On a Problem of Optimal Search

Transcription

1 Zeitschrift for Operations Research, Band , Seite Physico-Verlag, Wiirzburg. On a Problem of Optimal Search By H. W. Gottinger, Bielefeld 1 ) Eingegangen am 26. November 1975 Revidierte Fassung eingegangen am 4. Mai 1976 Summary: A class of one-dimensional search problems is considered. The formulation results in a functional-minimization equation of the dynamic programming type. In the case of a uniform a priori distribution for the location of the hidden object an optimality criterion is established and the optimal search procedwe is found. Zusammen[ossung: Nach allgemeinen Betrachtungen iiber das Spektrum von Suchproblemen wird eine Klasse eindimensionaler Suchprobleme betrachtet. Die Formulierung fiihrt zu einer Funktio nalgleichung vom Typ der dynamischen Optimierung. In dem Fall einer uniformen a priori Vertei lung fur das Auffmden eines versteckten Objekts wird ein Optimalitatskriterium begriindet und das optimate Suchverfahren angewandt. 1. Introduction The standard search problem, treated statistically (see de Groot [ 1970)),is as follows: Suppose that an object is hidden in one of r possible locations (r;;;, 2) and let P; be the prior probability that the object is in location i. The searcher must find the object but he can search in one location at a time. Hence he must devise a sequential search procedure which specifies at each stage that a certain one of the r locations is to be searched. Such optimization problems are also of eminent interest in operations research, for general references see Dobbie [ 1968), Pollock [1971) or a recent book by Stone [1975]. An important subclass of such problems, restricted to onedimensional search will be considered in this paper. It involves a functional minimization technique of the dynamic programming type. It has been pointed out by a referee of this journal that a continuous analogue of this problem has been treated earlier by Murakami [1971 ). -. The problem treated in this paper has some interesting ramifications. One appealing nt:rpretation (due to Bolle, Hamburg) is to conceive a sequential weighing scheme in Which a selection among genuine and false coins will be made. Given some prior prob- 1. ) Prof. Dr. Hans w. Gottinger, Institut ftir Mathematische Wirtschaftsforschung der Universitiit Bielefeld. Postfach Bielefeld.

2 224 H. W. Gottinger ability on the weights of the genuine (or false) coins the problem is to minimize the number of weighings in order to detect the correct coin over the search period. Bolle has also pointed out that a similar problem in game-theoretic formulation has been treated by Johnson (1964]. Another aspect of the problem worth mentioning is the following. Theoretically, a person can search as long as he wants to provided his expected gain at each stage exceeds his expected cost at that stage. However, for a finite number of objects to be searched and for a fixed time period available for search.there is an absolute upper bound for search which constitutes itself a forced search rule. This upper bound, in the present problem, is roughly KN log N where N is the number of items to be detected and K is a constant independent of N but somewhat dependent on the combinatorial structure of the problem. Under these circumstances the search problem considered here comes very close to the sorting problem as treated by Mo"is [ 1969]. 2. The Problem and the Optimization Equation We consider a storage unit of an information system consisting of N cells, with in fonnation stored in tabular fonn. That is, the record r (r} stored in cell i is in the form of a pair [x;, lfj (x;)], the file being arranged in strictly ascending order of the argument x 1 An example of such an arrangement is a dictionary. Given a particular argument x, we find lfj (x) by searching for the cell containing (x, I{J (x)]. The search proceeds by comparing x against the arguments in a sequence of cells i,, i2,... This sequence is to be chosen so as to minimize the average number of comparisons required for locating the correct cell, given an a priori distribution for the location of the correct cell. In other words, for a given search cost associated to each comparison we want to minimize the expected cost of search over the sequence under consideration. We begin with the following assumptions. (i) In a comparison of x against x;, only three possible outcomes exist, namely, (ii) Let X be an integer-valued random variable denoting the location of x. We assume that the prior probabilities Pk = Prob (X= k) are given, with N ~ pk = 1. k=l Furthermore lets be the set of integers 1 through N, and lets' be a non-empty subset of S We assume that the posterior probability distribution of X is unchanged except for renormalization; i.e. Pk Prob (X= k I XES']=--, kes' (2) P(S') (1) where P (S') = :t p.. ies' I = 0, kfl.s'

