# Tight Bounds for Parallel Randomized Load Balancing (TIK Report Number 324) Revised November 5, 2010 Revised February 26, 2010

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5 Table 1: Comparison of parallel algorithms for m = n balls. Committing balls into bins counts as half a round with regard to time complexity. algorithm symmetric adaptive choices rounds maximum bin load messages naive [15] yes no O ( log n ) log log n n ( ) par. greedy [1] yes no O log n log log n O(n) par. greedy [1] yes no Θ ( log log n ) log log log n Θ ( log log n ) log log log n O ( log log n ) log log log n O ( n log log n ( log log log n ( collision [37] yes no 2 r O log n ) ) 1/r log log n O(n) A 2 b yes yes O(1) (exp.) log n + O(1) 2 O(n) Ab(r) yes yes O(1) (exp.) r + O(1) log (r) n log (r+1) n + r + O(1) O(n) Ac(l) yes yes O(l) (exp.) log n log l + O(1) O(1) O(ln) A( log n) no yes O(1) (exp.) O(1) 3 O(n) ) 4

7 may take the local topology of its current state into account. A different lower bound technique is by Kuhn et al. [21], where a specific locally symmetric, but globally asymmetric graph is constructed to render a problem hard. Like in our work, [21] restricts its arguments to graphs which are locally trees. The structure of the graphs we consider imposes to examine subgraphs which are trees as well; subgraphs containing cycles occur too infrequently to constitute a lower bound. The bound of Ω(log n) from [14], applicable to hashing in a certain model, which also argues about trees, has even more in common with our result. However, neither of these bounds needs to deal with the difficulty that the algorithm may influence the evolution of the communication graph in a complex manner. In [21], input and communication graph are identical and fixed; in [14], there is also no adaptive communication pattern, as essentially the algorithm may merely decide on how to further separate elements that share the same image under the hash functions applied to them so far. Various other techniques for obtaining distributed lower bounds exist [13, 25], however, they are not related to our work. If graph-based, the arguments are often purely information theoretic, in the sense that some information must be exchanged over some bottleneck link or node in a carefully constructed network with diameter larger than two [24, 34]. In our setting, such information theoretic lower bounds will not work: Any two balls may exchange information along n edge-disjoint paths of length two, as the graph describing which edges could potentially be used to transmit a message is complete bipartite. In some sense, this is the main contribution of this paper: We show the existence of a coordination bottleneck in a system without a physical bottleneck. The remainder of this technical report is organized as follows. In Section 4, we state and solve the aforementioned load balancing problem in a fully connected system. The discussion of the related symmetric balls-into-bins algorithm A b is postponed to Section 5, as the more general proofs from Section 4 permit to infer some of the results as corollaries. After discussing A b and its variations, we proceed by developing the matching lower bound in Section 6. Finally, in Section 7, we give algorithms demonstrating that if any of the prerequisites of the lower bound does not hold, constant-time constant-load solutions are feasible. 3 Preliminary Statements Our analysis requires some standard definitions and tools, which are summarized in this section. Definition 3.1 (Uniformity and Indepence) The (discrete) random variable X : Ω S is called uniform, if P [X = s 1 ] = P [X = s 2 ] for any two values s 1, s 2 S. The random variables X 1 : Ω 1 S 1 and X 2 : Ω 2 S 2 are independent, if for any s 1 S 1 and s 2 S 2 we have P [X 1 = s 1 ] = P [X 1 = s 1 X 2 = s 2 ] and P [X 2 = s 2 ] = P [X 2 = s 2 X 1 = s 1 ]. A set {X 1,..., X N } of variables is called independent, if for any i {1,..., N}, X i is independent from the variable (X 1,..., X i 1, X i+1,..., X n ), i.e., the variable listing the outcomes of all X j X i. The set {X 1,..., X N } is uniformly and independently at random (u.i.r.) if