Scheduling content multicast trees

with minimal repetitions

Sandford Bessler Lukas F. Lang Nysret Musliu FTW Telecommunications Research Center Vienna

Email: bessler@ftw.at Vienna University of Technology

Email: lukas.lang@tuwien.ac.at Vienna University of Technology Email: musliu@dbai.tuwien.ac.at

Abstract-This paper discusses an approach for distributing a large number of multimedia (video) titles to users located at the leaves of a tree network. The proposed approach is to build multicast trees, over which the content files are sent to user disks, followed by delayed local playback of the content. The work concentrates on exact and heuristic methods for an efficient tree packing in several download periods. In order to satisfy all the requests accumulated in a certain time period, we choose to minimize the number of times a multicast tree is split because of capacity constrains. We show that this objective provides lowest link utilization and good convergence both in the integer formulation and in the developed heuristics (of type tabu search, iterated local search, and variable neighborhood search). The model can be also used in a planning setting to answer the question on the maximum user request rate that can be supported by a certain network topology.

Keywords: content distribution, integer programming, tree packing, multicast trees, metaheuristics, iterated local search, variable neighborhood search.

I. INTRODUCTION

Despite the continuous expansion of backbone and access broadband infrastructure, an internet wide deployment of video on demand service with full size movies poses a formidable challenge for network operators. Meanwhile, most cable and network providers have adopted a closed model that serves the own user population, it is based on a relatively small number of local and popular content titles, but enables the full control of the content distribution process. On an internet scale, among the P2P approaches for content distribution networks (CDN) the most successful one uses the BitTorrent technology [3]. From a network operator perspective however, the performance level of such a system cannot be managed.

In this work we refer to the following content distribution scenario: large content files (movies) are downloaded to the user disks, whereas the playback occurs locally, at a later time. The most efficient technology today for this scenario is BitTorrent: instead of downloading the same content from the server for each user again and again, it is theoretically possible to do it once, the further distribution being performed autonomously by the participating peers. Is there another possibility to achieve this download efficiency of BitTorrent and provide in this way a scalable solution with respect to the number of titles ? The solution proposed in this paper takes into consideration the actual available link capacities as planned by the ISP and builds multicast trees in each of several

978-1-4244-6705-111 0/$26.00 ©20 1 0 IEEE

download time periods. When several content trees compete for the same link capacity, some users in the tree cannot be satisfied, therefore they have to be served in another time period, meaning that the content has to be sent again from the server through the whole network. Our contribution is to explore and compare efficient tree packing strategies: the chosen objective to be minimized is the number of times a multicast tree (for a certain content t) is split into subtrees. The number of time periods is a quality parameter, not too large, otherwise the waiting time for delivery would be unnecessarily high. In an operational system the planning would thus follow a rolling horizon approach.

A. Related work

Recently, experiments were performed with a so called push-VODI in which the requested content is downloaded directly to the user disks during the low utilized network hours (at night), and using multicast transmission. The video content (that can be in this case also high definition content) is played later on, locally. Commercial DVD rental systems such as Netflix2 have a similar model, selling a package with limitation on the number of titles per week or viewing hours per week.

The key for a scalable video on demand service is the use of multicast, a technology extensively studied in the last 20 years, in particular in the context of IPTV streaming and multiparty conferencing. Most optimization problems that involve multicast algorithms have been focused on routing and caching at the intermediate nodes (see [16], [11] for surveys on multicast optimization problems and VoD services). Specifically, the allocation of multicast groups to existing bandwidth in a network can be modeled either as a bin packing problem [8], or as a multiple knapsack problem [2].

In meshed network graphs, the construction of minimal costs multicast trees is always related to finding the multicast nodes, or the so called Steiner points. However, in a hierar­chical network, if the routing to the terminals is kept fixed, the problem is simpler, as a tree is determined only by the multicast group definition. In [7] the authors formulate a graph theoretical Steiner tree packing problem that aims at building a maximal number of edge-disjoint (therefore coexisting) Steiner

lwww.stratacache.com 2www.netflix.com

trees. In [2] and [8] no admission control is performed, i.e. it is assumed that a feasible solution exists and a multicast packing problem is solved. The objective in [2], [8] is the minimization of the total load of the most congested link, where the link load is the sum of the traffic demand of all multicast groups that traverse that link.

