Parallel On-Line Searching

Parallel On-Line Searching∗ Patrick Jaillet ‡ and Matthew Stafford¶ July 2000

Abstract We consider the problem where p searchers explore m > p half-lines, all joined at a common origin. The searchers travel at unit speed and the goal is to find a single object, or “exit” whose location is bounded away from the origin. We consider two minmax measures of effectiveness for a search strategy. The first is the worst case distancecompetitive ratio. This is the smallest value that exceeds the fraction (total distance covered until discovery)/(distance of exit from origin) for every exit location in the search space. The second is the worst case time-competitive ratio. This is the smallest value that exceeds the fraction (time until discovery)/(distance of exit from origin) for every exit location in the search space. Note that the denominator of this competitive ratio coincides with the amount of time it would take a searcher with perfect knowledge to reach the exit. In this paper we propose optimal deterministic on-line strategies for solving these problems.

Keywords : parallel on-line search, incomplete information

∗

Research funded in part by ONR, grants N00014-94-1-0257 and N00014-96-1-0190, and by NSF, grant DMI-9713682. Second author also wishes to thank SBC Technology Resources for supporting this work. ‡ MSIS Department, The University of Texas at Austin, Austin, TX 78712, USA, and Mathematics Department, ENPC, 75343 Paris cedex 07, France ¶ SBC Technology Resources, 9505 Arboretum Blvd, Austin, TX 78759 USA, and Operations Research, The University of Texas at Austin, Austin, TX 78712, USA

1

1

Introduction

1.1

Motivation and overview

There are many situations in which present actions influence our “well being” in the future. The difficulty in these situations is that we have to make our decision based only on the past and the current task we have to perform. It is not even clear how to measure the quality of a proposed decision strategy. The approach usually taken is to devise some probabilistic model of the future and act on this basis. This is the starting point of the theory of Markov Decision Processes (see for example [6]). The approach around which this paper is based is to compare the performance of a strategy that operates with no knowledge of the future (on-line) with the performance of the optimal clairvoyant strategy that has complete knowledge of the future (off-line). This requires no probabilistic knowledge of the future and is therefore a “worst case” measure of quality. This new approach, called “competitive analysis” in [12], takes the following pessimistic approach in analyzing the performance of an on-line algorithm: an on-line algorithm is good only if its performance on any sequence of requests is within some (desired) factor of the performance of the off-line algorithm. In particular, a good algorithm would certainly perform well in the presence of an unknown distribution. In this paper we consider generic searching problems in some given space. Searching unknown graphs and planar regions is central to many areas of computer science and operations research. In the field of operations research, these problems are simple and stylized versions of general problems for which an object or a boundary has to be located by successive moves in a largely unknown search space (see [2, 3, 4, 5, 7]). This paper is concerned with the special case where p searchers explore m > p half-lines (hereafter called branches), all joined at a common origin. The searchers travel at unit speed; the goal is to find a single object, or “exit”. We consider two minmax measures of effectiveness for a search strategy. The first is the worst case distance-competitive ratio. This is the smallest value that exceeds the fraction (total distance covered until discovery)/(distance of exit from origin) for every exit location in the search space bounded away from the origin. The second is the worst case time-competitive ratio. This is the smallest value that exceeds the fraction (time until discovery)/(distance of exit from origin) for every exit location in the search space bounded away from the origin. Note that the denominator of this time-competitive ratio coincides with the amount of time it would take a searcher with perfect knowledge to reach the exit. Note also that the two minmax measures are 2

identical when p = 1. For both minmax measures we have restricted the exit location to be bounded away from the origin. That is, we assume that there exists a strictly positive constant ζ > 0 such that the exit distance d from the origin verifies d ≥ ζ. Without this assumption, the exit location could be placed arbitrarily close to the origin and lead to infinite competitive ratio for any strategy, a trivial case. Throughout this paper we are choosing the value ζ = 1. This is largely a matter of convenience. Converting to a different bound amounts to rescaling our measure of distance (and if one strategy has better competitive ratio than another before a uniform rescaling, then the same is true after the rescaling). Finally, we are restricting our attention to deterministic strategies; any searcher which reaches the exit recognizes it with probability 1 and can instantaneously notify the other searchers that the search is complete. Under both minmax measures, a strategy and its (distance or time)-competitive ratio is said to be optimal if it has the smallest provable (distance or time)-competitive ratio. Hereafter, we refer to this class of problems as the “m-branch, p-searcher” problem. The paper is organized around two main parts. In Section 3, we analyze the distancecompetitive ratio. Working within the class of partitioning strategies, we develop a nontrivial upper bound for the competitive ratio. We also exhibit strategies that realize this upper bound and conjecture that these strategies are optimal. This is a direct and relatively straightforward extension of the work contained in [10]. In Section 4, we tackle the time-competitive ratio case and propose optimal strategies among those with the “monotone-increasing” property (to be defined later). We first treat the 3-branch, 2-searcher problem in full detail before considering more general cases.

1.2

Previous work and related problems

In case of a single searcher, the previous problem has been analyzed in detail under various conditions by Jaillet [8] and Jaillet and Stafford [10]. In addition to finding optimal deterministic on-line algorithms for the m-branch, 1-searcher problem, we have considered the “value” of getting additional information before starting the search: We have provided optimal on-line strategies when (i) the exit is known a priori to be within a given distance D from the origin, or when (ii) probabilistic information such as “the exit has a probability pi of being on branch i” is given a priori to the searcher. Baeza-Yates, Culberson, and Rawlins [1] discuss on-line strategies for a problem very similar to the m-branch 1-searcher problem, in which the exit is at an integer distance from 3

the origin. When m = 2 they give a proof that the strategy of alternatively moving on each branch, each time doubling the previous distance, is optimal. When m > 2 they propose the following strategy: move in the integrally increasing powers of m/(m − 1) in a cyclic manner, visiting branches in the same order over and over again. They show that, among all “monotone-increasing” cyclic strategies, this one is optimal. In [10], we provide a rigorous proof that a continuous version of this strategy does indeed belong to the class of overall optimal unrestricted strategies. In Kao, Reif and Tate [11], the authors present an optimal randomized algorithm for the single searcher with m = 2. A different but related problem has also appeared in the literature under the name of the “Layered Graph Traversal” problem. A layered graph is a connected weighted graph whose nodes are partitioned into sets (i.e., layers) L0 = {s}, L1 , L2 , . . . and all edges connect nodes in consecutive layers. The edges between layer Li and layer Li+1 are all revealed when the searcher visits some nodes in Li (this is the main difference with our problems). This problem is introduced in Papadimitriou and Yannakakis [14] and is solved optimally for the case of 2 disjoint paths, by using the results of [1]. None of these previous works have considered the problem of multiple searchers/travelers working in parallel, which is the main focus of the present paper. Finally Jaillet [9] considers a similar on-line searching problem in which the exit is a moving target whose purpose is to avoid being located by the searcher. Optimal strategies are proved for the 2-branch, 1-searcher moving target problem, and conjectured for the m-branch, p-searcher cases.