and only if it is independent and consists of uniform random variables. Two sets of random variables X = {X 1,..., X N } and Y = {Y 1,..., Y M } are independent if and only if all X i X are independent from (Y 1,..., Y M ) and all Y j Y are independent from (X 1,..., X N ). We will be particularly interested in algorithms which almost guarantee certain properties. Definition 3.2 (With high probability (w.h.p.)) We say that the random variable X attains values from the set S with high probability, if P [X S] 1 1/n c for an arbitrary, but fixed constant c > 0. More simply, we say S occurs w.h.p. The advantage of this stringent definition is that any polynomial number of statements that individually hold w.h.p., also hold w.h.p. in conjunction. Throughout this paper, we will use 6

8 this lemma implicitly, as we are always interested in sets of events whose sizes are polynomially bounded in n. Lemma 3.3 Assume that statements S i, i {1,..., N}, hold w.h.p., where N n d for some constant d. Then S := N i=1 S i occurs w.h.p. Proof. The S i hold w.h.p., so for any fixed constant c > 0 we may choose c := c + d and have P (S i ) 1 1/n c 1 1/(Nn c ) for all i {1,..., N}. By the union bound this implies P [S] 1 N i=1 P [S i] 1 1/n c. Frequently w.h.p. results are deduced from Chernoff type bounds, which provide exponential probability bounds regarding sums of Bernoulli variables. Common formulations assume independence of these variables, but the following more general condition is sufficient. Definition 3.4 (Negative Association) The random variables X i, i {1,..., N}, are negatively associated if and only if for all disjoint subsets I, J {1,..., N} and all functions f : R I R and g : R J R that are either increasing in all components or decreasing in all components we have E(f(X i, i I) g(x j, j J)) E(f(X i, i I)) E(g(X j, j J)). Note that independence trivially implies negative association, but not vice versa. Using this definition, we can state a Chernoff bound suitable to our needs. Theorem 3.5 (Chernoff s Bound) Let X := N i=1 X i be the sum of N negatively associated Bernoulli variables X i. Then, w.h.p., (i) E[X] O(log n) X O(log n) ) (ii) E[X] O(1) X O ( log n log log n (iii) E[X] ω(log n) X (1 ± o(1))e[x]. In other words, if the expected value of a sum of negatively associated Bernoulli variables is small, it is highly unlikely that the result will be of more than logarithmic size, and if the expected value is large, the outcome will almost certainly not deviate by more than roughly the square root of the expectation. In the forthcoming, we will repeatedly make use of these basic observations. In order to do so, techniques to prove that sets of random variables are negatively associated are in demand. We will rely on the following results of Dubhashi and Ranjan [10]. Lemma 3.6 (i) If X 1,..., X N are Bernoulli variables satisfying N i=1 X i = 1, then X 1,..., X N are negatively associated. (ii) Assume that X and Y are negatively associated sets of random variables, and that X and Y are mutually independent. Then X Y is negatively associated. (iii) Suppose {X 1,..., X N } is negatively associated. Given I 1,..., I k {1,..., N}, k N, and functions h j : R Ij R, j {1,..., k}, that are either all increasing or all decreasing, define Y j := h j (X i, i I j ). Then {Y 1,..., Y k } is negatively associated. This lemma and Theorem 3.5 imply strong bounds on the outcome of the well-known ballsinto-bins experiment. Lemma 3.7 Consider the random experiment of throwing M balls u.i.r. into N bins. Denote by Yi k, i {1,..., N}, the set of Bernoulli variables being 1 if and only if at least (at most) k N 0 balls end up in bin i {1,..., N}. Then, for any k, the set {Yi k} i {1,...,N} is negatively associated. 7