3 On a Problem of Optimal Search 225 LetT ((pk),n] formally denote the minimum average number of comparisons per successful search, given N cells and prior distribution (pk).it is clear that the search procedure starts with the selection of a cell for the first comparison. Suppose cell n is selected and xis compared with Xn. The following situation then results: a) With probability Pn,x = Xn and the search terminates. n-l b) With probability Pn l = l: p., x < x and x must be contained in the first n- I /=1 ' n cells. If we renumber the first n- 1 cells backwards starting with cell n- 1, the new distribution becomes, Pn-k Pk = p, k = 1,..., n - 1. n-1 (3) N c) With probability 1 - P = :E p 1, x > xn. Upon renumbering the lastn- n n i=n+l cells, we find the new distribution to be, Pn+k pk= l-p,k=l,...,n-n. n (4) It is clear that whichever cell is optimal for the first choice, succeeding choices must remain optimal for the overall sequence to be optimal. Therefore, T (.,N) must satisfy the following functional equation: T[(pk},N) =I~~ <:N {I +Pn-! T [ (;:::),n-1]+ (1-Pn} (S) r[(:~~ ).N-n]}, NH Equation ( 5) is in the formalism of dynamic programming, Bellman [ 195 7], yielding as solutions the objective T ((pk),n] and the optimal policy n* [(pk),n] 2 ). As initial conditions we set p 0 = o, T (., O) = o, and T (., 1) = 0. (Note that this last condition is implied by the fact that if there is only one cell no comparision is necessary. This is a consequence of a).) 2 )The proof of equation (5) except for simple (deterministic) models is not strajght.forward and involves in the general c:ase (i.e., for models with uncountable state space but discrete transi tion law) extensive measure-theoretic tools. A rigorous proof for the general c:ase has been given by Hinderer [1970], chapt.ll.

4 226 H. W. Gottinger 3. Optimal Solutions for Uniform Distribution If pk = ~, k = 1, 2,...,N, explicit solution of (5) can be found. In this case, it is clear that T (.,. ) and n* (.,. ) are functions of N only. With a slight change in nota tion we can rewrite (5) as The solution T (N) of ( 6) is given by (2k+l +2m- I) T(2k+l+ 2m- I)= 2k+l (k- t) + 2mk +3m+ 1, k = 0, 1, 2,... k m = 1, 2,..., 2 (7} (2k+l +2m} T(2k+l +2m)= 2k+l (k- ~)+(2m+ 1) k+3(m+ 1), m = 1, 2,..., 2k :.._ 1. The policy n* (N) which yields the minimum is not unique. In fact, the multiplicity of solutions can be quite large. The complete set of solutions is n* (2k m) = 2k + j, j = 0, 1,..., 2m+ 1, m < 2k l ,..k + 1 2k ~ 2k m ~:,...,, m :P n*(2k+l +2m-1)=2k +2j,j=0,1,...,m, m~2k l (8). = - 2k l 2k l > 2k-l. 1 m,...,,m For example, consider N = = 25. Then n (N) = 10, 12, 14, 16. The policy solution is interesting and somewhat surprising. Intuitively, one would expect that the optimal solution n (N) should be such as to divide the remaining N- 1 cells into nearly equal subsets, i.e.,n - n :! n*- 1. Thus, the large multiplic ity of solution is not expected. Furthermore, in some cases the midpoint is in fact not a solution. For example, for N = 25, the point n = 13 divides the remaining 22 cells equally, but is not among the solutions.

