Directed File Transfer Scheduling

Transcription

1 Directed File Transfer Scheduling Weizhen Mao Deartment of Comuter Science The College of William and Mary Williamsburg, Virginia Abstract The file transfer scheduling roblem was introduced and studied by Coffman, Garey, Johnson, and LaPaugh. We extend their model to directed networks and study the comlexity of the roblem under different conditions. We discover that even though the general roblem is still NP-comlete, adding directions in the file transfer grah makes the roblem easy in the sense that the roblem in some secial cases is solvable in olynomial time while its undirected counterart was roved NPcomlete. Introduction The roblem of scheduling file transfers in a network so as to minimize the makesan of the schedule was first introduced by Coffman et al. []. In their model, an instance of the roblem consists of a weighted undirected multigrah G = (V, E), called a file transfer grah. The vertices of G corresond to the nodes in the network, and the edges corresond to the files to be transferred between the nodes. Each vertex v in V is labeled with a ositive integer (v), which denotes the maximum number of file transfers that v can engage in simultaneously. Each edge e in E is labeled with a ositive real t(e), which denotes the amount of time needed to transfer that file. Under the assumtions that each file is transferred directly between the vertices that are its endoints, i.e. forwarding is not allowed, and that once the transfer of a file begins, it continues until it comletes, i.e. reemtion is not allowed, we are asked to schedule the file transfers in G so that the makesan, which is the time interval between the beginning of the first transfer and the comletion of the last transfer, is minimized. This roblem is called File Transfer Scheduling (F T S). In this aer, we extend their model to directed file transfer grahs. Let G = (V, A) be a digrah with multi-arcs allowed. For each vertex v in V, s(v) reresents the maximum number of file transfers that v can engage in as a sender, and r(v) reresents the maximum number of file transfers that v can engage in as a receiver. For each arc a in A, denoted by (u, v) if it goes from vertex u to vertex v, t(a) reresents the amount of time needed to transfer the file from the source u to the destination v. We also assume that forwarding and reemtion are not allowed. The Directed File Transfer Scheduling (DF T S) is to find a schedule of the file transfers such that the sending and receiving constraints are resected and the makesan is minimized. We organize this aer as follows. In Section, we will resent a olynomial-time algorithm for the roblem when all file transfers require the equal amount of time. In Section 3, we will study the case when the file transfer time is arbitrary. And finally, we make the conclusion in Section 4. Without confusion, we shall also use F T S and DF T S to denote the corresonding decision roblems. Equal file transfer time Permission to coy without fee all or art of this material is granted rovided that the coies are not made or distributed for direct commercial advantage, the ACM coyright notice and the title of the ublication and its date aear, and notice is given that coying is by ermission of the Association for Comuting Machinery. To coy otherwise, or to reublish, requires a fee and/or secific ermission. c ACM /93/0003/99 $.50 In this section, we assume that all files in the file transfer grah require the same amount of time to be transferred from their sources to their destinations. Without loss of generality, we assume t(a) =, a A. We define the following grah coloring roblem: Given a digrah G = (V, A), with each vertex v being weighted by two osi- 99

2 tive integers s(v) and r(v). We are asked to use the minimum number of colors to color the arcs in G such that for any vertex v, at most s(v) outgoing arcs have the same color and at most r(v) incoming arcs have the same color. It is obvious that there is a direct reduction from DF T S with equal file transfer time to this roblem, which indicates that if there is a olynomial-time algorithm for the grah coloring roblem, then there is a olynomialtime algorithm for DF T S with equal file transfer time. Lemma Given a digrah G = (V, A). For any v V, let in(v) be the number of arcs that end at v, and out(v) be the number of arcs that start at v. Let D(G) = max v V { in(v)/r(v), out(v)/s(v) }. Then the minimum number of colors needed to color G is D(G). Proof