4.3 Implementation in Various Models of Computation
4.3.2 Implementation
In this section, we describe for each of the above three models and the standard RAM model of computation how our algorithm can be implemented in them. We first recall our results from Corollary 4.15: Algorithm GradientTransship can be executed using a total of O((ε−2+ log α + log λ)α2λ2log n) operations of the following types:
1. oracle call
2. computation of (W−1ATπ)maxfor a given π 3. computation of ∇Φβ(π) for given β and π
4. scalar product ˜bTh for given ˜b and h
5. comparisons of and multiplications with scalar values
We will thus describe for each of the models how to implement these steps. Fortunately, this is rather straightforward.
Broadcast Congested Clique model
We start with items 2 and 3. Every node v can within one round learn the values πw for
all its neighbors w. Thus, it can locally compute za : (W−1ATπ)a for all arcs a ∈ δinG(v).
Then, every node can broadcast max{za : a ∈ δinG(v)} to all other nodes and thus every
node can learn the value of (z)max (W−1ATπ)maxwithin a constant number of rounds. Let
us recall the definition of Φβ(π) : lseβ(W−1ATπ) and the form of its gradient ∇Φβ(π)
AW−1∇ lseβ(W−1ATπ). Note that for v ∈ V, the gradient takes the form
(AW−1∇ lseβ(z))v 1 βÍ a∈Aeβza Õ a∈δin(v) eβza wa − Õ a∈δout(v) eβza wa . (4.15)
Thus, as every node knows its incident edges, it can locally compute eβza for all its incident
edges a ∈ δin(v) ∪δout(v) and thus it can broadcast the valueÍ
a∈δin(v)eβza. Then, every
node can learnÍ
a∈Aeβza Ív∈VÍa∈δin(v)eβzaand thus compute (AW−1∇ lseβ(z))vaccording
to (4.15) locally. Items 4 and 5 are even more straightforward, as we can assume that the values of ˜b and h are globally known every node can do these computations locally for free. For item 1, i.e, the implementation of the oracle, we have already described in Section 4.2.2 that the optimal solution to Psym ⊥(S, w, ∇Φβ(π)) on an α-spanner S gives an α-apprixmate solution for the problem Psym ⊥(G, w, ∇Φβ(π)). It suffices to compute this α-spanner S
once initially as the weights of all oracle problems are identical, namely w. Whenever, the algorithm calls the oracle, we can then make sure that every node knows ∇Φβ(π) and can
locally compute an optimal solution to Psym ⊥(S, w, ∇Φβ(π)). For the initial computation of
an α-spanner, note that already Baswana and Sen gave a randomized method to construct a 2k − 1 spanner of expected size O(kn1+1/k) within O(k2) rounds in the Broadcast Congest
model [BS07, Theorem 5.1].This strategy can be made deterministic. In [CPS17], Censor- Hillel et al. show a comparable result. They show how to, in the Broadcast Congested Clique model, deterministically construct a (2k − 1)-spanner with O(kn1+1/klog n) edges in O(k log n) rounds. Thus, for k log(n), this result directly gives a spanner that fulfills our requirements. We conclude that after an initial computation of the spanner in O(log2n) rounds, every iteration can be implemented within constant number of rounds. Thus, we obtain the following theorem.
Theorem 4.29 (Broadcast Congested Clique). 1. There is an algorithm that, for any0 < ε ≤
1, computes a (1 + ε)-approximate primal/dual solution pair to the asymmetric transshipment
problem in theBroadcast Congested Clique model within ˜O(ε−2λ2polylog n) rounds of communication.
2. There is an algorithm for the asymmetric single source shortest path problem that, for any
0 < ε ≤ 1, computes a (1 + ε)-approximate shortest path for every node in the Broadcast Congested Clique model within ˜O(ε−2λ2polylog(n, kwk∞)) rounds of communication.