The admission control form of the multicast routing problem has been extensively studied both in the offline and the online version. The algorithms in [1] and [6] have in common that they try to merge components (subtrees) starting from the bottom of the tree upwards, in such a way that the merging of two components provides the maximum profit, meaning that the resulting tree serves most users by using least additional resources. In this case the common links of two multicast subtrees express the savings in the resources.

In contrast to the aforementioned works, we investigate the scheduling over several time periods, as the requests have been accumulated and should be packed optimally. All requests must be allocated. One naive solution approach would be to minimize the number of time slots (bins), which corre­sponds to a multidimensional bin packing problem. Besides the discontinuities in the "number of bins" objective function, the resulting variable number of periods implies a variable, uncontrolled delivery time (and quality of service). Our idea for a fine-grained objective function is based on the following observation: any splitting of a content tree within the schedul­ing horizon is a waste of bandwidth resources, therefore minimize the number of splits.

The remaining of the paper is structured as follows: In Section II an integer formulation is given, Section III describes several local search procedures and defines appropriate neigh­borhoods for local search. In Section IV we give numerical results and compare the performance of exact and heuristic solutions approaches. We conclude and point to further re­search questions.

II. INTEGER MODEL OF THE TREE PACKING PROBLEM

We assume that the content consists of a collection of video titles that follow a Zipf distribution on their popularity [11], [6]. Denote with E the set of links used in the distribution network, and including the access links to the user terminals. The simplifying assumption we make is that the routing path from the server to each terminal is fixed and known. There can exist several media servers, as long as it is clear which server serves a given user. This setting is typical for a hierarchical distribution network, as found at most DSL network providers. As a consequence of the fixed routing, the matrix r is given.

The approach aims at minimizing the redundancy caused by retransmitting a content, at least for the duration of S periods. Table I summarizes the used notation.

To simplify the scheduling problem, we set the download duration of any title to the time slot length, so that, in order to accommodate different content sizes, the download rate of each content dt, t E T is calculated accordingly. For each time slot we solve therefore a knapsack problem over the link

E set of edges S set of periods U set of users Q set of content requests T set of titles found in Q u(q) user the issues request q

k ( q) title index in request q dt download rate for content t E T xg variable: I if the q-request is satisfied in period s z[ variable: 1 if the content t is scheduled for period s

rt' routing matrix: I if title t is routed on edge e c; capacity of edge e in period s

Table I NOTATION

capacity resources (3). We require in (2) that each request has to be served and be allocated to some period s E 8.

The integer program formulation of the repeated content download problem (RCDP) is easily written.

RCDP: min L z%

tET,sES

LX� = 1,q E Q sES

Lr�dtz%::::: c�,Ve E E,s E 8 tET

z%::::: L X�,VtET,SE8 qEQ\k(q)=t

(1)

(2)

(3)

(4)

z%'21/(8·T) L X�,VtET,SE8 (5) qEQ\k(q)=t

The last two constraints (4) and (5) express the following dichotomy: a content tree variable zt is set true, only if any request for that content t is allocated in period s.

In contrast to bin-packing problem, we keep the number of bins fixed but aim to minimize the number split trees. T is the number of actual requested titles which is much smaller than the number of existing titles, and of course smaller than Q. The problem size in terms of storage required for the variables is therefore proportional to (ITI + IQI) 181 R::: Q . 8.

One result concerning the solution properties of the problem is given in the following

Lemma In general, the optimal solution of problem RDCP contains split trees.

Proof For the proof we construct the following instance: each of two users requests the same 2m titles, where each title requires one unit of capacity. A feasible solution with the link capacities is shown in the Figure 1. Following inequalities hold: j < k < m. It is easy to see that j titles cannot be multicast in the s = 1, therefore they are sent in s = 2. Moreover all solutions with j, j + 1, ... , k tree splits are feasible, and the optimum solution has j splits (there are

actually ( 2; ) solutions) with the same objective 2m + j.

time stot S, time slot s,

Figure I. Split solution for 2m multicast trees, two users and two time slots.

