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Algorithms and Data Review Fall 2010

Algorithms and Data Review Fall 2010. Karl Lieberherr. Problem Solving. Polya: How to Solve It. Algorithmic Patterns. Greedy Divide and Conquer Dynamic Programming Network Flow Intractability: Reductions and characterization of NP. Algorithmic thinking. Making claims about algorithms

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Algorithms and Data Review Fall 2010

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  1. Algorithms and Data ReviewFall 2010 Karl Lieberherr CS 4800 Fall 2010

  2. Problem Solving • Polya: How to Solve It CS 4800 Fall 2010

  3. Algorithmic Patterns • Greedy • Divide and Conquer • Dynamic Programming • Network Flow • Intractability: Reductions and characterization of NP CS 4800 Fall 2010

  4. Algorithmic thinking • Making claims about algorithms • Choosing from a predefined set of claims • claims that are refuted • claims that are defended • Used: Scientific Community Game for Algorithmic domains • make learning interactive and second order • students who are better at the game are better at algorithms CS 4800 Fall 2010

  5. Summary of SCG Definitions Domain Problem Solution valid(Problem, Solution) quality(Problem, Solution) →[0,1] Claim(Domain) Problems: PowerSet(Domain.Problem) q: Quality = [0,1] r: Resource = N+ Rules of the Scientific Community: propose and oppose, be an active scholar, rules for reputation accumulation. Tournaments CS 4800 Fall 2010

  6. Formal: HSR • Domain: • Problem: (n,k), k <= n. • Solution: Decision tree to determine highest safe rung. • quality(problem, solution): depth of decision tree / number of rungs • valid(problem, solution): at most k left branches, ... CS 4800 Fall 2010

  7. Formal: HSR • Claim(Domain): • Alice claims ({(25,2)},9/25,5 seconds) • {(25,2)}: set of problems (singleton) • 9/25: quality • 5 seconds: resource • Refutation Protocol: • Bob refutes: only one problem: (25,2) • Alice: solves problem by providing decision tree t. • predicate: t is a valid decision tree for (25,2) of depth 9 CS 4800 Fall 2010

  8. Protocol variationsExample: Independent Set • Domain • Problem: Directed graph G=(V,E) • Solution: V’ subset of V • Claim • Problems: all directed graphs with < 10^6 nodes • quality: |V’|/|V| • Protocol • Alice to Admin: G, AlgA(G) • Admin to Bob: G • Bob to Admin: AlgB(G) • Admin to both: AlgB(G), r[G,AlgB(G),AlgA(G)] Alice claims that her algorithm finds a solution at least as good as Bob’s. CS 4800 Fall 2010

  9. Protocol variationsExample: Knapsack • Domain • Problem: Knapsack problem K(n,Items(Weights,Values),W) • Solution: subset of Items • Claim • Problems: all knapsack problems with W < 10^3 • quality: |V’|/|V| • Protocol • Alice to Admin: K, AlgA(K) • Admin to Bob: K • Bob to Admin: AlgB(K) including AlgB(K).resource • Admin to both: AlgB(K) including AlgB(K).resource, r[K,AlgB(K),AlgA(K)] Alice claims that her algorithm finds a solution at least as good as Bob’s and using fewer resources. CS 4800 Fall 2010

  10. Languages for expressing algorithms • Decision trees • Finite state machines • Pseudo code • Programming languages CS 4800 Fall 2010

  11. Prove properties about algorithm • Find an invariant of the algorithm that helps to prove what you want. • Example: Ford-Fulkerson • Dynamic Programming: formulate recurrence for optimal solution in terms of optimal solutions for sub problems. • Formulate algorithm so that it is easier to prove its correctness. CS 4800 Fall 2010

  12. Intractability • Problems that don’t have an algorithmic solution. E.g. is grammar ambiguous? • Problems that don’t have an efficient algorithmic solution, e.g., NP-complete problems. • Reduction: Y <= X : Y is reducible to X, Y reduces to X: an algorithm for X implies an algorithm for Y • Polynomial Reduction: Y <=p X: a polynomial algorithm for X implies a polynomial algorithm for Y. CS 4800 Fall 2010

  13. NP-complete Problem X • X in NP • for all Y in NP, Y <=p X. Every problem in NP can be reduced to X. Circuit Satisfiability is NP-complete. CS 4800 Fall 2010

  14. More NP-complete problems • If Y is NP-complete and X is in NP with the property that Y <=p X, then X is NP-complete. CS 4800 Fall 2010

  15. Strategy for proving problem X to be NP-complete • Prove that X in NP. • Choose a problem Y that is NP-complete. • Prove that Y <=p X. • Cook and Karp (translation) reductions (page 473) • Example of Cook reduction: search (FSAT) reduces to decision (SAT) CS 4800 Fall 2010

  16. Interview question • Given a set S and a predicate pred defined on subsets of S, find a subset ss of S satisfying pred. • Example pred: false on all subsets not having 3 elements. For 3 element subsets pred({a,b,c}) iff a+b+c=0. CS 4800 Fall 2010

  17. Order Statistics • Given a sequence of n numbers, find the kth largest element. • O(n*log(n)) • O(n*log(k)) • O(n): median of the median CS 4800 Fall 2010

  18. Expected number of satisfied constraints under random assignment • Maximum Satisfiability • Maximum Boolean CSP CS 4800 Fall 2010

  19. Problem Snapshot • Boolean CSP: constraint satisfaction problem • Each constraint uses a Boolean relation. • e.g. a Boolean relation 1in3(x y z) is satisfied iff exactly one of its parameters is true. • Boolean MAX-CSPa multi-set of constraints. Maximize satisfied fraction. CS 4800 Fall 2010

  20. 3p(1-p)2 for MAX-CSP({22}) CS 4800 Fall 2010

  21. all the look-ahead polynomials for T Ball CS 4800 Fall 2010

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