9 Proof. Using Lemma 3.6, we can pursue the following line of argument: 1. For each ball j {1,..., M}, the Bernoulli variables {B i j } i {1,...,N} which are 1 exactly if ball j ends up in bin i, are negatively associated (Statement (i) from Lemma 3.6). 2. The whole set {Bj i i {1,..., N} j {1,..., M}} is negatively associated (Statement (ii) of Lemma 3.6). 3. The sets {Y k i } i {1,...,N} are for each k N 0 negatively associated (Statement (iii) of Lemma 3.6). The following special case will be helpful in our analysis. Corollary 3.8 Throw M N ln N/(2 ln ln n) balls u.i.r. into N bins. Then w.h.p. (1 ± o(1))ne M/N bins remain empty. Proof. The expected number of empty bins is N(1 1/N) M. For x 1 and t x the inequality (1 t 2 /x)e t (1 + t/x) x e t holds. Hence, with t = M/N and x = M we get ) ( (1 o(1))e M/N (1 (M/N)2 e M/N 1 1 ) M e M/N. N N Due to the upper bound on M, we have Ne M/N (ln n) 2 ω(log n). Lemma 3.7 shows that we can apply Theorem 3.5 to the random variable counting the number of empty bins, yielding the claim. Another inequality that yields exponentially falling probability bounds is typically referred to as Azuma s inequality. Theorem 3.9 (Azuma s Inequality) Let X be a random variable which is a function of independent random variables X 1,..., X N. Assume that changing the value of a single X i for some i {1,..., N} changes the outcome of X by at most δ i R +. Then for any t R + 0 we have 4 Parallel Load Balancing P [ X E[X] > t ] 2e t2 /(2 N i=1 δ2 i ). In this section, we examine the problem of achieving as low as possible congestion in the complete graph K n if links have uniform capacity. In order to simplify the presentation, we assume that all loops {v, v}, where v V := {1,..., n}, are in the edge set, i.e., nodes may send messages to themselves. All nodes have unique identifiers, that is, v V denotes both the node v and its identifier. We assume that communication is synchronous and reliable. 10 During each synchronous round, nodes may perform arbitrary local computations, send a (different) message to each other node, and receive messages. We will prove that in this setting, a probabilistic algorithm enables nodes to fully exploit outgoing and incoming bandwidth (whichever is more restrictive) with marginal overhead w.h.p. More precisely, we strive for enabling nodes to freely divide the messages they can send in each round between all possible destinations in the network. Naturally, this is only possible to the extent dictated by the capability of nodes to receive messages in each round, i.e., ideally the amount of time required would be proportional to the maximum number of messages any node must send or receive, divided by n. This leads to the following problem formulation. 10 This is convenient for ease of presentation. We will see later that both assumptions can be dropped. 8

11 7. Delete all messages for which confirmations have been received and all currently held copies of messages. Remark 4.3 Balancing message load and counting the total number of messages (Steps 1 to 4) is convenient, but not necessary. We will later see that it is possible to exploit the strong probabilistic guarantees on the progress of the algorithm in order to choose proper values of k. Definition 4.4 (Phases) We will refer to a single execution of the loop, i.e., Steps 1 to 7, as a phase. Since in each phase some messages will reach their destination, this algorithm will eventually terminate. To give strong bounds on its running time, however, we need some helper statements. The first lemma states that in Steps 5 and 6 of the algorithm a sufficiently large uniformly random subset of the duplicates will be received by their target nodes. Lemma 4.5 Denote by C v the set of copies of messages for a node v V that A s generates in Step 5 of a phase. Provided that C v ω(log n) and n is sufficiently large, w.h.p. the set of messages v receives in Step 6 contains a uniformly random subset of C v of size at least C v /4. Proof. Set λ := C v /n 1. Consider the random experiment where C v = λn balls are thrown u.i.r. into n bins. We make a distinction of cases. Assume first that λ [1/4, 1]. Denote for k N 0 by B k the random variable counting the number of bins receiving exactly k balls. According to Corollary 3.8, B 1 λn 2 (B 0 (1 λ)n) ( 2 λ 2(1 + o(1))e λ) n = 2 λ 2(1 + o(1))e λ C v λ w.h.p. Since λ 1/4, the o(1)-term is asymptotically negligible. Without that term, the prefactor is minimized at λ = 1, where it is strictly larger than 1/4. On the other hand, if λ < 1/4, we may w.l.o.g. think