III. METAHEURI STIC FORMULATION OF RDCP

In this section, we present a metaheuristic formulation of the RCDP and show the application of local search techniques such as Tabu Search, Variable Neighborhood Search and Iter­ated Local Search. Firstly, the objective function and secondly various neighborhood structures will be defined. Finally, the metaheuristics are compared with each other and with the exact ILP solution in Section IV.

A. Evaluation function

In order to evaluate a possible solution candidate x E X which was explored in the neighborhood, we define the polynomial f : X --+ lR as evaluation function to express the fitness of the candidate. As the heuristic implementation also allows non-feasible assignments w.r.t. to the constraints (2)-(5), the evaluation function is designed to consist of three parts: The first part aims at reducing the overall congestion, as defined in Equation (3). The congestion A� on link e E E in period s E S is defined as

A� = L rfdtz% tET

and the overall exceeding can be written as a function e X --+ lR as

e(x) = L max{O, A� - ee }, eEE,sES

which should be minimized. Moreover, we want to experiment with variable number of periods lSI needed to broadcast all requests. The third part of the objective, which is the number of times a title is repeatedly distributed, should also be kept as small as possible. After adding a multiplicative constant to each part to allow a fine-grained control of the optimization procedure, the objective function f can now be written as the sum of the the three weighted parts as

minf(x) = o:e(x) + ,BISI + , L zf. (6) tET,sES

The objective consists of conflicting criteria so that by reducing the number of periods, the splitting of large trees in favor of avoiding congested links pays off. Therefore, a title is distributed more often and also the network utilization increases as less edges can be shared by multicast trees. It is highly dependable on an operator's network and cost

structure how to set the weights properly. Two major cases can be distinguished: A low number of repetitions reduces the average utilization of the network and increase the user's waiting time for a title. Approaching a high operating level in terms of working load will automatically increase the number of multicast trees to a certain point. Assuming that lSI is fixed and the metaheuristic yields a solution with e(x) = 0, then f is equivalent to the ILP objective defined in Section II.

As every request has to be fulfilled within some period, a lower bound for f can be calculated. Assuming that there exists an optimal assignment, i.e. the number of repetitions is zero and all multicast trees can be scheduled to one period without bandwidth exceeds, the lower bound of an instance of the RCDP is

LB(x) =,B + ,ITI·

In the next section, we will introduce the proposed meta­heuristics and appropriate neighborhood structures to take care of the different characteristics of the objective.

B. Local search

In the last section, we discussed the objective of the problem. The goal is find an assignment within an admissible number of computational steps which is globally optimal with respect to the evaluation function f : X --+ lR where X is the set of solutions. In [9] we have shown that the problem is NP-hard and therefore we propose the application of local search techniques such as Tabu Search, Variable Neighborhood Descent and Iterated Local Search to the problem and show that even for larger input sizes acceptable running times can be achieved. The minimization variant of a general optimization problem can be written as min f (x). Furthermore, a general neighborhood function is defined as N : X --+ P(X) where P(X) = {U I U � X} is the power-set of X. Consequentially, Nm(x) denotes the set of neighbors of x which can be reached by a move m and is further referred as to a neighborhood structure Nm.

Tabu Search was first proposed by Fred Glover [5] and targets at exploring a wide range of neighbors by "locking" recently accepted neighbors in order to escape local op­tima [12]. Our implementation uses a Recency Based Memory

so that a move is considered tabu if it tries to relocate a tree which was part of a move within the last t iterations (t is determined experimentally). Nevertheless, we endorse strong improvements by accepting solutions which are tabu but better than every solution found so far.