Mao [4] has roved the lemma for s(v) = r(v) =. We are going to generalize the roof to arbitrary s(v) and r(v). It is clear that in order to color G we have to use at least D(G) colors. To rove the lemma, we need to show that D(G) colors are enough, i.e. G is D(G)-colorable. We rove this by induction on the number of arcs. When A =, D(G) =. G is obviously -colorable. Assume that G is D(G)-colorable when A = n. Suose A = n. Define G to be G with one arc removed. Without loss of generality, we assume the arc is (, ). Therefore, G = (V, A ), where A = A {(, )}. Because A = n, G is D(G )- colorable by inductive hyothesis. After we color G using D(G ) colors, (, ) is the only uncolored arc in G. We will study the following cases and show that there is always a way to color (, ) such that only D(G) colors are used for G. Case. If D(G) = D(G ) +, then we can color (, ) with a new color. Therefore, G is D(G)- colorable. Case. If D(G) = D(G ), we need to rove that (, ) can be colored without using a new color. Let C be the set of colors used in G. Let C k be the set of colors used r(k) times for the arcs coming in to vertex k and C k+ be the set of colors used s(k + ) times for the arcs going out from vertex k +. Consider the following coloring rocedure. Ste i. C C and C C, otherwise case haens. If (C C ) (C C ) φ, then there exists some color in C, but not in C and C, which can be used to color (, ). Therefore G is D(G)- colorable. If (C C ) (C C ) = φ, then there exists a color, not in C (since C C), but in C (otherwise (C C ) (C C ) φ). Let (3, ) be the arc which comes in to vertex and is colored with. Uncolor (3, ), and color (, ) with. The situation is now as deicted in Figure. We need to modify C such that it includes if is used s() times at vertex, and modify C 3 such that it excludes. It is clear that C is unchanged. Figure Ste ii. C C (since C is unchanged in ste i) and C 3 C (since C 3 ). If (C C ) (C C 3 ) φ, then there exists some color in C, but not in C and C 3, which can be used to color (3, ). Therefore, G is D(G)-colorable. If (C C ) (C C 3 ) = φ, then there exists a color, not in C (since C C), but in C 3 (otherwise (C C ) (C C 3 ) φ), and (since C 3, C 3 ). Let (3, 4) be the arc which goes out from vertex 3 and is colored with. Uncolor (3, 4), and color (3, ) with. See Figure for this situation. Before we go to next ste, modify C such that it includes if is used r() times at vertex, and modify C 4 such that it excludes. C 3 is unchanged in this ste. 4 Figure Ste iii. Similar to ste i, if (C C 3 ) (C C 4 ) φ, then G is D(G)-colorable. If (C C 3 ) (C C 4 ) = φ, then C 3, but C 4. Let (5, 4) be the arc which comes in to vertex 4 and is colored with. Uncolor (5, 4), and color (3, 4) with. Finally, modify C 3 and C 5. Ste iv. Similar to ste ii, if (C C 4 ) (C C 5 ) φ, then G is D(G)-colorable. If (C C 4 ) (C C 5 ) =

3 φ, then C 4, but C 5. Let (5, 6) be the arc which goes out from vertex 5 and is colored with. Uncolor (5, 6), and color (5, 4) with. Finally, modify C 4 and C 6. Reeat the rocedure described in ste iii and ste iv and in each ste always uncolor the arc that has never been uncolored before, until either G is D(G)-colorable due to the fact that the two endoints of the uncolored arc have a common usable color or all the available arcs have been uncolored before. We claim that in the second case G is still D(G)-colorable. k According to Coffman et al. [] and later Choi et al. [], F T S with equal file transfer time is NPcomlete even though it becomes olynomial when the file transfer grah is a biartite grah, tree, ath or cycle. From the above discussion, we find that DF T S with equal file transfer time is solvable in olynomial-time for arbitrary directed file transfer grahs. Intuitively, this is because that the direction constraint imosed on the grah and the relaxed constraint of dual vertex caacities eliminate many of the ossible cases that need to be considered in the exhaustive search of the otimal schedule, which as a result changes the roblem from NP-comlete to olynomial. k+ k 3 Arbitrary file transfer time Figure 3 Without loss of generality, we assume as shown in Figure 3 that we come to a oint to color (k, k + ) which was colored with and was later uncolored because was given to (k, k ). Define A to be the set of the arcs that end at k + and are colored with. According to the coloring rocedure, we intend to uncolor one arc in A, and color (k, k+) with. Since all arcs in A have been uncolored before, the old