of the balls as being thrown sequentially. In this case, the number of balls thrown into occupied bins is dominated by the sum of C v independent Bernoulli variables taking the value 1 with probability 1/4. Since C v ω(log n), Theorem 3.5 yields that w.h.p. at most (1/4+o(1)) C v balls hit non-empty bins. For sufficiently large n, we get that w.h.p. more than (1/2 o(1)) C v > C v /4 bins receive exactly one ball. Now assume that instead of being thrown independently, the balls are divided into n groups of arbitrary size, and the balls from each group are thrown one by one uniformly at random into the bins that have not been hit by any previous ball of that group. In this case, the probability to hit an empty bin is always as least as large as in the previous setting, since only non-empty bins may not be hit by later balls of a group. Hence, in the end again a fraction larger than one fourth of the balls are in bins containing no other balls w.h.p. Finally, consider Step 5 of the algorithm. We identify the copies of messages for a specific node v with balls and the nodes with bins. The above considerations show that w.h.p. at least C v /4 nodes receive exactly one of the copies. Each of these nodes will in Step 6 deliver its copy to the correct destination. Consider such a node w V receiving and forwarding exactly one message to v. Since each node u V sends each element of C v with probability 1/n to w, the message relayed by w is drawn uniformly at random from C v. Furthermore, as we know that no other copy is sent to w, all other messages are sent with conditional probability 1/(n 1) each to any of the other nodes. Repeating this argument inductively for all nodes receiving exactly one copy of a message for v in Step 5, we see that the set of messages transmitted to v by such nodes in Step 6 is a uniformly random subset of C v. The proof of the main theorem is based on the fact that the number of copies A s creates of each message in Step 5 grows asymptotically exponentially in each phase as long as it is not too large. 10

13 bounded by ( ) k kmv n (ke Ω( log n) ) k e Ω(k log n) Ω(log n) e = n Ω(1). Since m v is non-increasing and k non-decreasing, we conclude that all messages will be delivered within the next O(1) phases w.h.p. With this at hand, we can provide a probabilistic upper bound of O(log n) on the running time of A s. Theorem 4.8 Algorithm A s solves Problem 4.2. It terminates within O(log n) rounds w.h.p. Proof. Since the algorithm terminates only if all messages have been delivered, it is correct if it terminates. Since in each phase of A s some messages will reach their destination, it will eventually terminate. A single phase takes O(1) rounds. Hence it remains to show that w.h.p. after at most O(log n) phases the termination condition is true, i.e., all messages have been received at least once by their target nodes. Denote by k(i) the value k computed in Step 5 of phase i N of A s, by m v (i) the number of messages a node v V still needs to receive in that phase, and define M r (i) := max v V {m v (i)}. Thus, we have k(i) = n/m r (i), where k(1) = 1. According to Lemma 4.6, for all i it holds that w.h.p. M r (i + 1) max{(1 + o(1))e k(i)/4 M r (i), e log n n}. Thus, after constantly many phases (when the influence of rounding becomes negligible), k(i) starts to grow exponentially in each phase, until M r (i + 1) e log n n. According to Lemma 4.7, A s will w.h.p. terminate after O(1) additional phases, and thus after O(log n) phases in total. 4.2 Tolerance of Transient Link Failures Apart from featuring a small running time, A s can be adapted in order to handle substantial message loss and bound the maximum number of duplicates created of each message. Set i := 1 and k(1) := 1. Given a constant probability p (0, 1) of independent link failure, at each node v V, Algorithm A l (p) executes the following loop until it terminates: 1. Create k(i) copies of each message. Distribute these copies uniformly at random among all nodes, but under the constraint that (up to one) all nodes receive the same number of messages. 