Furthennore, we implemented Iterated Local Search (ILS)

which is discussed by Louren<;o, Martin and Stlitzle [10] and brings the ability to diversify the search procedure. Starting from an initial solution, a local search procedure is done. Later, a number of p random perturbations (p is experimentally determined) from the neighborhood were applied, a local search perfonned and a local minimum obtained. Assuming that the "jump" and the subsequent search lead to an overall improvement, the candidate is always accepted if it is the best so far in terms of fitness. The power of ILS lies in

both its simple implementation and the fast execution, as the perturbations can be generated and applied very fast.

Moreover, we applied Variable Neighborhood Descent

(VND) [4] which is a variant of Variable Neighborhood Search

[14]. The heuristic exploits the fact that a local optimum w.r.t. a neighborhood structure is not necessarily optimal to another neighborhood, but a global optimum is optimal W.r.t. all neighborhoods. It starts from the smallest neighborhood and iteratively increases it by exploring the next larger neigh­borhood structure. However, the ordering of the exploration N1 C N2 C . . . c Nmax is crucial and is chosen in such a way, that smaller and computationally less complex neighborhoods will be searched first.

C. Neighborhood structures

1) Merge neighborhood: As the name reveals, the merge

neighborhood consolidates two or more multicast trees broad­casting the same title from either the same period S or from multiple periods Sl, S2," ., Sn E S. Both aim at improving the objective by either decreasing the number of repetitions of the respective title or by reducing the congestion through shared edges. Imagine an assignment that violates the capacity constraint (3) on some upper edge e in period s. By merging two trees, the congestion on edge e decreases if both trees were already scheduled for the same period. In case of merging trees from different periods, the congestion of e stays the same if the new larger tree is scheduled for one of the two periods. If the new tree is assigned to another period, the congestion of e decreases in both source periods. In any case the number of trees in the assignment is reduced. In Section III-A, we consider the number of repetitions as a measure of fitness. Therefore, three major cases for the size of the neighborhood can be distinguished: a) all repetitions arise from different titles, b) only one title is broad casted mUltiple times, or c) anything in between. Case a) gives an upper bound for the size, namely O(IT x SI - ITI) neighbors.

Unfortunately, merges of large trees are likely to produce further capacity exceeds, which are to be handled by the min­conflict neighborhood:

2) Min-coriflict neighborhood: In order to address a very contributive part of the evaluation function, namely the number of exceeding edges and the accumulated overflow, we will introduce another neighborhood structure. We call this the min-coriflict neighborhood as it tries to repair non-feasible assignments [13]. The neighborhood yields candidates by first identifying conflicting edges and in further consequence splitting trees which participate to this conflict. First, we define Fs � E as the set of edges so that for every edge in Fs constraint (3) is violated in period s, i.e. the sum of demands of multicast trees traversing that edge exceeds its capacity:

W F "" td s s v e E s: � re tZt > ceo

tET (7)

Subsequently, from all multicast trees assigned to period s, neighbors were generated by removing users which have a path containing an exceeding edge e E Fs . This operation is

referred to as subset move as it splits a tree into two disjoint trees and the new subtree is subject to reassignment to another period. A more descriptive way to think of subtrees would be graph connectivity. Assuming that we remove a conflicting edge from a tree scheduled for period s, a new component would be created. The resulting component corresponds to the (maximum) subtree, which again has to be connected to the source to yield valid paths to the leafs. The resulting subtree is now scheduled for another period.

i]EF i,' 'l = 0, 0

\ ( 1 \ 1 \ \

o 00 00 0

Figure 2. Conflict Heuristics

)

Figure 2 illustrates a possible outcome of the min-conflict structure. Edge e is indicted as conflicting and therefore Ti will be split into two disjoint trees T2 and 73. From the example shown, we can conclude that there exists at least one other tree issuing demand on edge e, which is also considered as a possible candidate for the subset operation. Furthermore, this neighborhood structure yields numerous candidates including shifts and swaps of conflicting trees to other periods, as further described in the next section.