2

Background: The Single Searcher Case

Let us combine some results of [10] about the single searcher case and present them in the form of a theorem. The distinction between distance and time is irrelevant here so we will drop the reference to distance in our statements. For sake of future notational simplicity, we label the m branches 0, 1, . . . , m − 1. Any potentially optimal search strategies can be uniquely described by (see [10] for a full justification): (i) a sequence of search extents (xk )k≥0 , where xk represents the distance between the origin and the returning point during search k (a search here is formally defined as leaving the origin on one branch, exploring this branch up to a returning point - possibly infinite, and, in case of a finite returning point, returning to the origin). So x0 of search 0 corresponds to the first search extent; 4

(ii) an assignment of which branch is visited during each search. Theorem (Jaillet and Stafford [10]). For m ≥ 2, the optimal competitive ratio for the m-branch, 1-searcher problem is:  m m−1 r∗ (m, 1) = 1 + 2m . (1) m−1
 m k+1 The search extent sequence xk = 1 + (k + 1)/(m − 1) m−1 where search k is done on branch k (mod m), k ≥ 0, is an optimal strategy. For m = 1 the optimal competitive ratio

is obviously r∗ (1, 1) = 1.

3 3.1

Distance-Competitive Ratio Case Fundamentals

Let Z be the search space, i.e., the set of all points belonging to the m branches. Definition 1. A (deterministic) strategy can be described as a map from the non-negative reals R+ into Z p . For a given strategy S : R+ → Z p , S(d) is interpreted as “the positions of the p searchers in the space Z after d total units of distance have been traversed.” Suppose that a strategy S is given. Then for each z in the search space, the total distance DS (z) until the first visit to z (by any searcher) is uniquely determined. If S never visits z, DS (z) is taken to be infinite. The distance-competitive ratio map of strategy S is given by r(z) = rS (z) = DS (z)/kzk, where kzk represents the distance from the source to z. For reasons explained before, we need to bound the position of z away from the origin. We therefore view r(·) as a map Z˜ → R, where Z˜ := {z ∈ Z | kzk ≥ 1}. Our definition of worst-case performance then becomes: Definition 2. The distance-competitive ratio rS of a strategy S is sup{z∈Z} ˜ DS (z)/kzk.

3.2

Partitioning strategies

In this section we restrict ourselves to partitioning strategies, i.e., strategies dividing the branches into disjoint sets to be searched by single searchers. Theorem 1. Among the class of partitioning strategies, the optimal distance-competitive ratio for the m-branch, p-searcher problem, denoted rd∗ (m, p), is not greater than  m − p + 1 m−p p + 2(m − p + 1) . m−p 5

(2)

A strategy of assigning p−1 searchers to p−1 single branches (one searcher for each branch) and one searcher to the remaining m−p+1 branches realizes the competitive ratio given here. Moreover, the upper bound given in the formula is tight for the m-branch, (m − 1)-searcher problem, i.e. rd∗ (m, m − 1) = m − 2 + r∗ (2, 1).

(3)

Proof. Assume searcher i has been assigned mi branches, 0 ≤ i ≤ p − 1, and that indices are such that m0 ≥ m1 ≥ · · · ≥ mp−1 . Let Zi be the corresponding search space for searcher i. The best individual strategy for searcher i is to use the results of Theorem 2 with m = mi . Let z e be the position of the exit. If z e ∈ Zi then searcher i will discover it after a total individual distance of r∗ (mi , 1)kzk in the worst case. Intuitively, searcher 0 puts forth the most exertion to search its space, since r∗ (m0 , 1) is maximal among the individual competitive ratios. If searchers with mi < m0 were to move constantly at the same speed as searcher 0, the searched extents in their subspaces would grow faster than that of searcher 0; this is clearly a waste. Thus we consider strategies in which searcher 0 moves constantly, but searchers with mi < m0 are sometimes idle. We formalize this notion. For i = 0, 1, . . . , p − 1, let Ei := min{extent searched thus far on branch b}, where the minimization is taken over all branches b contained in the subspace Zi . In words, Ei is the searched extent in subspace i at any given juncture. We will develop a strategy by stipulating that • for i = 0, 1, . . . , p − 1, searcher i executes a strategy that achieves (individual) competitive ratio r∗ (mi , 1). • searcher 0 travels continually; whenever any other searcher is not idled, it travels at the same speed as searcher 0. • for i = 1, 2, . . . , p − 1, we have Ei = E0 at all times. It is clear that, for any strategy S meeting these criteria, the competitive ratio satisfies rS ≤

p−1 X

r∗ (mi , 1).

(4)

i=0

For definiteness, we exhibit the following “compliant” strategy. Note that whenever searcher 0 reaches a turning-back point, E0 stops increasing, entering a period of constancy. At each 6

such point, each searcher i for which mi < m0 will be idled, resuming motion only when it is necessary in order to be in the correct position when the next period of increase in E0 begins. (So if mi < mj < m0 , searcher j will resume motion sooner than searcher i.) Note that when a searcher is assigned to a single branch, the individual competitive ratio, as defined in Section 2, is identically 1 on that branch. Thus, for the partition which assigns m − p + 1 branches to one searcher and one branch each to the remaining p − 1 searchers, Inequality (4) is tight. Our theorem’s assertion regarding the m-branch, m − 1searcher case is now proven because the only non-trivial partition of the branch set assigns 2 branches to one searcher and 1 branch to each of the other searchers. For each partition of the branch set, Inequality (4) gives an upper bound for rd∗ (m, p). We claim that the least of these bounds is given by the partition that assigns m − p + 1 branches to one searcher and a single branch to each of the remaining searchers. For this claim it suffices to show that r∗ (j, 1) + r∗ (k, 1) ≥ 1 + r∗ (j + k − 1, 1) whenever j, k ≥ 2. The last inequality follows quickly from the fact that ρ00 (x) < 0 for x ≥ 2, where  x x−1 . ρ(x) := x x−1 The negativity of the second derivative follows in turn from Jensen’s Inequality.

3.3

Open problem

Conjecture 1. The distance-competitive ratio and corresponding strategy in Theorem 1 are optimal among all possible strategies.