2. To each node, forward one copy of a message destined to it (if any has been received in the previous step; any choice is feasible). 3. Confirm any received messages. 4. Forward any confirmations to the original sender of the corresponding copy. 5. Delete all messages for which confirmations have been received and all currently held copies of messages. 6. Set k(i + 1) := min{k(i)e k(i) (1 p)4 /5, log n} and i := i + 1. If i > r(p), terminate. Here r(p) O(log n) is a value sufficiently large to guarantee that all messages are delivered successfully w.h.p. according to Theorem Lemmas 4.5 and Lemma 4.6 also apply to A l (p) granted that not too much congestion is created. 12

14 Corollary 4.9 Denote by C v the set of copies of messages destined to a node v V that A l (p) generates in Step 1 of a phase. Provided that n C v ω(log n) and n is sufficiently large, w.h.p. the set of messages v receives in Step 2 contains a uniformly random subset of C v of size at least (1 p) 4 C v /4. Proof. Due to Theorem 3.5, w.h.p. for a subset of (1 o(1))(1 p) 4 C v all four consecutive messages in Steps 1 to 4 of A l (p) will not get lost. Since message loss is independent, this is a uniformly random subset of C v. From here the proof proceeds analogously to Lemma 4.5. Corollary 4.10 Fix a phase of A l (p) and assume that n is sufficiently large. Denote by m v the number of messages node v V still needs to receive and by k the number of copies of each message created in Step 1 of that phase of A l (p). Then v will w.h.p. get at least one copy of all but max{(1 + o(1))e (1 p)4 k(i)/4 m v (i), e log n n} of these messages. Proof. Analogous to Lemma 4.6. Again we need to show that all remaining messages are delivered quickly once k is sufficiently large. Lemma 4.11 Suppose that n is sufficiently large and fix a phase of A l (p). If k Ω(log n), v will w.h.p. receive a copy of all messages it has not seen yet within O(1) more phases of A l (p). Proof. If v still needs to receive ω(1) messages, we have that C v ω(log n). Thus, Corollary 4.9 states that v will receive a uniformly random subset of fraction of at least (1 p) 4 /4 of all copies destined to it. Hence, the probability that a specific message is not contained in this subset is at most ) Ω(log n) (1 p)4 (1 n Ω(1). 4 On the other hand, if the number of messages for v is small say, O(log n) in total no more than O((log n) 2 ) copies for v will be present in total. Therefore, each of them will be delivered with probability at least (1 o(1))(1 p) 4 independently of all other random choices, resulting in a similar bound on the probability that at least one copy of a specific message is received by v. Hence, after O(1) rounds, w.h.p. v will have received all remaining messages. We now are in the position to bound the running time of A l (p). Theorem 4.12 Assume that messages are lost u.i.r. with probability at most p < 1, where p is a constant. Then A l (p) solves problem 4.2 w.h.p. and terminates within O(log n) rounds. Proof. Denote by C v the set of copies destined to v V in a phase of A l (p) and by M r (i) the maximum number of messages any node still needs to receive and successfully confirm at the beginning of phase i N. Provided that n C v, Corollary 4.10 states that { } M r (i + 1) max (1 + o(1))e (1 p)4 k(i) /4 M r (i), e log n n. The condition that C v n is satisfied w.h.p. since w.h.p. the maximum number of messages M r (i) any node must receive in phase i {2,..., r(p)} falls faster than k(i) increases and k(1) = 1. Hence, after O(log n) many rounds, we have k(i) = log n and M r (i) e log n n. Consequently, Corollary 4.11 shows that all messages will be delivered after O(1) more phases w.h.p. It remains to show that the number of messages any node still needs to send decreases faster than k(i) increases in order to guarantee that phases take O(1) rounds. Denote by m w v (i) the number of messages node w V still needs to send to a node v V at the beginning of phase i. Corollary 4.9 states that for all v V w.h.p. a uniformly random fraction of at least (1 p) 4 /4 of the copies destined to v is received by it. We previously used that the number of successful messages therefore is stochastically dominated from below by the number of non-empty bins 13