5 On a Problem of Optimal Search Proof of Optimality In this section we shall prove that the solutions of (6) are indeed given by (7) and (8). The proof proceeds in three stages. First, it is shown that the right-hand side of (6) is minimized by a specific choice of policy n* (N). Next, T (N) will be derived. Finally, the multiplicity of the policy solution is found. A. If we let f (N) =NT (N), ( 6) is simplified and can be rewritten as min /(N) =N + 1 <. n<.n {f(n- 1) + f(n- n)}. (9) We begin by proving the following theorem: Theorem 1: Under the conditions f (0) = f (I) = 0, the minimization in (9) is achieved with n = n* (N), where fqr all positive integers m, n* (4m - 2) = n* (4m- 1) = n* (4m) = n* (4m + 1) =2m. (10) Proof: It is seen that Theorem I is equivalent to the following set of equations with m ranging over all positive integers: /(4m-2) = 4m f (2m - 1) +/{2m - 2) (1 la) /(4m- I) = 4m f (2m - I) +/(2m - 1) (lib) /(4m) =4m +/(2m- 1) +/(2m) (II c) /(4m + 1) = 4m f(2m- 1) +/(2m+ 1). (lid) We proceed by induction. First, by enumerating all possibilities, we find that ( 11) is true form= 1. Next, we assume (11) to be true form= 1,..., K, and prove the following lemma: Lemma: Equation (11) being valid form= 1, 2,...,K, implies!(n+ 1) - /('I+ I) > f(n), f(n)>/(n-1)-/(n-2), n= 2,...,4K n = 1, 2,...,4K (12) (13) /(2n) - f(2n- 1) > f(2n + I)-f(2n), n = 1, 2,..., 2K (14) Proof: If ( 11) is true for m = 1,..., K, then [/(4m +I)-/(4m)]- [f(4m -1)-/(4m - 2)] = U'(2m +I)+ f(2m- 2)]- rf(2m) + f(2m -1)], m = 1, 2,...,K. (15)

6 228 H. W. Gottinger Now under the same assumption, [compare (11c) and (9)], /(2m)+/(2m -1) = min {f(n -1) + f(4m- n)} 1 <; m <.K. (16) I~n~4m Therefore, it follows that /(2m+ I)+ /(2m--2);;;, /(2m)+ /(2m- 1); 1 ~ m ~ K (17) and f(4m + 1)-{(4m)~ f(4m -1)-f(4m- 2), 1 ~ m <.K. (18) Similarly, we fmd that f(4m- 1)-f(4m- 2) ={(4m-3)-f(4m- 4) f(4m)-{(4m- 1);;:, f(4m- 2)-f(4m- 3) /(4m-2)-{(4m-3)=/(4m-4)-/(4m-5), m~k. (19) (20) (21) Relationships (18)- (21) imply (12), and together with the fact that /{2)-f(l) > 0 and {(3)-/{2) > 0 imply (13). Now, if ( 11) is valid for m = 1,..., K, then implies and /(2m)-/(2m- I)> /(2m+ 1)-f(2m) f(4m- 2)-f(4m- 3) > f(4m -1)-f(4m- 2) f(4m)- f(4m- 1) > f(4m + 1)-f(4m), for m = 1, 2,...,K. Therefore, (14) is implied by {(2)-/(1) > {(3)- /(2). This latter is easily verified. Now, we proceed with the main part of the proof for Theorem 1. First we write f(4k + 2) as f(4k + 2) = 4K <;n~~k + l {f(n-1) + f(4k + 2-n)} = 4K + 2 +min { I.:;: <:;K [f(2n- 1) + f(4k + 2-2n)), (22) min } 1 <; n ~ k (f (2n) + f ( 4K + 1-2n) J.

7 On a Problem of Optimal Search 229 By ( 12) of the lemma, (22) is reduced to /(4K + 2) = 4K + 2 +min {ff(2k + 2) + f(2k -1)], ff(2k) + f(2k + 1)}} = 4K f (2K) + f (2K + 1), where the last step follows from ( 12). We note that we have extended (1la) to m = K + 1, and ( 12) to n = 4K + 1. Similarly, by the use of(l2) and (13), f(4k + 3) can be written f(4k + 3) = 4K + 3 +min {2/(2K + 1), f(2k) + f(2k + 2)}. (23) It follows from (14) that!(2k + 2)- f(2k + 1) > {(2K + 3)-f(2K + 2) and it follows from ( 12) that Therefore, [(2K + 3)- f(2k + 2) ~ f(2k + 1)-f(2K). f(2k + 2) + f(2k) > 2/(2K + 1) and from (23), _and [(4K + 3) = 4K /(2K + 1). (24) Following a procedure nearly identical to the above, we can show that /(4K + 4) = 4K {(2K + 1) + {(2K + 2) (25) {(4K + 5) = 4K {(2K + 1) + {(2K + 3). By induction, Theorem 1 follows. B. The functional form of f(n) is given by the following theorem: Theorem 2: Equation (9) is satisfied if and only if k = 0,1,... k m = 0,1,...,2, (26) (27a) =2k+l(k-~)+(2m+l)k+3m+3, k=o,l,... (27b) m = 0,1,...,2k-1.