4

Time-Competitive Ratio Case

The material is organized as follows. After Section 4.1 on notations and statement of the main result, we first show in 4.2 that we can restrict the space of search strategies to those for which no searcher is ever idled during the exploration. This is in marked contrast to the case of minimizing distance-competitive ratio. Amplified, the assumption states that each searcher, starting at the origin, proceeds directly to some point of maximum distance therefrom (the extent of the search), returns directly to the origin, and immediately begins the next search. Thus it makes sense to speak of a sequence of search extents (indexed in the order of departure from the origin). In 4.2, we then show that search strategies should be branch-increasing, i.e., such that the subsequence of searches of any individual branch is strictly increasing. 7

In 4.3 we apply the no-idleness property to show that, for the purpose of strategy evaluation, one can replace the time-competitive ratio map with a sequence of worst-case time-competitive ratios. In Section 4.4 we show that if we further restrict the space of search strategies to those with non-decreasing sequences of search extents, then the “least-searched-so-far” rule can be used to determine which branch to be explored when a new search is ready to depart the origin. It is not difficult to see that the “no-idleness” and “branch-increasing” properties together with the “non-decreasing sequence” assumptions (itself implying the “least-searchedso-far” rule) lead to cyclic searches, i.e., strategies in which the searcher to perform search k is always determined by k (mod p) and the branch to be searched is determined by k (mod m). This is the starting point of Section 4.5 which eventually leads to an alternative formulation as a sequence of linear programming problems. This permits us to find an optimal strategy in 4.6 for m = 3 and p = 2. In conclusion the only restriction we have imposed but not totally justified is the nondecreasing sequence of search extents. We feel strongly that there is no loss of generality in making this restriction, but this remains an open question (it was proved for p = 1 in [10]).

4.1

Fundamentals and statement of main result

Let Z be the search space, i.e., the set of all points belonging to the m branches. Suppose that a strategy S is given. Then for each z in the search space, the time TS (z) of the first visit to z (by any searcher) is uniquely determined. If S never visits z, TS (z) is taken to be infinite. Given that the exit is located at z, the time at which a searcher with perfect information reaches the exit coincides with the distance of z from the origin (recall that travel proceeds at unit speed). We denote the latter quantity by kzk. The time-competitive ratio map of strategy S is then given by r(z) = rS (z) = TS (z)/kzk. Again we need to bound the position of z away from the origin. We therefore view r(·) as a map Z˜ → R, where Z˜ := {z ∈ Z | kzk ≥ 1}. Our definition of worst-case performance then becomes: Definition 3. The time-competitive ratio of the strategy S is defined to be supz∈Z˜ r(z) and is denoted by rS . For strategies S and T , we say that S dominates T if rS ≤ rT . As stated in the introductory remarks of Section 4, because of the “no-idleness” property, a searcher’s time horizon can be segmented into discrete searches. Each of these begins at the 8

origin, proceeds directly to a (local) maximum distance from the origin called the extent of the search, and returns directly to the origin. Definition 4. Let S be a strategy with sequence {xk } of search extents (indexed in the order of departure from the origin). We say that S ∈ Γincr if the following hold: (i)

xk → ∞

(ii)

xk ≥ 1 for all k

(iii) S is branch-increasing (iv)

The xk ’s form a non-decreasing sequence.

Insisting that a strategy has properties (i), (ii), and (iii) of the above definition entails no loss of generality, as we will see in later sections. While property (iv) seems entirely natural to us, we currently can prove dominance only in the one-searcher case. With these preliminaries out of the way, we can now make a precise statement of our main theorem. Conjecture 2. Among the class of strategies in Γincr , the optimal time-competitive ratio for the m-branch, p-searcher problem is rt∗ (m, p) = 1 + 2 The search extent sequence xk =



m m−p

m − p h m im/p . p m−p

(k+1)/p

(5)

where search k is done on branch k (mod m)

by searcher p (mod m), k ≥ 0, is an optimal strategy. Theorem 2. For the case of m = 3 branches, p = 2 searchers, the above conjecture holds.

4.2

Justifying the “no idleness” and “branch-increasing” properties

The idea of this section is simple when thought of visually. Suppose we pick a single searcher and graph its distance from the origin as a function of time. What we show is that without loss of generality, we can assume that this graph consists of isosceles triangles sitting side by side and based along the time-axis. (The altitude of any of these triangles is the extent of the corresponding search.) First we clarify some details of the model. Each searcher can travel at unit speed or remain motionless; it does not consume any time to start, stop, or change directions. A strategy must specify the position of each searcher at each time t ∈ R+ . This is done for searcher i by naming a branch and giving a distance d(i, t) from the origin. The branch can only change at t values for which d(i, t) = 0; the branch b(i, t) is ambiguous at such times. This ambiguity can be removed by saying 9

that b(i, t) is right continuous as a function of t (with the discrete topology on the set of branches). To avoid pathologies, each type of instruction the searcher is given (travel away from the origin, travel toward the origin, hold your position) is presumed to persist for an interval of time. Furthermore, we restrict ourselves to strategies for which the lengths of these time intervals are bounded away from 0. So the distance d(i, t) is piecewise linear with slope 1, −1, or 0. Any time at which a searcher’s state changes among elements of the set {traveling away from origin, traveling toward origin, at rest} is termed a decision epoch. Since the elapsed time between any two such epochs is bounded away from 0, the set of decision epochs has no finite accumulation points. Once a strategy S is specified, the discovery time of each point is determined. That is, given a point z in the search space Z, V (z) := {t ∈ R+ | some searcher visits point z at time t} is well-defined. Our assumptions imply that V (z) has a minimum element. Setting TS (z) equal to this minimum element makes our definition of the discovery time map (from Section 4.1) rigorous. Definition 5. For strategies S and T , we will say that S  T if TS (z) ≤ TT (z) for all z ∈ Z. In words, if pursuing S never locates the exit later than pursuing strategy T , then T is not preferred to S. Remark 1. The assertion that S  T is very strong; in particular, it clearly implies that S dominates T . We emphasize that the arguments which make up the balance of this section work with the “” relation. As such, they do not depend on our choice of defining the dominance relation in terms of worst-case ratios. Claim. If S 0 is formed from S by removing wait period(s), S 0  S. The claim is easily verified once we clarify the terminology. Suppose that [a, a+δ] ⊂ R+ is a wait period for searcher i. In other words, searcher i’s distance d(i, ·) (from the origin) is constant on [a, a + δ] but non-constant at a− and (a + δ)+ . To remove the wait period [a, a + δ], we replace d(i, t) by d(i, t + δ) for all t > a and perform exactly the same change of variable on the branch function b(i, ·). The functions d(i, ·) and b(i, ·) remain unchanged on [0, a]. 10