16 Theorem 4.14 A g solves Problem 4.1 w.h.p. in ( ( )) Ms + M r O + log n log n n rounds. Proof. Denote by M r (i) the maximum number of messages any node still needs to receive at the beginning of phase i N. The first execution of Step 2 of the algorithm will take M s /n rounds. Subsequent executions of Step 2 will take at most M r (i 1)/n rounds, as since the previous execution of Step 2 the number of messages at each node could not have increased. As long as M r (i) > n, each delivered message will be a success since no messages are duplicated. Denote by m v (i) M r (i) the number of messages that still need to be received by node v V at the beginning of the i th phase. Observe that at most one third of the nodes may get 3m v (i)/n 3M r (i)/n or more messages destined to v in Step 5. Suppose a node w V holds at most 2n/3 messages for v. Fix the random choices of all nodes but w. Now we choose the destinations for w s messages one after another uniformly from the set of nodes that have not been picked by w yet. Until at least n/3 many nodes have been chosen which will certainly deliver the respective messages to v, any message has independent probability of at least 1/3 to pick such a node. If w has between n/3 and 2n/3 many messages, we directly apply Theorem 3.5 in order to see that w.h.p. a fraction of at least 1/3 o(1) of w s messages will be delivered to v in that phase. For all nodes with fewer messages, we apply the Theorem to the set subsuming all those messages, showing that if these are ω(log n) many again a fraction of 1/3 o(1) will reach v. Lastly, any node holding more than 2n/3 messages destined for v will certainly send more than one third of them to nodes which will forward them to v. Thus, w.h.p. (1/3 o(1))m v messages will be received by v in that phase granted that m v (i) ω(log n). We conclude that for all phases i we have w.h.p. that M r (i+1) max {n, (2/3 + o(1))m r (i)}. If in phase i we have M r (i) > n, Steps 5 and 6 of that phase will require O (M r (i)/n) rounds. Hence, Steps 5 and 6 require in total at most log M r/n i=1 ( ) Mr (i) O O n ( Mr n ) i=0 M r ( ) i o(1) O ( Mr rounds w.h.p. until M r (i 0 ) n for some phase i 0. Taking into account that Steps 1, 3, 4, and 7 take constant time regardless of the number of remaining messages, the number of rounds until phase i 0 is in O((M s + M r )/n) w.h.p. 14 In phases i i 0, the algorithm will act the same as A s did, since M r n. Thus, as shown for Theorem 4.8, k(i) will grow asymptotically exponentially in each step. However, if we have few messages right from the beginning, i.e., M r o(n), k(i 0 ) = k(1) = n/m r might already be large. Starting from that value, it requires merely O(log n log (n/m r )) rounds until the algorithm terminates w.h.p. Summing the bounds on the running time until and after phase i 0, the time complexity stated by the theorem follows. Roughly speaking, the general problem can be solved with only constant-factor overhead unless M r n and not M s n, i.e., the parameters are close to the special case of Problem 4.2. In this case the solution is slightly suboptimal. Using another algorithm, time complexity can be kept asymptotically optimal. Note, however, that the respective algorithm is more complicated and since log n grows extremely slowly will in practice always exhibit a larger running time. Theorem 4.15 An asymptotically optimal randomized solution of Problem 4.2 exists. 14 To be technically correct, one must mention that Lemma 3.3 is not applicable if log M r/n is not polynomially bounded in n. However, in this extreme case the probability bounds for failure become much stronger and their sum can be estimated by a convergent series, still yielding the claim w.h.p. n ) 15