color of them must be. It is clear that the old color of (k, k + ) is also. So we have r(k + ) A +. Therefore, C k+, and C k. We can color (k, k + ) with. This comletes the roof. Theorem DF T S with equal file transfer time is solvable in time O( A ). Proof First, let us consider the corresonding grah coloring roblem. Define G = (V, A ), where A = {(a, b)} for some (a, b) in A, and G i = (V, A i ), where A i = A i {(a, b)} for some (a, b) in A but not in A i. Obviously, G A = G. Using the method described in the roof of the lemma, we can color G, G,..., G A in the order given. Assume T (n) is the time required to color a grah with n arcs. We have T () =, and T (n) = T (n ) + n. So T (n) = O(n ). Therefore, any digrah G = (V, A) can be colored in time O( A ). In DF T S, if G is the file transfer grah, then the minimum makesan is D(G), and the schedule can be constructed in time O( A ). Let us consider DF T S with arbitrary file transfer time. We use the aroach similar to that in [] to discuss the comlexity of the roblem for different grah structures, such as cycle, ath, tree, biartite grah, and general grah. Theorem in time O( A ) if the file transfer grah is a directed cycle, and either s(v) = r(v) = for any v V or no multi-arcs are resent. Proof Let V = {,,..., n} and A = {(, ), (, 3),..., (n, n), (n, )}. If s(v) = r(v) = for any v V, and multi-arcs may resent in G, the files between any two adjacent vertices in the cycle have to be transferred sequentially, and files not sharing sources and destinations can be transferred simultaneously. For (u, v) A, let T (u, v) be the sum of the transfer time of files from u to v. Then, the minimum makesan is max (u,v) A {T (u, v)}. If s(v) and r(v) are arbitrary ositive integers, and multi-arcs are not allowed, then the minimum makesan is max a A {t(a)}. In both cases, the schedule can be easily constructed in time O( A ). Corollary. in time O( A ) if the file transfer grah is a directed ath, and either s(v) = r(v) = for any v V or no multi-arcs are resent. Proof If s(v) = r(v) = for any v V, the minimum makesan is obviously max (u,v) A {T (u, v)}. If s(v) and r(v) are arbitrary ositive integers, and 0

4 multi-arcs are not allowed, the minimum makesan is max a A {t(a)}. Theorem 3 DF T S with arbitrary file transfer time is NPcomlete if the file transfer grah is a directed cycle with multi-arcs and arbitrary s(v) and r(v). Proof Consider the NP-comlete Multirocessor Scheduling (MS) [3], in which we are given n indeendent tasks with rocessing time,..., n, resectively, and are asked to schedule the tasks on m( ) identical rocessors so that the makesan of the schedule is minimized. We can define a olynomial transformation from MS to DF T S as follows. Let V = {,,..., k} and (i, i+) A with t(i, i+) = for i =,,..., k. From k to, add an arc for each task in MS and label the arc with the rocessing time of the task. Let r() = m and s(k) = m. Files (, ),..., (k, k) can be transferred simultaneously, and at any time m of the n files from k to can be transferred. It is not hard to see that MS has a schedule with makesan B or less if and only if DF T S of the grah in Figure 4 has a schedule with makesan B or less. r()=m.... n k Figure 4 k s(k)=m Since DF T S is in NP and there is a olynomial transformation from an NP-comlete roblem to it, DF T S in this secial case is NP-comlete. Corollary 3. DF T S with arbitrary file transfer time is NPcomlete if the file transfer grah is a directed ath with multi-arcs and arbitrary s(v) and r(v). Proof Similar to the roof of Theorem 3 excet that the file transfer grah is the grah in Figure 4 with the arcs (, ), (, 3),..., (k, k) being removed. Theorem 4 in time O( A ) if the file transfer grah is a directed tree and s(v) = r(v) = for any v V. Proof Without loss of generality, assume that the directions of arcs are from arents to children. Files sharing the source have to be transferred sequentially, and files not sharing the source can be transferred simultaneously. Let T (v) be the sum of the transfer time of files to be transferred from v, then the minimum makesan is max v V {T (v)}. Theorem 5 DF T S is NP-comlete if the file transfer grah is a directed tree with arbitrary s(v) and r(v). Proof Consider the transformation from MS. Assume that n tasks have to be scheduled on m( ) rocessors. Define G = ({,..., n + }, {(n +, i) : i =,..., n}), where t(n +, i) = i for i =,..., n, and let s(n + ) = m. Since at any time only m of n files can be transferred out from n +, MS