3) Shift neighborhood: The two previously described neighborhoods aim at constructing larger trees and at reducing the number of exceeds by splitting. No neighborhood has yet been defined to allow the reassignment of a complete tree from period Si to another period Sj. Such shift operations lead to re-ordering of tree assignments and can be crucial to escape from local optima. The shift neighborhood Nshift yields neighbors of a candidate x E X such that a tree being assigned to period Si in x is scheduled for another period Sj in y E NShift(x) and Si i- Sj. Let us denote by k the number of randomly selected and reassigned trees then a k-shift neighbor y E Nshift results by the subsequent application of k shifts. In further consequence, we extend the shift neighborhood to yield k-swap neighbors. Let m be the number of trees and S be the number of periods in an assignment X. Each tree can now be scheduled for S - 1 periods, as the neighborhood relation should be irreflexive, i.e. x 'I- N(x). Assuming that a tree can be selected only once for a shift within a neighbor of x, the size of the k-shift neighborhood is O(mk sk).

Moreover, we want to determine an upper bound for the size of the k-swap neighborhood. Let us assume that for each k =

{I, . . . , l!Jf J} two trees were selected. Again, a tree shouldn't be selected more than once, and cannot be swapped with itself. Consequentially, the number of possible swaps in a k-swap neighborhood is bound by O(m2k). Hence, we limit k by 1. Another reason for the limitation is that the same result could be achieved by applying multiple subsequent swap and shift moves.

4) Compress neighborhood: None of the neighborhood structures described above has the strong ability to reduce the number of periods used in an assignment effectively because all of them only touch only a small number of trees in each operation and furthermore, the application of several subsequent moves which could lead to an assignment having one period less is very unlikely. A possibility to reduce the number of periods used lSI is to reschedule all trees of a certain period 8 E S to the remaining periods S \ {8}.

Therefore, let us introduce the compress neighborhood, which is a relation Ncompress : X --+ P(X) with x E X, Y E

Ncompress(x) such that ISyl = ISxl - 1. In other words, the number of periods used has decreased by one. In an assignment y E Ncompress(x), all trees which were scheduled for the last period 8n in x are now randomly assigned to some period 81,82, ... 8n-1 in y with the same probability. The size of Ncompress is 0(1). Nevertheless, a move in this neighborhood is likely to involve a great number of trees and to result in a large number of computations. The effectiveness of the neighborhood clearly depends on the coefficient (3 of the evaluation function (see Section III-A) so that, for high (3 it possibly accepts a high number of exceeding links in the remaining periods. Furthermore, the compress neighborhood is used only if the number of periods is allowed to vary.

IV. NUMERICAL RE SULT S

We have generated instances and solved them by using both the CPLEX solver and the described heuristics. The distribution tree topologies used for tests have three levels: for example to reach 1000 users, each node has 10 children nodes, etc. The link capacities of the tree network are the same for all the links within a level and are described by a triple e.g. (100, 20,10). In a real implementation those capacities would be set by the ISP depending of the time of the day. The set of requests Q that have to be served in S periods determine the load of the system. In order to create the set Q, we allocate each request randomly to a user u and to a title index t. The requests were created according to a Zipf distribution with skew factor B= 0,372. The distribution starts from title index t=1O end ends with t=13000. (The reason for excluding very popular titles is that, those titles could be obtained from a small content cache placed in all leaf nodes directly connected to the users).

We performed a number of 30 runs per instance and metaheuristic and calculated the average runtime and solution quality of the best solution found. Each run was stopped after 30 iterations without improvement. Furthermore, VND uses TS as embedded heuristic (in both plain TS and VND the tabu length t was set to 12) whereas ILS uses simple hillclimbing. In ILS, 2 random perturbations were performed. In all cases, the maximum number of embedded search iterations was kept short. See [9] for detailed setup and results. The simulation runs were performed in order to answer following questions:

A How does the runtime of the IP and heuristics compare? B What is the scalability and solution quality of the

heuristics?

C What is maximum number of requests supported by an instance?

A. Integer programming versus metaheuristics

Table II shows the runtimes of the integer RDCP and of the average best metaheuristic [Heur] instances. The IP program stopped when 1 % gap was reached. The heuristic is almost two order magnitudes faster than the IP running under AMPLlCPLEX. The most instances are solved without tree splits. The reason for this behavior lies in the instance generation: in order to obtain splits, the link capacities have to show strong asymmetries between the time slots as we have shown in the Figure 1.