17 We postpone the proof of this theorem until later, as it relies on a balls-into-bins algorithm presented in Section 7. 5 Relation to Balls-into-Bins The proofs of Theorems 4.8, 4.12 and 4.14 repeatedly refer to the classical experiment of throwing M balls u.i.r. into N bins. Indeed, solving Problems 4.1 can be seen as solving n balls-into-bins problems in parallel, where the messages for a specific destination are the balls and the relaying nodes are the bins. Note that the fact that the balls are not anonymous does not simplify the task, as labeling balls randomly by O(log n) bits guarantees w.h.p. globally unique identifiers for anonymous balls. In this section we will show that our technique yields strong bounds for the well-known distributed balls-into-bins problem formulated by Adler et al. [1]. Compared to their model, the decisive difference is that we drop the condition of non-adaptivity, i.e., balls do not have to choose a fixed number of bins to communicate with right from the start. 5.1 Model The system consists of n bins and n balls, and we assume it to be fault-free. We employ a synchronous message passing model, where one round consists of the following steps: 1. Balls perform (finite, but otherwise unrestricted) local computations and send messages to arbitrary bins. 2. Bins receive these messages, do local computations, and send messages to any balls they have been contacted by in this or earlier rounds. 3. Balls receive these messages and may commit to a bin (and terminate). 15 Moreover, balls and bins each have access to an unlimited source of unbiased random bits, i.e., all algorithms are randomized. The considered task now can be stated concisely. Problem 5.1 (Parallel Balls-into-Bins) We want to place each ball into a bin. The goals are to minimize the total number of rounds until all balls are placed, the maximum number of balls placed into a bin, and the amount of involved communication. 5.2 Basic Algorithm Algorithm A s solved Problem 4.2 essentially by partitioning it into n balls-into-bins problems and handling them in parallel. We extract the respective balls-into-bins algorithm. Set k(1) := 1 and i = 1. Algorithm A b executes the following loop until termination: 1. Balls contact k(i) u.i.r. bins, requesting permission to be placed into them. 2. Each bin admits permission to one of the requesting balls (if any) and declines all other requests. 3. Any ball receiving at least one permission chooses an arbitrary of the respective bins to be placed into, informs it, and terminates. 4. Set k(i + 1) := min{k(i)e k(i) /5, log n} and i := i + 1. Theorem 5.2 A b solves Problem 5.1, guaranteeing the following properties: 15 Note that (for reasonable algorithms) this step does not interfere with the other two. Hence, the literature typically accounts for this step as half a round when stating the time complexity of balls-into-bins algorithms; we adopted this convention in the related work section. 16

18 It terminates after log n + O(1) rounds w.h.p. Each bin in the end contains at most log n + O(1) balls w.h.p. In each round, the total number of messages sent is w.h.p. at most n. The total number of messages is w.h.p. in O(n). Balls send and receive O(1) messages in expectation and O( log n) many w.h.p. Bins send and receive O(1) messages in expectation and O(log n/ log log n) many w.h.p. Furthermore, the algorithm runs asynchronously in the sense that balls and bins can decide on any request respectively permission immediately, provided that balls messages may contain round counters. According to the previous statements messages then have a size of O(1) in expectation and O(log log n) w.h.p. Proof. The first two statements and the first part of the third follow as corollary of Theorem 4.8. Since it takes only one round to query a bin, receive its response, and decide on a bin, the time complexity equals the number of rounds until k(i) = log n plus O(1) additional rounds. After O(1) rounds, we have that k(i) 20 and thus k(i + 1) min{k(i)e k(i)/20, log n}, i.e., k(i) grows at least like an exponential tower with basis e 1/20 until it reaches log n. As we will verify in Lemma 6.8, this takes log n + O(1) steps. The growth of k(i) and the fact that algorithm terminates w.h.p. within O(1) phases once k(i) = log n many requests are sent by each ball implies that the number of messages a ball sends is w.h.p. bounded by O( log n). Denote by b(i) the number of balls that do not terminate w.h.p. until the beginning of phase i N. Analogously