Broadcast Congest model
In order to transfer our results from the Broadcast Congested Clique to the Broadcast Congest model, we use two different previous results. The first is a classical result in distributed computing. The idea is to pipeline information over a BFS tree.
Lemma 4.30 (cf. [Pel00]). Suppose each v ∈ V holds mv ∈Z≥0messages ofO(log n) bits each. Let
M :Í
v∈V mv. Then all nodes in the graph can receive allM messages within O(M+ D) rounds.
For the single source shortest path case we use the following more recent result:
Lemma 4.31 ([HKN16]). Given any weighted undirected graph G and source node s ∈ V, there is an ˜
O( √
n)-round deterministic distributed algorithm in the Broadcast Congest model7 that computes
an overlay networkG0 (V0, E0) with edge weights w0: E0→ {1, . . . , poly n} and some additional information for every node with the following properties.
• |V0|
˜O(√nε−1) and s ∈ V0.
• For ε0 Θ(ε), each node v ∈ V can infer a (1 + ε)-approximation of its distance to s from
(1 + ε0)-approximations of the distances between s and each t ∈ V0.
Using the above results, we obtain the following theorem.
Theorem 4.32 (Broadcast Congest). 1. There is an algorithm that, for any0 < ε ≤ 1, com- putes a(1+ ε)-approximate primal/dual solution pair to the asymmetric transshipment problem in theBroadcast Congest model within ˜O(nε−2λ2) rounds of communication.
2. There is a Las Vegas algorithm that, for any0 < ε ≤ 1, computes a (1 + ε)-approximate shortest path tree to the asymmetric single source shortest path problem in the Broadcast Congest model inO(˜
√
n+ D)ε−2λ2) rounds.
Proof. 1. We obtain the result by using Lemma 4.30 and simulating the broadcast con-
gested clique in every iteration. It holds that M O(n log n) in this case and thus the result follows from the first part of Theorem 4.29.
2. It is easy to see that the construction from Henzinger et al. in Lemma 4.31 also works in the asymmetric setting. Hence the bound follows using the second part of Theorem 4.29 on the overlay network.
7All communication of the algorithm in [HKN16] meets the constraint that in each round, each node sends the same message to all neighbors (which is the difference between the Broadcast Congest and the standard Congest model).
4.3. Implementation in Various Models of Computation 67
Multipass Streaming model
The main observation is that we can apply the same approach as before with O(n log n) space as this enables us to initially compute the spanner and store it during the entire computation. Similarly, we can store the variables related to the nodes within O(n log n) space. The gradient ∇Φβ(π) can be evaluated in a single pass. It follows that ε−2λ2polylog n
passes suffice and O(n log n) space is enough.
Theorem 4.33 (Multipass Streaming). 1. There is an algorihtm that, for any 0 < ε ≤ 1, computes a deterministic(1 + ε)-approximation to the asymmetric transshipment problem in theMultipass Streaming model in ˜O(ε−2λ2polylog n) passes with O(n log n) space. 2. There is a Las Vegas algorithm that, for any0 < ε ≤ 1, computes a (1 + ε)-approximate shortest
path tree to the asymmetric single source shortest path problem in theMultipass Streaming model inO(ε˜ −2λ2polylog(n, kwk∞)) passes with O(n log n) space.
RAM model
We turn back to the classical RAM model of computation and conclude the first part of the thesis with this. We will see that we can apply the results from Chapter 3 for implementing the oracle for the gradient descent algorithm and obtain an interesting result for the RAM model in this way. We need one more technical lemma that shows that we can implement an oracle for Psym ⊥(G, w, ∇Φβ(π)) using an oracle for Pasym(G, w, PT∇Φ
β(π)), where P is a
specific projection matrix:
Lemma 4.34. Let π ∈ Rn be a γ-approximate solution to Preci(G, w, b), i.e., bTπ 1 and
(W−1ATπ)max ≤ γ(W−1ATπ∗)max. Let P (I − πbT) and let x, h be an η-approximate pri- mal/dual solution pair toPasym(G, w, PT∇Φβ(π)). Then (x, πT∇Φβ(π)), Ph/(1+γ), is an α(1+γ)- approximate primal/dual solution pair toPsym ⊥(G, w, ∇Φβ(π)).