has a schedule with makesan B or less if and only if DF T S of the grah in Figure 5 has a schedule with makesan B or less. Theorem 6 n+ s(n+)=m... n n n Figure 5 DF T S with arbitrary file transfer time is NPcomlete if the file transfer grah is a directed biartite grah. Proof According to Coffman et al. [], F T S of a biartite grah with single orts ((v) =, v), single edges, and arbitrary file transfer time is NPcomlete. Given a biartite grah G = (V V, E), where each edge has one vertex in V and one vertex in V, consider the corresonding directed biartite grah G by adding direction from u to v if (u, v) E, and u V, v V. Assume that in G s(v) = r(v) = for any v V V. We notice that in the schedule for G when file (u, v) is being transferred no file from source u or to destination v can be transferred, which haens to be the exactly same case in the schedule for G. Therefore, G has a schedule with makesan B or less if and only if G has a schedule with makesan B or less. Corollary 6. DF T S with arbitrary file transfer time is NPcomlete. To summarize, Table and Table show the comlexity results for F T S [] and DF T S, resec- n 0

5 tively, when the file transfer time is arbitrary. To kee the tables small, we use the following abbreviations: G =general grah or digrah, B =biartite grah or digrah, T =tree, P =ath, E =evenlength cycle, O =odd-length cycle, and finally, Single means that the file transfer grah has only single edges or arcs, and Multile means that multi-edges or multi-arcs are allowed. (v) = Arbitrary (v) Single Multile Single Multile G NPC NPC NPC NPC B NPC NPC NPC NPC T NPC NPC NPC NPC P O( E ) O( E ) O( E ) NPC E O( E ) O( E ) O( E ) NPC O O( E ) NPC O( E ) NPC Table s(v) = r(v) = Arbitrary s(v), r(v) Singles Multile Single Multile G NPC NPC NPC NPC B NPC NPC NPC NPC T O( A ) O( A ) NPC NPC P O( A ) O( A ) O( A ) NPC O O( A ) O( A ) O( A ) NPC E O( A ) O( A ) O( A ) NPC 4 Conclusion Table In this aer, we have studied the comlexity of the Directed File Transfer Scheduling under different conditions. We have roved that if all files have the same transfer time the roblem can be reduced to a grah coloring roblem which has a olynomial solution, and that if files have arbitrary transfer time the roblem is NP-comlete for general digrahs, biartite digrahs, directed trees with arbitrary s(v) and r(v), directed aths or cycles with multi-arcs and arbitrary s(v) and r(v). In comarison with the results of undirected File Transfer Scheduling, we have found that adding directions to the file transfer grah sometimes makes the roblem easy. As its undirected counterart, there are a wide variety of directions for future research in Directed File Transfer Scheduling. Coffman et al. [] roved for F T S that the List Scheduling (LS) heuristic gives a schedule of makesan at most 3 times that of the otimal schedule and that Decreasing List Scheduling (DLS) heuristic gives a schedule of makesan at most.5 times that of the otimal schedule. Within our model, an interesting question we see is to analyze the worst-case erformance bounds of LS and DLS when they are alied to DF T S with arbitrary file transfer time for general digrahs and see whether they can achieve better bounds on DF T S than on F T S. Whitehead [5] studied the comlexity issue of F T S with forwarding. An imortant extension of our model is to relace the s(v) and r(v) constraints with a network grah which contains ossible routes from sources to destinations via intermediate nodes. In this modified model, in order for a file in the file transfer grah to be transferred from u to v, we must find a ath from u to v in the network grah for it. Acknowledgement I wish to thank Rahul Simha for bringing the original file transfer scheduling roblem to my attention, Adam Rifkin for heling me with the L A TEX format, and the three anonymous reviewers for their valuable suggestions. References [] H.A. Choi and S.L. Hakimi, Scheduling File Transfers for Trees and Odd Cycles, SIAM J. on Comut., 6 (987), [] E.G. Coffman, Jr., M.R. Garey, D.S. Johnson and A.S. LaPaugh, Scheduling File Transfer, SIAM J. on Comut., 4 (985), [3] M.R. Garey and D.S. Johnson, Comuters and Intractability: A Guide to the Theory of NP- Comleteness, W. H. Freeman and Comany, San Francisco, 979. [4] W. Mao, Arc-Coloring of a Digrah is Polynomial, Technical Reort, Deartment of Comuter Science, College of William and Mary, 99. [5] J. Whitehead, The Comlexity of File Transfer Scheduling With Forwarding, SIAM J. on Comut., 9 (990),