to Theorem 4.14 we have w.h.p., so in particular with probability p 1 1/n 2, that { } b(i + 1) (1 + o(1)) max b(i)e k(i) /4, e log n n (1) for all i {1,..., r O(log n)} and the algorithm terminates within r phases. In the unlikely case that these bounds do not hold, certainly at least one ball will be accepted in each round. Let i 0 O(1) be the first phase in which we have k(i 0 ) 40. Since each ball follows the same strategy, the random variable X counting the number of requests it sends does not depend on the considered ball. Thus we can bound ( E(X) < 1 n (1 p) i ( r )) log n + p b(i) k(i) n i=1 i=1 ( o(1) + 1 i0 ) r b(i)k(i) + b(i)k(i) n O(1) + 1 n O(1) + 1 n i=1 ( r i=i 0+1 r i=i 0+1 O(1) + b(i 0)k(i 0 ) n O(1). b(i 0 )k(i 0 ) i=i 0+1 ( { max e k(j) /4+ k(j) /5}) i i 0 + e log n n j i 0 b(i 0 )k(i 0 )e (i i0)( k(i0)/20 1) i=1 e i Thus, each ball sends in expectation O(1) messages. The same is true for bins, as they only answer to balls requests. Moreover, since we computed that r b(i) k(i) O(n) i=1 17 )

20 Corollary 5.6 For any r N, A b can be modified into an Algorithm A b (r) that guarantees a maximum bin load of log (r) n/ log (r+1) n+r+o(1) w.h.p. and terminates within r+o(1) rounds w.h.p. Its message complexity respects the same bounds as the one of A b. Proof. In order to speed up the process, we rule that in the first phase bins accept up to l := log (r) n/ log (r+1) n many balls. Let for i N 0 Y i denote the random variables counting the number of bins with at least i balls in that phase. From Lemma 3.7 we know that Theorem 3.5 applies to these variables, i.e., Y i O(E(Y i ) + log n) w.h.p. Consequently, the same is true for the number Y i Y i+1 of bins receiving exactly i messages. Moreover, we already observed that w.h.p. bins receive O(log n/ log log n) messages. Thus, the number of balls that are not accepted in the first phase is w.h.p. bounded by ( n ( ) ( ) i ( n 1 polylog n + O n (i l) 1 1 ) ) ( ) n i ( e ) i polylog n + O n i n n i i=l+1 polylog n + n i=l Ω(l) i i=l polylog n + 2 Ω(l log l) n, where in the first step we used the inequality ( n k) (en/k) k. Thus, after the initial phase, w.h.p. only n/(log (r 1) n) Ω(1) balls remain. Hence, in the next phase, A b (r) may proceed as A b, but with k(2) (log (r 1) n) Ω(1) requests per ball; we conclude that the algorithm terminates within r + O(1) additional rounds w.h.p. The observation that neither balls nor bins need to wait prior to deciding on any message implies that our algorithms can also be executed sequentially, placing one ball after another. In particular, we can guarantee a bin load of two efficiently. This corresponds to the simple sequential algorithm that queries for each ball sufficiently many bins to find one that has load less than two. Lemma 5.7 An adaptive sequential balls-into-bins algorithm Aseq exists guaranteeing a maximum bin load of two, requiring at most (2 + o(1))n random choices and bin queries w.h.p. Proof. The algorithm simply queries u.i.r. bins until one of load less than two is found; then the current ball is placed and the algorithm proceeds with the next. Since at least half of the bins have load less than two at any time, each query has independent probability of 1/2 of being successful. Therefore, it can be deduced from Theorem 3.5, that w.h.p. no more than (2+o(1))n bin queries are necessary to place all balls. 6 Lower Bound In this section, we will derive our lower bound on the parallel complexity of the balls-into-bins problem. After presenting the formal model and initial definitions, we proceed by proving the main result. Subsequently, we briefly present some generalizations of our technique. 6.1 Definitions A natural restriction for algorithms solving Problem 5.1 is to assume that random choices cannot be biased, i.e., also bins are anonymous. This is formalized by the following definition. Problem 6.1 (Symmetric Balls-into-Bins) We call an instance of Problem 5.1 symmetric parallel balls-into-bins problem, if balls and bins identify each other by u.i.r. port numberings. We call an algorithm solving this problem symmetric. 19

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