I would like to thank the leaders of the Department of Computer Science and Technology during my Ph.D. In the second contribution of the thesis, we investigate the trade-off of round messages in distributed ST construction.
Steiner tree
In this thesis, we focus on solving a global optimization problem called Steiner tree in the distributed environment. 64] provide a website with continuously updated state-of-the-art results for many variants of the problem.
Applications
Forming an ST between these nodes provides the best evolutionary tree to predict the evolutionary chain of the related species. Real-world problems in network design (e.g. fiber optic networks), content distribution networks (video on demand, streaming multicast) [8], protein-protein interaction networks [33], leak detection systems [117], phylogenetic tree reconstruction [97], to name a few can be modeled as cases of the PCST problem.
Motivation
What is the best approximation factor that can be achieved to solve the ST problem while incurring a round complexity cost close to that of the best known MST algorithm. Can we achieve a distributed algorithm for the ST problem in the CONGEST model whose round and message complexities coincide with those of the singular optimality results of the MST algorithms while maintaining an approximation factor of at most 2.
Objectives
Can we design a non-trivial distributed algorithm for the PCST problem while maintaining a constant approximation factor. We will design a primal-dual-based deterministic distributed algorithm for the PCST problem while maintaining an approximation factor of at most 2.
Contributions
- Improved distributed approximation for ST in the CONGEST model 11
- Round-Message Trade-off in distributed ST construction
- Distributed approximation algorithms for ST in the CCM
- STCCM-A: A 2(1 − 1/`)-factor ST algorithm using ˜ O(n 1/3 )
- STCCM-B: A 2(1 − 1/`)-factor ST algorithm using O(S +
- Primal-dual based distributed approximation algorithm for PCST
- Exact algorithms
- Approximation algorithms
The polylogarithmic factors associated with the circuit and message complexity of the modified DST algorithm are O(logn) and O(log2n), respectively. One of the well-known standard approaches to solve the ST problem is to use the proper LP relaxation.
Distributed ST algorithms
The circuit and message complexity of the K-SPH-based algorithm according to Bauer and Varma [12] are O(Dn) and O(nt), respectively. The time and message complexities of the SPH-based algorithm are O(Dt) and O(nt), respectively.
Sequential PCST algorithms
GW-algorithm
The growth phase maintains a forest containing a set of candidate edges selected for PCST construction. The growth phase also maintains a set of components whose possible states can be active or inactive.
Perturbation algorithms
Subsequent iteration of the loop feeds the GW algorithm where the graph is perturbed as in the previous iteration. This algorithm successfully outperforms the direct use of the GW algorithm and computes optimal solutions for multiple instances.
Memetic and ILP Based algorithms
The last step in the iteration is a perturbation algorithm that changes the premium for some nodes. Since the ST calculation is an MST heuristic, the approximation ratio of the resulting ST is 2 of the optimal [57].
Distributed PCST algorithms
The time and message complexity of the UTA version of the tree algorithm are O(h) and O(m), respectively. The deterministic lower bounds of the circuit and message complexity of the MST problem in the CONGEST model are Ω(D+√ . n/logn) [115] and Ω(m) [86], respectively.
SPF construction
Distributed SPF algorithm
We also assume that r, the root node, knows the height (denoted as h) of the BFS tree. Since the height of the BFS tree (h(r)) can be at most D, the round complexity of computing h(r) is O(D) in the worst case. Since h is the height of the BFS tree, it is guaranteed that from the time of any change of local state in a node (which generates a hechoi message), r this information (by receiving a hechoi message ) received after at most h rounds.
An illustrative example of the SPF algorithm
The above two cases guarantee that the delay messages generated at each network node eventually arrive. In case r receives no message for h + 1 consecutive rounds, this ensures that no change has occurred at any node in the network in the last h + 1 rounds.
Correctness of the SPF algorithm
-terminal BFS tree edge 8. nil nil nil nil nil nil nil nil nil. zero zero zero zero zero zero zero zero zero zero zero zero zero. zero zero zero zero zero. B zero zero zero zero zero zero zero zero zero zero zero zero. zero zero zero zero zero zero zero zero zero zero zero zero zero. Therefore, no further messages related to the SPF algorithm will be sent or transported in the network after at most S+ 2h+ 1 rounds of execution.
DST algorithm
- Preliminaries
- Outline of the DST algorithm
- An illustrative example of the DST algorithm
- Correctness of the DST algorithm
The worst-case approximation ratio of the DST algorithm derives from the correctness of Algorithm H. It is clear that the overall message complexity of the DST algorithm is dominated by step 1 and step 3. We have already shown earlier that step 1 (i.e. . , the SPF construction) of the DST algorithm is deadlock-free.
Round-message trade-off in Distributed ST construction
Modified DST algorithm
Since step 4 also requires O(S) rounds, the total round complexity of the modified DST algorithm is O(S+ (D+√ . n) logn). Note that the polylogarithmic factor associated with the round complexity of the modified DST algorithm is at most logn. Combining these two steps, we get that the message complexity of the modified DST algorithm is O(Sm +mlogn +nlogn ·log∗n).
Summary
It is clear that the overall message complexity of the modified DST algorithm is dominated by step 1 and step 3. 37] studied the lower bounds of the circular complexity of the subgraph detection problem in CCM. For graphs with S = ω(n1/3logn), the STCCM-A algorithm performs better than the STCCM-B algorithm in terms of circuit complexity.