Now it is clear that, for any given z ∈ Z, the portion of V (z) ∩ [a + δ, ∞] which is contributed by searcher i is shifted δ units to the left. If the interval [a, a + δ] is contained in the visit set V (z), it may be replaced by its left endpoint. But otherwise no points of V (z) disappear completely and none are moved to the right. So the claim is verified. Next we claim that equivocation (to be defined shortly) can be replaced by wait periods. Searcher i equivocates on the interval [a, a + δ] if d(i, a) = d(i, a + δ) = supt∈[a,a+δ] d(i, t), d(i, a + ) and d(i, a + δ − ) are both strictly less than this supremum for all  sufficiently small, and d(i, t) > 0 on [a, a + δ]. Here we are thinking of a period where there is some retreat towards the origin, but not a return to the origin. Observing that it is not possible to discover the exit during (the interior of) an interval of equivocation, we conclude that inf V (z) cannot be changed for any z by replacing an interval of equivocation by a wait period of the same duration. So the discovery time map is unchanged by this procedure. Suppose that a and a + δ are times of successive visits to the origin by searcher i. In other words the distance function d(i, ·) is 0 at the endpoints of the interval [a, a + δ] and strictly positive on the interior. Then maxt∈[a,a+δ] d(i, t) will be called the extent of the search performed on this time interval (by searcher i). We can now state one of the main results of the section: Lemma 1. For purposes of assessing optimality, we can assume without loss of generality that between successive visits to the origin, each searcher travels directly to the associated extent and returns directly to the origin. This is in the sense that any strategy S containing wait and/or equivocation periods can be replaced by a strategy S 0 which is devoid of wait and equivocation periods and which satisfies S 0  S. Proof. We describe a procedure which can be performed for each searcher individually without reference to the positions of the other searchers. The argument which follows makes sense if our searcher returns infinitely often to the origin. If not, matters are even easier: from the time t0 of the last departure from the origin, this searcher never changes branches. So we know that, beginning at time t0 , we can send the searcher away from the origin at unit speed forever. Doing so will not increase the discovery time of any point. Here is the general argument: First we replace all periods of equivocation by wait periods as prescribed. To avoid an infinite regress, we should make sure that an equivocation period so replaced is not (strictly) contained in any longer equivocation period. This process is well-defined: there will be finitely many replacements on any compact time interval [0, Λ]; the results of performing the procedure on [0, Σ] and [0, Λ] are consistent on the intersections 11

of these intervals provided that Σ and Λ are decision epochs which are not interior to any period of equivocation. Therefore the described process yields a piecewise linear function which is well-defined on R+ . The new set of decision epochs is contained in the original set. Also note that the new graph is comprised of non-decreasing segments from origin to extent and non-increasing segments from extent to origin. Next, wait periods are removed. The well-definedness is now harder to write down, partly because the set of decision epochs will now not be contained in the original one. However, there is no real difficulty: to find out what the resulting graph looks like on the compact set [0, λ], sum twice the extents of consecutive searches starting with the first until the result σ reaches or exceeds λ. Let Σ be the time at which the last of these searches ends. Then Σ − σ must be the total duration of wait periods during the time interval [0, Σ]. Whatever the particulars of the graph on [0, Σ], we know that it will be replaced by a segment which now covers the time interval [0, σ] ⊇ [0, λ], and is made up of the familiar triangles whose altitudes coincide with the extents on [0, Σ]. Note in particular that removing a wait period cannot bring an equivocation period into existence, so that there is no infinite regress. Returning to the time-competitive objective, we observe that the same type of reasoning used for removing wait periods gives us something else for free. Remark 2. In the event that two consecutive excursions by the same searcher are on the same branch, we can remove the search whose extent is the smaller of the two without increasing the discovery time of any point. Similarly, if an extent is less than or equal to that of a previous search of the same branch, the redundant search can be removed without increasing the discovery time of any point. A search whose extent is less than or equal to 1 can be viewed as a special case of this. This last remark implies that we can assume from now on that strategies are branchincreasing, in the sense that the subsequence of searches of any individual branch is strictly increasing.

4.3 4.3.1

Simplification of the time-competitive ratio map An important property of the ratio map

We will argue that, for purposes of assessing worst-case performance of a strategy, we may as well replace the time-competitive ratio function by a sequence of time-competitive ratios. 12

This relies on a piecewise monotone property of the time-competitive ratio map which we now describe. Intuitively, here is the content of this section: there is a (locally) worst case corresponding to an exit location just past the extent of any search. Suppose that execution of some strategy began at time 0, and that at the current time t0 , a given searcher is at the origin commencing a new search (of branch b0 , say). The currently-departing search will discover the exit provided it is on branch b0 with distance-from-origin y ∈ (xprev , xcurr ]. Here xcurr is the extent of the current search, and xprev is the extent of the previous search on the same branch (regardless of “who” performed the previous search. 1 ) Let I := (xprev , xcurr ] × {b0 } ⊂ Z; this is just the interval in question as a subset of the search space rather than as a subset of R+ . We write an element of I as an ordered pair: z = (y, b0 ). Our generic notation for the time-competitive ratio map is now r(·, ·). Remark 3. The time-competitive ratio is decreasing on I (formally, this is as a function of its first argument). Therefore r(x+ prev , b0 ) is the least upper bound for r|I , the restriction of r to I. Writing z = (y, b) ∈ I ⊂ Z we in fact have the following implicit formula for r(z): t0 + y = r(z)y.

(6)

This verifies the remark: on I, the time-competitive ratio map is essentially y 7→ (t0 + y)/y = t0 /y + 1.

(7)

The correctness of these formulae depends on the fact that the strategies we consider are free of wait and equivocation periods (these were defined in Section 4.2). Applying the above argument to an analogous interval with left endpoint xcurr (written schematically as (xcurr , xnext ] × {b0 }, say), we conclude that the points of discontinuity of r(·, ·) coincide with the extents of searches and therefore form a countable set. Note well that although we have dropped the subscript from rS (·, ·) for readability, the time-competitive ratio map in general and its set of discontinuities in particular depend on the choice of strategy S. 1

In the case that this is the first search of branch b0 , xprev is taken to be 1. Otherwise “the previous

search” will be taken to mean the search of largest extent on branch b0 to be completed before the current search. For the moment we are only interested in the case where xcurr in fact exceeds this previous extent.

13

4.3.2

The sequence of time-competitive ratios

The observation that the set of discontinuities of the time-competitive ratio map is in oneto-one correspondence with the set of extents of searches (and in turn with the set of “bad exit locations” for a given strategy) motivates us to index the set of extents. On the face of it, there might be some difficulty in choosing between an indexing for each searcher and an indexing for each branch. As hinted earlier, there is a better choice: a global indexing on the basis of elapsed time. Our goal is to make performance evaluation more tractable by reducing it to looking at a sequence of local suprema for the time-competitive ratio map. To this end, the symbol xk will denote the extent of the k th search to depart the origin overall. It will prove beneficial to use as index set N := {0, 1, 2, . . . }. After numbering the searchers 0, 1, . . . , p − 1, we resolve ties in cyclic order of searcher index (we will call this the lexicographic tie-breaking rule; we observe that this simply picks one of a family of strategies with the same performance and therefore incurs no loss of generality. The performance is the same in the sense that equations (6) and (7) are instantiated with exactly the same data). The branches will be numbered 0, 1, . . . , m − 1. We write n(k) := index of next search to explore the same branch as the k th search. Certainly we can write down the sequences {xk } and {n(k)} if we are given an “individual description” of each searcher’s assignments. Conversely, we can recover these individual descriptions from {xk } and {n(k)}, up to a permutation of the branches’ subscripts. For, each search with an index which is not in the image of n(·) is the first to explore some branch. Provided the original strategy satisfies the lexicographical tie-breaking rule if applicable, there is no other ambiguity. We therefore claim that the following definition is well-posed, even if we are only given the sequences {xk } and {n(k)} initially, as long as we choose an assignment of searches to branches that is consistent with these sequences. Definition 6. Let bk denote the branch which is explored by the k th search to depart the origin and let xk denote its extent. Then rk is defined to be r(x+ k , bk ) for k ≥ 0. We also set r−1 := t0 + 1, where t0 is the initiation time of the first search of the last unexplored branch. Remark 4. Suppose that xk is less than the previously-explored extent on branch bk . Then it is not possible for the exit to be discovered at (x+ k , bk ) by a subsequent search (it is also not possible for the k th search to discover the exit). So rk is not of interest in this case:

