node v ANR DESCARTES, Poitier Oct. 4th, 2017 12 / ANR DESCARTES 2017 57

Introduction Preliminaries Building a QPTAS O ( p log n ) -approximation algorithm Conclusion and Perspective Query Model for Trees – Query : a node v – Reply : true , if v is the target ANR DESCARTES, Poitier Oct. 4th, 2017 12 / ANR DESCARTES 2017 57

Introduction Preliminaries Building a QPTAS O ( p log n ) -approximation algorithm Conclusion and Perspective Query Model for Trees – Query : a node v – Reply : true , if v is the target otherwise, return a neighbor u of v which is closer to the target x ANR DESCARTES, Poitier Oct. 4th, 2017 12 / ANR DESCARTES 2017 57

Introduction Preliminaries Building a QPTAS O ( p log n ) -approximation algorithm Conclusion and Perspective Example 1 Query e Target : f ANR DESCARTES, Poitier Oct. 4th, 2017 13 / ANR DESCARTES 2017 57

Introduction Preliminaries Building a QPTAS O ( p log n ) -approximation algorithm Conclusion and Perspective Example 1 Query e Target : f ANR DESCARTES, Poitier Oct. 4th, 2017 14 / ANR DESCARTES 2017 57

Introduction Preliminaries Building a QPTAS O ( p log n ) -approximation algorithm Conclusion and Perspective Example 1 Query c Target : f ANR DESCARTES, Poitier Oct. 4th, 2017 15 / ANR DESCARTES 2017 57

Introduction Preliminaries Building a QPTAS O ( p log n ) -approximation algorithm Conclusion and Perspective Example 1 Query c Target : f ANR DESCARTES, Poitier Oct. 4th, 2017 16 / ANR DESCARTES 2017 57

Introduction Preliminaries Building a QPTAS O ( p log n ) -approximation algorithm Conclusion and Perspective Example 1 Query g Target : f ANR DESCARTES, Poitier Oct. 4th, 2017 17 / ANR DESCARTES 2017 57

Introduction Preliminaries Building a QPTAS O ( p log n ) -approximation algorithm Conclusion and Perspective Example 1 Query g Target : f ANR DESCARTES, Poitier Oct. 4th, 2017 18 / ANR DESCARTES 2017 57

Introduction Preliminaries Building a QPTAS O ( p log n ) -approximation algorithm Conclusion and Perspective Example 1 Query f Target : f ANR DESCARTES, Poitier Oct. 4th, 2017 19 / ANR DESCARTES 2017 57

Introduction Preliminaries Building a QPTAS O ( p log n ) -approximation algorithm Conclusion and Perspective Example 1 Found Target : f ANR DESCARTES, Poitier Oct. 4th, 2017 20 / ANR DESCARTES 2017 57

Introduction Preliminaries Building a QPTAS O ( p log n ) -approximation algorithm Conclusion and Perspective Example 1 : cost of locating the target ANR DESCARTES, Poitier Oct. 4th, 2017 21 / ANR DESCARTES 2017 57

Introduction Preliminaries Building a QPTAS O ( p log n ) -approximation algorithm Conclusion and Perspective Example 1 : cost of locating the target ANR DESCARTES, Poitier Oct. 4th, 2017 22 / ANR DESCARTES 2017 57

Introduction Preliminaries Building a QPTAS O ( p log n ) -approximation algorithm Conclusion and Perspective General Graph Variation General Graph : Query u ANR DESCARTES, Poitier Oct. 4th, 2017 23 / ANR DESCARTES 2017 57

Introduction Preliminaries Building a QPTAS O ( p log n ) -approximation algorithm Conclusion and Perspective General Graph Variation General Graph : Query u Reply a v 2 N ( u ) , s.t. v is on the shortest path to the target ANR DESCARTES, Poitier Oct. 4th, 2017 24 / ANR DESCARTES 2017 57

Introduction Preliminaries Building a QPTAS O ( p log n ) -approximation algorithm Conclusion and Perspective Search Strategy Problem in Trees Setting Tree T = ( V , E , w ) with root r ( T ) Cost of query to vertex v : w : V ! R + , max v w ( v ) = 1 Cost of search strategy A on tree T : worst-case cost of finding a target Optimal strategy : search strategy with minimal cost on T , costs OPT ( T ) ANR DESCARTES, Poitier Oct. 4th, 2017 25 / ANR DESCARTES 2017 57

Introduction Preliminaries Building a QPTAS O ( p log n ) -approximation algorithm Conclusion and Perspective Search Strategy Problem in Trees Setting Tree T = ( V , E , w ) with root r ( T ) Cost of query to vertex v : w : V ! R + , max v w ( v ) = 1 Cost of search strategy A on tree T : worst-case cost of finding a target Optimal strategy : search strategy with minimal cost on T , costs OPT ( T ) Our Problem : Input : tree T = ( V , E , w ) Compute : Optimal strategy for generalized binary search query model ANR DESCARTES, Poitier Oct. 4th, 2017 25 / ANR DESCARTES 2017 57

Introduction Preliminaries Building a QPTAS O ( p log n ) -approximation algorithm Conclusion and Perspective Search Strategy Problem in Trees Setting Tree T = ( V , E , w ) with root r ( T ) Cost of query to vertex v : w : V ! R + , max v w ( v ) = 1 Cost of search strategy A on tree T : worst-case cost of finding a target Optimal strategy : search strategy with minimal cost on T , costs OPT ( T ) Our Problem : Input : tree T = ( V , E , w ) Compute : Optimal strategy for generalized binary search query model Application Aspects Locating buggy nodes in network models Finding specific data in organized databases ANR DESCARTES, Poitier Oct. 4th, 2017 25 / ANR DESCARTES 2017 57

Approximation Strategies for Generalized Binary Search in Weighted - PowerPoint PPT Presentation

Introduction Preliminaries Building a QPTAS O ( p log n ) -approximation algorithm Conclusion and Perspective Approximation Strategies for Generalized Binary Search in Weighted Trees Dariusz Dereniowski 1 , Adrian Kosowski 2 , Przemys aw

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