Ferdinando Cicalese
ABSTRACT
This paper focuses on competitive function evaluation in the context of computing with priced information. A function f is given together with a cost cx for each variable x of f. The cost cx has to be paid to read the value of x. The problem is to design algorithms that query the values of the variables sequentially in order to compute the function while trying to minimize the total cost incurred. Competitive analysis is employed to evaluate the performance of the algorithms. We describe a novel approach for devising efficient algorithms in this setting. We apply our approach to several classes of functions which have been studied in the literature of computing with priced information. In all cases considered, our approach provides algorithms that achieve better bounds than the best known algorithm for the same class of functions.More precisely, for the class of monotone boolean functions, we give a polynomial time algorithm with extremal competitiveness (k+l - √ min(k,l)) where k (l) denotes the minimum number of variables that one must read, in the worst case, in order to prove that the function under consideration evaluates to 1 (0). This dramatically improves upon the best known result which is an exponential time 2 max(k, l)-competitive algorithm. For the subclass of monotone boolean functions known as Threshold Trees we further improve our bounds and give a polynomial time algorithm with extremal competitive ratio 1.618 max(k, l).We then apply our methodology to classes of non-boolean functions. We consider the case of the so called Game Trees. We improve upon previously published results for this class of functions providing a polynomial time algorithm with extremal competitive ratio 1.5 γ(f), where γ(f) is a lower bound on the extremal competitive ratio of any deterministic algorithm.Finally, we consider the case when f is the function min (minimum). In this case, we are able to determine the optimal competitiveness for the problem. In fact we provide an algorithm with an (n-2)-competitive ratio, which matches the known lower bound.
- R. Carmo, T. Feder, Y. Kohayakawa, E. Laber, R. Motwani, L. O'Callaghan, R. Panigrahy, and D. Thomas. Querying priced information in databases: the conjunctive case. 2004. submitted.Google Scholar
- M. Charikar, R. Fagin, V. Guruswami, J. Kleinberg, P. Raghavan, and A. Sahai. Query strategies for priced information. JCSS: Journal of Computer and System Sciences, 64:785--819, 2002. Google ScholarDigital Library
- A. Gupta and A. Kumar. Sorting and selection with structured costs. In Proc. of IEEE FOCS 2001, pages 416--425, 2001. Google ScholarDigital Library
- A. Gupta, D. O. Stahl, and A. B. Whinston. Pricing of services on the internet. In IMPACT: How IC2 Research Affects Public Policy and Business Markets. Quorum Books, forthcoming.Google Scholar
- V. Guruswami and E. Laber. Email exchanges. 2003.Google Scholar
- J. Hartline, E. Hong, A. Mohr, E. Rocke, and K. Yasuhara. As reported in {2}. 2001.Google Scholar
- S. Kannan and S. Khanna. Selection with monotone comparison costs. In Proc. of ACM-SIAM SODA-03, pages 10--17, 2003. Google ScholarDigital Library
- S. Khanna and W. Tan. On computing functions with uncertainty. In Proc. of PODS 2001, pages 171--182, 2001. Google ScholarDigital Library
- E. Laber. A randomized competitive algorithm for evaluating priced AND/OR trees. In Proc. of STACS 2004, pages 501--512, 2004.Google ScholarCross Ref
- E. Laber, O. Parekh, and R. Ravi. Randomized approximation algorithms for query optimization problems on two processors. In Proc. of ESA 2002, pages 649--661, 2002. Google ScholarDigital Library
- M. Saks and A. Wigderson. Probabilistic Boolean decision trees and the complexity of evaluating game trees. In Proc. of IEEE FOCS 1986, pages 29--38, Toronto, Ontario, Canada, 27--29 Oct. 1986. IEEE.Google ScholarDigital Library
- M. Tarsi. Optimal search on some game trees. Journal of the ACM, 30(3):389--396, July 1983. Google ScholarDigital Library
Index Terms
- A new strategy for querying priced information
Recommendations
On the competitive ratio of evaluating priced functions
Let f be a function on a set of variables V. For each x ∈ V, let c(x) be the cost of reading the value of x. An algorithm for evaluating f is a strategy for adaptively identifying and reading a set of variables U ⊆ V whose values uniquely determine the ...
Online algorithms for 1-space bounded 2-dimensional bin packing and square packing
In this paper, we study 1-space bounded 2-dimensional bin packing and square packing. A sequence of rectangular items (square items) arrive one by one, each item must be packed into a square bin of unit size on its arrival without any information about ...
Online Results for Black and White Bin Packing
In online bin packing problems, items of sizes in [0, 1] are to be partitioned into subsets of total size at most 1, called bins. We introduce a new variant where items are of two types, called black and white, and the item types must alternate in each ...
Comments