14

although we claim that rk is still well-defined, it is not a point of discontinuity of the timecompetitive ratio map r(·, ·) and therefore is not a local supremum (unless the branch-extent pair (xk , bk ) has appeared before - but even in this case rk gives no new information). Lemma 2. The supremum of the time-competitive ratio map for a given strategy is the same as supk≥−1 rk , the supremum of the time-competitive ratio sequence for the same strategy. Proof. Let us consider the first search of some branch b0 , and let t0 denote its initiation time. In the terminology of Section 4.3.1, and in particular of Remark 3, xprev is just 1 (clearly, we can assume without loss of generality that every search extent, including the current one, exceeds 1). So the worst case time-competitive ratio for this search is t0 + 1 (cf. (7)). Rather than recording this initial-search ratio for each branch, we simply let r−1 be the largest of these, since we are interested in worst case performance. For every z ∈ Z which is beyond the extent of the first search on its particular branch, r(z) < rk for some k ≥ 0. The equality of supz∈Z˜ r(z) and supk≥0 rk now follows directly from the definition of supremum. It is clear that one needs a sequence of extents in order to characterize a strategy. Our choice to use n(·) as the other significant piece of specifying a strategy is perhaps less intuitive to the reader. However, as will become apparent in the next section, the n(·) “pointers” provide a convenient means of manipulating a strategy while maintaining some control over the time-competitive ratio sequence.

4.4

Justifying the “least-searched-so-far” rule

Upon completion of a search, how does a searcher decide what to do next? We have said that the searcher takes the next search extent in a specified sequence. Heretofore we have spoken in terms of also specifying a branch to be explored. Another choice suggests itself, however: for each currently empty branch (i.e., not containing a searcher), record the largest extent of any search who explored that branch up to the current time. Then choose to explore the branch for which this number is smallest. It turns out that this myopic rule, which we will term least-searched-so-far, is good in a global sense. Lemma 3. Let S ∈ Γincr . Then S is dominated by some strategy S 0 which (always) uses the least-searched-so-far rule to determine what branch will be searched by a searcher who has returned to the origin seeking a new assignment. 15

Proof. We first give the proof for the case that the sequence {xk } of search extents is strictly increasing. Fix a strategy S and let tj be the time at which the j th search begins, j = 1, 2, . . . Suppose (at time tj ) that the k1st search was the most recent of branch 1 and that the k2nd search was the most recent of branch 2. Suppose that xk1 > xk2 , but that strategy S searches branch 1 at time tj . In symbols, this says j = n(k1 ); also let j 0 := n(k2 ). If branches 1 and 2 are not the ones involved, renumber. We have (cf. (7)) rk1 =

tj + 1 and xk1

(8)

tj 0 + 1. xk2

(9)

rk2 =

To produce a dominating strategy S 0 , we interchange the roles of branches 1 and 2 for the search beginning at time tj and for every subsequent search (regardless of which searcher performs it). In other words we interchange n(k1 ) and n(k2 ), leaving the n(·) function intact elsewhere. It is essential to our argument that the inception times for searches are unchanged by this procedure: for example, a searcher departs the origin at time tj and again at time tj + 2xj under strategy S. This remains true for S 0 , as does the statement that searches depart the origin at times tj 0 , tj 0 + 2xj 0 , tj 0 + 2xj 0 + 2xn(j 0 ) and so on. Using primes to denote time-competitive ratios associated with the new strategy S 0 , the result is that rk0 1 =

tj 0 + 1 and xk1

(10)

tj + 1. xk2

(11)

rk0 2 =

Observe that rk1 < rk0 1 < rk2 and that rk0 2 < rk2 . What is absolutely key is that no other time-competitive ratios change in the passage from S to S 0 . For this allows us to conclude that their supremum cannot have increased. Without knowing that the search extents are still taken up in the same order and at the same times, we cannot make this conclusion. The foregoing argument suffices: a finite number of such interchanges is required to render a search that is least-searched-so-far compliant on any compact time interval [0, M ]. Furthermore, there are no inconsistencies introduced as M increases without bound.

16

Now we work in Γincr . In other words, we relax our hypothesis on the sequence {xk } from strictly increasing to non-decreasing. The only additional thing we have to show is that the interchange does not disrupt the branch-increasing property. For then (10) and (11) remain correct. After the interchange, the sequence of search extents along branch 2 looks like . . . xk2 , xj , xn(j) , . . . By hypothesis, xk2 < xk1 < xj , with the second inequality following from the branch-increasingness of the original strategy. The new sequence of search extents along branch 1 is . . . xk1 , xj 0 , xn(j 0 ) , . . . . We use the fact that the initial strategy resides in Γincr to conclude xk−1 < xj ≤ xj 0 . So the branchincreasing property is not violated on branch 1 or branch 2 (and we have made no changes at all along the other branches).

4.5 4.5.1

Cyclic strategies and linear programming A canonical search for a given sequence of extents

In this section we suppose that we are given a “target” time-competitive ratio r. The task is to devise a search strategy S for which every element of the time-competitive ratio sequence is bounded from above by r. In other words, we are not allowed to violate the time-competitive ratio r. Recall that p is the number of searchers (numbered 0, 1, . . . , p − 1) and that m is the number of branches (numbered 0, 1, . . . , m − 1). The following lemma, which is a corollary of Lemma 3, simplifies our task considerably. Lemma 4. Every strategy in Γincr is dominated by a cyclic strategy (which is also in Γincr ). By this we mean a strategy for which for all k the k th search explores branch k (mod m) and is performed by searcher k (mod p). Proof. For strategies in Γincr , it is clear that no searcher can return to the origin twice between consecutive returns of another. The lexicographic tie-breaking rule (first mentioned in Section 4.3.2) maintains the desired pattern in cases of ambiguity, without changing any time-competitive ratios. This verifies the identity-of-searcher portion of the claim. Again without affecting any time-competitive ratios, we can decree that each of the searches 0, 1, . . . m − 1 explores the like-indexed branch. Inductively we therefore see that, at the time of departure of the k th search (k > m − 1), branch k − m is least-searched-so-far. Lemma 3 perpetuates the pattern.

