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# lec11 - 6.006 Introduction to Algorithms to Algorithms...

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006 troduction Algorithms 6.006- Introduction to Algorithms Lecture 11 - Searching I Prof. Manolis Kellis CLRS 22.1-22.3, B.4

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Unit #4 – Games, Graphs, Searching, Networks 2
Unit #4 Overview: Searching Today: Introduction to Games and Graphs Rubik’s cube, Pocket cube, Game space Graph definitions, representation, searching Tuesday: Graph algorithms and analysis readth First Search Depth First Search Breadth First Search, Depth First Search Queues, Stacks, Augmentation, Topological sort Thursday: Networks in biology and real world Network/node properties, metrics, motifs, clusters Dynamic processes, epidemics, growth, resilience

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Graph Applications eb Web – crawling ocial Network Social Network – friend finder omputer Networks Computer Networks – internet routing – connectivity Game states – rubik’s cube, chess
oday: Solving Rubik’s cube Today: Solving Rubik s cube… youtube: 5inASBBYpWU … and finding God’s number

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Cracking the 3x3 Rubik’s cube Increasingly efficient algorithms exist for solving the cube using a fixed set of moves – 1981: 52 moves. Today: <30 moves In practice, shortcuts may be possible! – Human intuition can reveal patterns, not follow fixed algorithm How hard is Rubik’s cube: – Size of game space: count distinct positions, number of edges – 43,252,003,274,489,856,000 positions (4.3*10 19 ) How big is 43 quadrillion? – Number of atoms in the universe: 10 81 7 – Complexity of chess (Shannon number): ~10 47 – 19x19 go: #turns ~10 48 ; 10 10^48 <#games<~10 10^171
Searching for God’s number Date Lower bound Upper bound Gap God’s algorithm would always use the minimal July, 1981 18 52 34 April, 1992 18 42 24 May, 1992 18 39 21 number of moves God’s number : maximum umber of moves needed May, 1992 18 37 19 January, 1995 18 29 11 nuary 1995 0 9 number of moves needed by an optimal algorithm pper bound earing in January, 1995 20 29 9 December, 2005 20 28 8 April, 2006 20 27 7 Upper bound nearing in by increasingly faster general algorithms May, 2007 20 26 6 March, 2008 20 25 5 April, 2008 20 23 3 Lower bound given by hardest known positions quiring most moves August, 2008 20 22 2 July, 2010 20 20 0 requiring most moves The two met just last year!

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So, how did they do it? ( ) Start with 43,252,003,274,489,856,000 positions artition into 2 billion sets each with Partition into 2.2 billion sets , each with 19.5 billion positions , solve each separately educe 2 billion sets 5 8 million y Reduce 2.2 billion sets to 55.8 million by symmetry & set cover olve each set of 9 5 billion positions Solve each set of 19.5 billion positions
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lec11 - 6.006 Introduction to Algorithms to Algorithms...

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