Proof. Let us first check feasibility. Clearly, x ≥ 0 and Ax + (πT∇Φβ(π))b PT∇Φβ(π) as well
as bTPh 0 by the definition of P. Moreover,
(W−1ATPh)max≤ (W−1ATh)max+ (W−1ATπ)max|bTh|
(W−1ATh) max· 1 +(W −1ATπ) max|bTh| (W−1ATh) max ≤ (W−1ATh)max· 1 + γ(W−1ATπ∗)maxbTy∗ (W−1ATh) max· 1 + γ,
and thus also Ph/(1 + γ) is feasible. It remains to show the approximation guarantee. Note that 1TW−x ≤η(PT∇Φβ(π))Th implies that
1TW−x ≤ η · ∇Φβ(π)TPh η(1 + γ) · ∇Φβ(π)T
Ph (1 + γ) and thus (x, πT∇Φ
β(π)), Ph/(1 + γ) is an η(1 + γ)-approximate primal/dual pair.
The proof of Lemma 4.14 implies that any iterate π during the run of the algorithm is a 2α2λ2-approximation. Notice that the naive thought, that any iterate should be an α- approximation as the initial solution is and the algorithm makes progress, is trappy as the algorithm guarantees progress in terms of Φβ(π) and this does not directly imply progress in
terms of (W−1ATπ)
max. In summary, the above lemma shows that we can get a primal/dual
pair for Psym ⊥(G, w, ∇Φβ(π)) with approximation ratio αλ(2α2λ2+1) ≤ 3α3λ3by computing
Recall that the asymmetric transshipment problem is a special case of the directed (unca- pacitated) min-cost flow problem, as we have seen in Lemma 1.1 in the introduction. Thus, we may use the integer interior point method from Chapter 3 for solving Pasym(S, w, PT∇Φβ(π)) and obtain an oracle running in ˜O(m+ n3/2log(kb, wk∞)) time. The ˜O(m)-term is the result
of the spanner computation, see for example the algorithm of Baswana and Sen [BS07] that can be used for this purpose. The term ˜O(n3/2log(kb, wk∞)) is due to the min-cost flow
computation on the sparse spanner, where m ˜O(n), see Theorem 3.10. Hence, we obtain the following result.
Theorem 4.35. There is a randomized algorithm that, for any 0 < ε ≤ 1, computes a (1 + ε)-
approximation to the asymmetric transshipment problem in O(ε˜ −2λO(1)(m+ n3/2) log(kb, wk∞)) time with high probability.
We remark that in settings where m Ω(n3/2), our method is thus nearly-linear in m, thus in terms of m optimal up to log-factors, and dominates over the method due to Sherman [She17] that runs in O(m1+o(1)).
4.4
Conclusion
We have presented a gradient descent approach to the asymmetric transshipment problem and its special case the single source shortest path problem. When implemented in dis- tributed and streaming models of computation this leads very efficient algorithms including the first non-trivial bounds for asymmetric transshipment in the Broadcast Congest and Broadcast Congested Clique models. We obtain the first algorithms or the single source shortest path problem in these models with near-optimal number of communication rounds. This work leaves some interesting research questions open. Clearly, our result for the RAM model is interesting but also somewhat unsatisfactory. One would hope to find an algorithm with nearly linear running time in m for all settings of n and m and not only for m Ω(n3/2). Another very interesting question that arises is the following: Does a similar
approach work for the single source shortest path problem in the PRAM model? Finding a PRAM-algorithm for computing single source shortest paths or (1 + ε)-approximate shortest paths with nearly-linear work and poly-logarithmic depth would be the ultimate goal here.
69
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