17

All we have to know in order to execute a strategy S ∈ Γincr is a sequence {xk } of search extents. This lemma tells us how to keep track of the status of each searcher and each branch. Any ambiguities in recreating S do not matter at all to our performance measure. 4.5.2

A sequence of linear programs

The lemma of the preceding section allows us formulate a sequence of linear programs which represent our task of maximizing the overall searched extent at each stage while not violating the target ratio r. This “overall searched extent” is defined globally at any stage as the minimum of the explored extents on all the branches up to that stage. For a cyclic search with an increasing sequence of extents, we have the constraint tk+m + xk = rk xk ≤ rxk for all k ≥ 0.

(12)

Here tk+m is defined to be the time at which search k + m is initiated. Observe that search k + m will be the next search (after the k th ) to explore the same branch (namely, branch k mod m). The equation is essentially a rewrite of (6), Section 4.3.1 with y = xk , the previously explored extent on branch k mod m at time tk+m . Suppose we want to upperbound xj , and that the search extents up to and including xj−1 have been determined. The genesis of such a bound is that the searcher who performs the j th search has to execute its next responsibility (namely, the (j +p)th search) in a timely fashion. Observe in turn that the (j + p)th search is preceded on branch (j + p) mod m by search j + p − m, suggesting the following re-subscripting of (12): tj+p ≤ (r − 1)xj+p−m .

(13)

We will treat the case j + p − m < 0 later. The point of working with a cyclic search is that it gives us a simple way of keeping track of the history of searcher j mod p and therefore of substituting for tj+p as follows: 2(xj + xj−p + xj−2p + · · · + xj modp ) ≤ (r − 1)xj+p−m .

(14)

1 xj ≤ (r − 1)xj+p−m − (xj−p + xj−2p + · · · + xj modp ). 2

(15)

So we have

Since p < m, j is in fact the largest subscript that appears in this most recent inequality.

18

Definition 7. Let ρ := (r − 1)/2. For k = 0, 1, . . . we define Fk = Fk (r) ⊂ Rk+1 (the k th feasible region) to be the set of tuples (x0 , x1 , . . . , xk ) ≥ 0 which satisfy xj ≤ ρxj−m+p − (xj−p + xj−2p + · · · + xj modp ).

(Ij )

for j = 0, 1, . . . k. We adopt the convention that xj+p−m = 1 whenever the subscript j < m − p and that the parenthesized expression on the RHS should vanish if j < p. Definition 8. For k ≥ 0, the linear program (Pk ) = (Pk (r)) is given by min {xk−m+p+1 , xk−m+p+2 , . . . , xk }

max

x ∈ Fk = Fk (r)

subj to

x0 ≥ 1

(Pk )

xj − xj−1 ≥ 0 for j = 1, 2, . . . , k. First, note that even though (Pk ) is not written as a linear programing problem, it can nevertheless be easily transformed into one by introducing an extra variable y, transforming the objective function into max y, and adding the m − p constraints y ≤ xk−m+p+j , 1 ≤ j ≤ m − p. Now, in order to understand the objective function of (Pk ), note that the overall search extent after completion of the k th stage is the minimum of the m − p search extents xk−m+p+1 , xk−m+p+2 , . . . , xk (which correspond to the m − p branches last visited by the p searchers. The other p branches play no role: controlling the extent of the last p searches, they can, by definition, be ultimately searched ad infinitum without violating the target time-competitive ratio). Note also that the last k constraints of (Pk ) imply that, if imposed, the objective function reduces to max xk−m+p+1 . Since we will consider various relaxations of (Pk ) in the subsequent section, we prefer to leave the formulation as it is.

4.6

Optimal time-competitive ratio for m = 3 and p = 2

We have assembled the necessary infrastructure for a proof of the main theorem for m = 3 and p = 2. In the previous section, the target ratio r was thought of as fixed. Now we want to vary it: how far can we decrease r and still maintain feasibility for each linear program in the sequence? In this section, all search strategies are assumed to be in Γincr unless stated otherwise. 4.6.1

A recurrence relation

It makes sense to study the recurrence relation which results when the structural constraint (Ij ) is tight for all j. This corresponds to the strategy in which each successive search is 19

pursued to the maximum possible extent. The recurrence simplifies to the form xj = ρxj−1 − ρxj−3 for all j ≥ 3.

(16)

This can be seen easily by rewriting inequality (Ij−2 ) as an equality, solving for ρxj−3 , and noting that the result is the same as the parenthesized expression in the equality form of (Ij ). The sequence of search extents is now completely determined by the initial values x0 = ρ,

(17)

x1 = ρx0 = ρ2 ,

(18)

x2 = ρx1 − x0 = ρ3 − ρ.

(19)

To determine what values of ρ (and ultimately of r) admit an increasing, unbounded sequence of positive search extents, we need to examine the characteristic equation of the recurrence relation: fρ (z) := z 3 − ρz 2 + ρ = 0.

(20)

Suppose that fρ possesses 3 distinct roots. It is well-known (see e.g. [13]) that the general solution (xj )j≥0 to the associated recurrence is a linear combination of powers of these roots. A double root d introduces a term of the form (j + 1)dj+1 to the general solution xj and so on for higher multiplicities. ¿From this it follows that there is an increasing, unbounded sequence of extents solving (16) if and only if fρ has a real root greater than 1. For, every complex number not on the real axis has infinitely many powers with negative real part and infinitely many powers with positive real part. Note that, for ρ1 > ρ0 > 1 and any z > 1, fρ1 (z) < fρ0 (z). Combined with the observation that fρ (z) → ∞ as z → ∞ for any real ρ, this shows that ρ1 > ρ0 > 1, fρ0 has a real root > 1 =⇒ fρ1 has a real root > 1. Moreover, it is easy to see that such a real root exists for large enough values of ρ. To find the minimum value of the parameter for which fρ has a root greater than 1, we solve (20) for ρ, yielding ρ=

z3 . z2 − 1

20

(21)

Equating the derivative dρ/dz to 0 yields the critical value zcrit =

√

3 > 1. Since ∂ρ/∂z is

negative on the interval (1, zcrit ) and positive on (zcrit , ∞), we know that √ ρcrit = ρ(zcrit ) = 3 3/2

(22)

is indeed the desired minimum value of ρ. It is easy to check directly that zcrit is a double root at the parameter value ρcrit . Let us summarize what we have shown so far and what remains to be done: √ 1. If ρ ≥ 3 3/2, we have shown that there is an increasing, unbounded sequence of extents solving (16). Since this recurrence was based on making the constraint (Ij ) tight for all j, one simply needs to verify that the other constraints of (Pk ) are verified in order for this sequence of extents to be legitimate. We will do that specifically for √ the critical value ρcrit = 3 3/2. This will show that there exists a strategy in Γincr √ with time-competitive ratio 1 + 3 3. √ 2. If ρ < 3 3/2, we have shown that there is no increasing, unbounded sequence of extents solving (16). In order to show that there are then no strategies with timecompetitive ratio of 1+2ρ, one still needs to show that the problem (Pk ) itself (without having (Ij ) necessarily tight) eventually becomes infeasible for large values of k. Let us consider both cases in turn. 4.6.2

√ The case ρ = ρcrit = 3 3/2

In that case the characteristic equation (20) becomes z −

√
2 √
3 z + 3/2 and the general

solution (xj )j≥0 to the associated recurrence (16) is then of the form xj = a + b(j + √ √
√ j+1
j+1 1) 3 + c − 3/2 , j ≥ 0. The initial conditions x0 = 3 3/2, x1 = 27/4, and √ x2 = 69 3/8 lead to a = 8/9, b = 6/9, and c = 1/9. In conclusion the sequence of search extents is given by x∗j = 8/9 + 6(j + 1)/9


√

j+1

3

+ (1/9) −

√


j+1 3/2 .