8 230 H. W. Gottinger Proof: The "only if' part follows simply from the fact that no two functions can both be the minimum without being equal. To prove (27), we again use induction. That is, we verify (27) for k = 0 and assume it to be valid for k = 0, 1,..., K- 1. If it follows thereby that (27) is valid fork= K, then (27) must be true for all k. The detailed proof involves substitution of (27) in (11) and elementary manipulation, and will be omitted here. C. Theorem 1 is strengthened by the following result: {[n (N)- 1] + f[n- n (N)) = 1 if and only if min ~ n <.N {f(n- 1) + f(n- n)} (28) n* (2k+l+ 2m) =2"+i, j=0,1,...,2m+l,o<:m<2" 1 (29a) i = 2m-2k+ 1,..., 2k, 2k l:s;; m <. 2k- l, k=o, 1,... n*(2k+ 1 +2m-1) =2k+2j, j=o,l,...,m,o<;m~2k-l (29b). = _ 2 k-1 2 k-1 2 k-1 < m~ 2 k_l 1 m,...,, k = 0,1'... Proof: The "if' part is proved by substituting (29) and (27) in (28) and verify. In the process it is also shown that for 2k+ 1 ~ N <; 2k+2-1, the only solutions in the range 2" ~ n* ~ 2k+l are those given by (29}. Thus, it remains only to show that no value of n greater than 2k+ 1 or less than 2" is a solution. Consider N = 2k+l +2m, 0 ~ m < 2k l. Since we know that n* = 2" is a solution, we need only to show that (similar results follow for n > 2k+ 1 by symmetry) /(2"-1) + /(2"+ 2m)</(2"- 2) + /(2"+ 2m+ 1) (30) <{(2"- 3) + /(2"+ 2m+ 2) ~... The fust of inequalities in (30) is easily verified using (27). The remaining inequalities follow from (12). For 2k l < m..; 2" -1, we use n = 2k+l, and from (27) and (12) show that /(2k+l_l) +/(2m) </(2k+l) +/{2m -1) </(2k+l+ 1) + f(2m- 2) < For N = 2" m - 1, the proof follows nearly identical lines and will not be given here. (31)

9 On a Problem of Optimal Search 231 Acknowledgment I am very grateful to an anonymous referee for valuable comments and pointing out some mistakes in an earlier draft. Also I am indebted to F. Bolle (LehrstuhJ fiir Volkswirtschaftslehre, Univ. Hamburg) for providing the interpretation in Sect. 1. References Bellman, R.: Dynamic Programming, Princeton de Groot, M.H.: Optimal Statistical Decisions, (Chap. 14), New York Dobbie,J.M.: A Survey of Search Theory, Operations Research 16, 1968, Hinderer, K.: Foundations of Non-Stationary Dynamic Programming with Discrete Time Parameter, Lecture Notes in Operations Research and Mathematical Systems No. 33, Berlin- Heidelberg New York Johmon, S.M.: A Search Game, in: Advances in Game Theory, Dresher, M., et al., eds., Princeton Morris, R.: Some Theorems of Sorting, SIAM Jour. Appl. Math. 17 (1), 1969, 1-6. Murakami, M.: On a Search Problem, Jour. Oper. Research Soc. Japan 14,1971, Pollock, S.M.: Search Detection and Subsequent Action: Some Problems on the Interfaces, Operations Research 19,1971, Stone, L.D.: Theory of Optim~ Search, New York 1975.