Type Q U E T S CPU [sec] IP 2000 1000 1110 323 8 90 Heur 2000 1000 1110 323 8 0,07 IP 8000 4000 4420 729 8 330 Heur 8000 4000 4420 729 8 0,36 IP 16000 8000 8420 1106 8 1320 Heur 16000 8000 8420 1106 8 8,4

Table II RUNTIMES OF I P AND HEURISTICS INSTANCES

The notation for the instances used the triplet (Q,U,S), for instance 4000-2000-8 denotes an instance with Q=4000 requests, U=1000 active users and S=8 periods.

B. Scalability and solution quality of the heuristics

In our experiments, we have found that TS yields the best results W.r.t. the evaluation function, although VND converges convincingly faster even for larger instances (see Figure 4). Furthermore, TS is able to keep the number of repetitions lower than VND and ILS (see Figure 3). From that we conclude that intensification of the search process leads to better results for the RCDP.

2000

1800

1600

1400

"'

� 1200 '0

t 1000 Z

800

600

400

200 2000-1000-8

4-Min ...... T5 VNS -ll-ILS

4000-2000-8 8000-4000-6 16000-6000-6 32000-16000-8

Figure 3. Number of trees above Min (no repetitions)in the solutions obtained with the three heuristics applied to five growing problem instances

80

70 f 60

50

40

30

20

10

o =--J:-----�� 2000-1000-8 4000-2000-8 8000-4000-8 16000-8000-8 32000-16000-8

Figure 4. Runtime of the three heuristics applied to five growing problem instances

Q 2000 4000 8000 1 6000 32000

Irs 3336 5 1 57 781 6 1 1 761 1 79 1 5

Iv N D IILS I tTS [s] tv N D [s] tILS [s]

3336 3336 0,002 0,004 0,002 5 1 59 5 1 58 0,2 0,1 1 0,07 781 6 781 6 0,53 0,1 2 0,03 1 1 955 1 2096 8,41 5,45 9,64 1 84 1 6 1 8758 64,1 8 26,84 74,52

Table III RUNTIMES AND QUALITY OF THE METAHEURISTICS

C. Maximum load experiments

In this setting we apply an increasing number of requests to a certain distribution network (given topology, number of users, number of periods), until non-feasibility. Table IV shows that ILP finds optimal allocations up to Qmax.

The result Qmax can be used as a benchmark for the comparison with other packing algorithms and objectives.

In the Table IV we see also the major impact of S on the maximal nwnber of requests supported by the network: decreasing S from 8 to 7 periods, the maximum request rate decrease by about a third for our example.

Q 2800 2900 3000

3030 (Qmax) 1 900 2000

2220 (Qmax)

U 8000 8000 8000 8000 8000 8000 8000

Table IV

S Util 6 0,84 6 0,85 6 0,87 6 0,88 5 0,77 5 0,8 5 0,84

With the goal of scalability in the nwnber of scheduled requests and active users, we developed efficient heuristics and compared them with the exact IP solution. Using the variable neighborhood search heuristic, an instance with 32000 active users and one request per user in 8 hours is "scheduled" in less than 90 seconds. With professional hardware, there is no problem to support the scheduling of hundreds of thousands of users. For the actual distribution, research has been started in the direction of using P2P nodes in the operator CDN network, that implement application layer multicast and dynamic load balancing [15].

Further work is needed to evaluate the quality of a rolling schedule using the methods presented in this work. The schedule is modified incrementally by adding requests.

VI. ACKNOWLEDGEMENT

This work has been funded by the Austrian Government and the city of Vienna in the COMET program.

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V. CONCLU SIONS AND FURTHER RE SEARCH

In this work we present an approach for content distribution in which network providers control the scheduling of titles using optimized tree packing and (application layer) multicast technology.

[ 1 6] Chu-Fu Wang, Chun-Teng Liang, and Rong-Hong Jan. Heuristic algorithms for packing of multiple-group multicasting. Comput. Ope!: Res., 29(7):905-924, 2002.