(23)

√ Note that this sequence is easily seen to be increasing and, since x∗0 = 3 3/2 > 1, it constitutes a legitimate sequence, feasible for all (Pk ), k ≥ 0. With that respect note that there are an infinite number of feasible strategies leading √ to a time-competitive ratio of 1 + 3 3. For instance, any sequence (x0j )j≥0 verifying the √ √ recurrence (16) and such that 1 ≤ x00 ≤ 3 3/2, x00 ≤ x01 ≤ 27/4, and x01 ≤ x02 ≤ 69 3/8 21

√ j+1 would work. As simple numerical examples one can take x0j = 3 or x00j = (1 + (j + √ j+1 1)/2) 3 . However, we will see in 4.6.4 that among all these strategies, the one given by (23) enjoys a specific and unique property. 4.6.3

√ The case ρ = 3 3/2 − ε

In that case the roots of the characteristic equation include a complex pair ω = α ± iβ, and a real root z3 ∈ (−1, 0) (for any ρ > 1, the characteristic equation has a unique real root in the interval (-1,0) via the intermediate value theorem and a simple check on the sign of the derivative on (−∞, 0)). The general real-valued solution to recurrence (16) can be found by p using the polar form of the complex pair, i.e., ω = t(cos θ ± i sin θ), where t = α2 + β 2 ,

and θ = arctan β/α. Note that ω n = tn (cos nθ ± i sin nθ), and since linear combinations of solutions are also solutions, we can use the independent real solutions tn cos nθ and tn sin nθ in order to express the general real-valued solution. In other words, we have for all j ≥ 0: xj (ρ) = atj+1 cos((j + 1)θ) + btj+1 sin((j + 1)θ) + cz3j+1 .

(24)

Using the initial conditions for x0 , x1 , and x2 , the unknowns a, b and c solve the following system of three equations: at cos θ + bt sin θ + cz3

= ρ

at2 cos 2θ + bt2 sin 2θ + cz32 = ρ2

As ρ → ρcrit

(25)

at3 cos 3θ + bt3 sin 3θ + cz33 = ρ3 − ρ. √ √ √ = 3 3/2, we have θ → 0, t → 3, and z3 → − 3/2, all in a continous fashion.

From the system (25), it is also not difficult, although somewhat laborious, to show that as √ ρ → 3 3/2, we also have a → 8/9, c → 1/9, and finally b sin((j + 1)θ) → 6(j + 1)/9, again in a continuous fashion. What we have thus shown is that for all j ≥ 0: lim xj (ρ) = x∗j .

ρ→ρcrit

(26)

As a consequence, for any fixed K, there exists a ρ sufficiently close to its critical value so
that the sequence xj (ρ) 0≤j≤K is increasing. Hence the maximum overall extent of this strategy approaches infinity as ρ → ρcrit .

Let us now show that for any ρ = ρcrit − ε, there exists a K such that the problem (PK ) is infeasible. Let us first explicitly rewrite the constraints x ∈ Fk = Fk (r). In addition to nonnegativity, these amount to the k + 1 constraints, 0 ≤ j ≤ k: xj ≤ ρxj−1 − (xj−2 + xj−4 + · · · + xj mod2 ). 22

Let ei denote the ith standard basis vector of Rk+1 (starting the indices at i = 0). Then these k + 1 constraints can be written in a matrix form as Bk x ≤ ρe0 , where Bk is as follows for k odd:  1 0 ... ...  −ρ 1 0 ...    1 −ρ 1 0   0 1 −ρ 1   0 1 −ρ 1   0 1 0 1   .. .. .. ..  . . . .   1 0 1 0  0 1 0 1

...

...

...

0

...

...

...

0

...

...

...

0

0

...

...

0

1

0

−ρ .. .

1 .. .

0 .. .

1

0

... .. . .. . .. .

0

1

...

0 1

 0  0   0   0   0  ..  .  ..  .   0 

−ρ 1

For k even, the form of Bk is similar. One can see for instance that the (k, 0) entry is 1 in this case (row and column indices each run from 0 to k). Deleting the first row and the first column of the above matrix gives the correct picture. By relaxing the last k constraints in (Pk ) the problem becomes: max

xk Bk x ≤ ρe0

subj to

−eT0 x ≤ −1

(Pk0 )

x ≥ 0. Note that if (Pk0 ) is infeasible then (Pk ) is infeasible. Farkas’ lemma says that Pk0 is infeasible iff the following system is feasible:  T  Bk | − e0

"

y

#

w

≥0

y

≥0

w  T  ρe0 | − 1

≥0

"

y w

23

#

< 0.

(F Dk0 )

Here y ∈ Rk+1 and w is a scalar. The last constraint reduces to ρy0 < w. For the system involving BkT , let us write it explicitly for k odd, the even case being analogous: y0 + y2 + y4 + . . . + yk−1 ≥ ρy1 + w y1 + y3 + y5 + . . . + yk ≥ ρy2 .. .

(27)

yk−2 + yk ≥ ρyk−1 yk−1 ≥ ρyk yk ≥ 0 Consider using back substitution, start with yk = ρ and replace all other inequalities in (27) by equalities. We then have yk−1 = ρ2 , yk−2 = ρ(ρ2 − 1) and for j = 1, 2, . . . , k − 4, k − 3: yj = ρyj+1 − ρyj+3 . Finally the first equality becomes: y0 = ρy1 − ρy3 + ω. Note that if we define xj = yk−j for all 0 ≤ j ≤ k − 1, we have (the first k terms of) the recurrence relation of Section 4.6.1.

√ For subcritical ρ arbitrarily close to ρcrit = 3 3/2, and appropriately chosen K, we can

0 as follows. Let then exhibit a feasible solution to the “Farkas dual” linear program F DK

K = K(ρ) be the smallest index j for which xj (ρ) < 0. (Here xj (ρ) is the generic solution of (24).) For 1 ≤ j ≤ K define yˆj = xK−j (ρ). By definition we then have yˆj > 0 for 1 ≤ j ≤ K, and ρˆ y1 − ρˆ y3 < 0. We can then choose yˆ0 = 0 and thus w = −(ρy1 − ρy3 ) > 0 0 (note that ρˆ to obtain a feasible solution for F DK y0 < w).

4.6.4

Optimality of the successive maximization solution

Consider the following relaxation of (Pk0 ) (and thus of (Pk )) in which we relax the constraint x0 ≥ 1: max subj to

xk Bk x ≤ ρe0 x ≥ 0.

24

(Pk00 )

Lemma 5. Let ρ > 1. If the successive maximization solution Bk x = ρe0 is feasible, i.e., such that x ≥ 0, then it is an optimal solution of (Pk00 ). Proof. Let x∗ be the solution of Bk x = ρe0 , and suppose tha x∗ ≥ 0. The dual (Dk00 ) of (Pk00 ) is min subj to

ρy0 BkT y ≥ ek

(Dk00 )

y ≥ 0. Consider the sytem BkT y ≥ ek , almost identical to (27). Rather than rewriting it, let us indicate that the w on the RHS of the first inequality of (27) is removed and that the RHS of the last inequality of (27) is 1 instead of 0. Then the procedure of turning inequalities into equalities leads to the same recurrence as before, via the same reasoning. If we now ∗ define yk−j = x∗j /ρ for all 0 ≤ j ≤ k, we then have BkT y ∗ = ek , y ∗ ≥ 0. Since x∗k = ρy0∗ , the

solutions x∗ and y ∗ are primal and dual optimal respectively. Remark 5. As a consequence of Lemma 5, the way to maximize the overall search extent until and including search k, for any k ≥ 0, is to maximize the local extent of all successive searches. Note that if xj (ρ) is the corresponding optimal solution of (Pk00 ) for a given ρ, then the overall search extent until and including search k is defined here as max0≤j≤k xj (ρ).
This corresponds to xk (ρ) if the sequence xj (ρ) 0≤j≤k is nondecreasing.
For a given ρ let k(ρ) be the smallest index k such that xj (ρ) 0≤j≤k is nondecreasing. If the searchers know that the exit is within a distance D ≥ 1 from the origin, the optimal

competitive ratio is 1 + 2ρ∗ , where ρ∗ = inf{ρ : xk(ρ) (ρ) ≥ D}, and the corresponding
optimal on-line strategy is given by xj (ρ∗ ) 0≤j≤k(ρ∗ ) . In the following table we have listed the optimal competitive ratio r∗ (D) for some values of D. Distance D

Optimal r∗ (D)

10

5.086

1000

5.901

10000

6.007

∞

6.196

Finally, it is in this sense that the successive maximization solution (23) when ρ = ρcrit is √ unique among all on-line strategies with optimal competitive ratio 1 + 3 3: It is the one that “covers the most ground during each search attempt”. 25

4.7

Time-competitive ratio for the general m-branch, p-searcher problem

One can easily extend the ideas contained in section 4.6.1. For general m and p, the following recurrence relation is analogous to the one given for the 3-branch, 2-searcher problem: xj = ρxj−m+p − ρxj−m .

(28)

The characteristic equation of the recurrence relation is now: fρ (z) := z m − ρz p + ρ = 0.

(29)

To find the minimum value of the parameter for which fρ has a root greater than 1, we again solve (29) for ρ, yielding ρ=

zm . zp − 1

Equating the derivative to 0 yields the critical value zcrit = ρcrit = ρ(zcrit ) =

(30)
m 1/p m−p

> 1, and

m − p h m im/p p m−p

(31)

is indeed the desired minimum value of ρ. It is easy to check directly that zcrit is a double root at the parameter value ρcrit . The generalization of sections 4.6.2, 4.6.3 and 4.6.4 is technically more challenging and requires various intermediate results, some of which we have verified numerically but not proved formally. Let us first mention an easy special case, for which optimality results can easily be obtained: Special case: m is a multiple of p, m = np: ¿From Lemma 4, searcher j will always visit the same branches, and no other searchers will visit these branches. In other words the set of m branches is partitioned into p sets of n branches, each set being searched by a unique searcher. One can then apply the results of Section 2. The overall time-competitive ratio will then be the same as the optimal competitive ratio for the n-branch, 1-searcher problem, i.e.:  n n−1 r∗ (n, 1) = 1 + 2n , n−1 which is what Theorem 2 gives when one replaces m by np. All the other optimality results for the bounded case of the single searcher easily apply to these special cases as well. Let us now assume that m is not a multiple of p but that they have a common divisor greater than 1, say k (i.e. m and p are not relatively prime). It should then be clear that 26

one can partition the “(m, p) problem” into k independent “(m/k, p/k) sub-problems”, and run k independent optimal strategies, one for each of the k sub-problems. Let us then assume from now on that m and p are relatively prime. Whenever the number of searchers, p, is one less than the number of branches, m, there is a well-defined “successive maximization” solution to the recurrence relation (28): it is possible to satisfy x0 ≤ ρ, and all subsequent inequalities, as an equality. Whenever m exceeds p by more than one, this is not the case, at least if we seek a solution (to the sequence of linear programs or to the associated recurrence) in which all terms are strictly positive. Therefore we will try for the next best thing: a solution to the sequence of linear programs in which the first p+1 constraints that can be tight, are tight. Clearly, the recurrence (28) requires p + 1 initial conditions. Once we have p + 1 consecutive tight constraints, we can satisfy all subsequent constraints as equalities by following the recurrence. For the solution to the recurrence to take the desired form, a necessary condition is that there are p − 1 characteristic roots smaller in modulus than the double root zcrit . These roots can then be used in conjunction with the double root for a solution in which the double root dominates asymptotically. This line of reasoning led us to investigate the root structure of the characteristic polynomial for small values of m and p. (Note that, for the case p = m − 1, our solution will end up using all of the roots.) Our experiments (using MATLAB) have confirmed (at least for small m and p) the following conjecture: Conjecture 3. Whenever m and p are relatively prime, the characteristic polynomial (29) has at least p − 1 roots of modulus smaller than the modulus of the double root zcrit Moreover in all our numerical experiments, solving the recurrence using the double roots and these p − 1 other roots, led to a valid solution xj whose asymptotically dominant term corresponds to the double root.

5

Concluding remarks

The effect of introducing additional (parallel) searchers to on-line search problems depends on the choice of the measure of effectiveness for a search strategy. If one chooses the worst case distance-competitive ratio criteria, then additional searchers do not bring as big of a reduction as for the time-competitive ratio criteria. For example from an optimal ratio of 14.5 for the 3-branch, 1-searcher problem, we obtain an optimal distance-competitive ratio

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of 9 for the 3-branch, 2-searcher problem. On the other hand the time-optimal competitive √ ratio for the 3-branch, 2-searcher problem becomes 1 + 3 3 = 6.196. Note that in this last case, we reduce the time-competitive ratio by a factor of more than 2 when we double the number of searchers. Also on-line strategies greatly differ depending on the criteria. For distance-related criteria, it seems better to have one searcher do most of the work. For time, an even load is more appropriate. In case of distance, we conjecture that partitioning strategies are optimal. We have shown in this paper that this is not necessarily the case for the time criteria.

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