SPF construction
Censor-Hillel et al.’s APSP algorithm in CCM
Then, all the resulting min-plus products are redistributed between nodes in such a way that each nodev ∈V receives n4/3 values. The distribution of the min-plus products is also performed using O(n1/3) rounds and O(n7/3) messages using Lenzen's deterministic routing scheme [89]. From the received n4/3 inputs, each node v ∈ V locally computes the row v of the resulting min-plus product W2 = W.
SPF-A algorithm
24] showed that repeated squaring of the weight matrix over the min-plus semiring can also be used to construct the routing table for each nodev ∈V. It is clear that the overall round and message complexities of the SPF-A algorithm are dominated by the first part of the algorithm. The correctness of the SPF-A algorithm follows directly from the correctness of the algorithm developed by Censor-Hillel et al.
STCCM-A algorithm
Outline of the STCCM-A algorithm
This step guarantees that each node v ∈ V knows which of the edges in δ(v) are in TM and for each such edge e it sets ES(e) to branch. Note that the round and message complexity of the algorithm proposed by Lotker et al. For correctness, we assume that a node in Z with the smallest ID is the root rt of the TM.
An illustrative example of the STCCM-A algorithm
The distances of the nodes to their respective sources are shown in the table. f ) Final Steiner tree (generalized MST) for Z of G. InGF, every non-terminalv is connected to a terminal s(v)∈Z whose distance is minimal in s(v) to any terminal another in G which is shown in the table of Figure 4.1(c). The modified Gc graph and the labeling of the edge weights according to the definition of Gc are shown in Figure 4.1(d).
Proof of the STCCM-A algorithm
The correctness of the above statement essentially follows from the correctness of Theorem 1 of Kou et al. The correctness of the deadlock freedom of step 3 essentially follows from the work of Lotker et al. Therefore, the deadlock freedom of all four steps together ensures deadlock freedom of the STCCM-A algorithm.
STCCM-B algorithm
SPF-B algorithm
In the second round of the algorithm, r receives the hechoi message sent by itself. Whenever each node in the network sets the initialize local variable flag to false, the algorithm terminates. In this state, r triggers the termination algorithm by sending hstopi messages to all nodes in the network.
Complexity of the STCCM-B algorithm
Summary
Like the D-PCST algorithm, the modified D-PCST algorithm is based on the use of the primal-dual technique. Pandit and Pemmaraju [109] showed that using the primal dual method, it is possible to solve the object matrix location problem with O(x) rounds with an approximation factor of O(f2/√x·g3/√x) optimal in the CONGEST model of distributed computing. In Section 5.2, we introduce the formulation of the PCST problem using integer programming (IP) and linear programming (LP).
D-PCST algorithm
Outline of the D-PCST algorithm
Initially, every component except the root component (which contains the special node r, the root) is in sleep state. The pruning phase does the following. i) prune all the non-root components. ii) keep trimming subcomponents from the root component until no improvement of the PCST is possible anymore. When the algorithm terminates, the root component is the required Steiner part, while the set of nodes not included in the root component contributes to the penalty part of the resulting PCST.
Phases of D-PCST
Note that if none of the edges of δ(v) are neither basic nor refind, then 1(v) is equal to . Therefore, ultimately the total price of component C (denoted as TP(C)) is also known to L(C). On the other hand, if neither of the merging components C or C0 is the main component, the resulting component C ∪C0 becomes active.
Proof of Correctness
Distributed Preservation of Dual Constraints
The initial values of the dual variables yC and yC0 are du and dv respectively. a) represents the addition of components C0 and C. The initial values of the dual variables yC and yC0 are respectively du and dv. The double variables yC and yC0 are updated to yC +(C) and yC0 +(C0) respectively. a) suggests deactivating components C.
Performance of the D-PCST Algorithm
Note that the pruning phase of the algorithm increases the PCST score calculated in the growing phase of the D-PCST algorithm. Note that the completion of the pruning phase ensures the overall completion of the D-PCST algorithm. We now prove the induction hypothesis for each instance of the growth phase of component C.
Fast PCST construction
Properties of the modified D-PCST algorithm
The growth phase of the modified D-PCST algorithm can be implemented using O(Dn) rounds and O(mn) messages. Since the height of the tree can be at most n−1, the circular complexity of strong pruning is O(n). The pruning phase of the modified D-PCST algorithm can be performed using O(n) rounds and O(n) messages.
Summary
An illustrative example of the D-PCST algorithm. a) state before deactivation of the active component {v7, v11}. For graphs with S =ω(n1/3logn), the STCCM-A algorithm performs better than the STCCM-B algorithm in terms of rounded complexity. For networks with constant or small unweighted diameter (D = o(n)), the modified D-PCST algorithm outperforms the original D-PCST algorithm in terms of rounded complexity.
Future Perspectives
Distributed Approximation Algorithms for Steiner Tree in CONGESTED CLIQUE", International Journal of Foundations of Computer Science (IJFCS), Vol. Primal-Dual Based Distributed Approximation Algorithm for Prize Collection Steiner Tree", Discrete Mathematics, Algorithms and Applications (DMAA). In Proceedings of the 24th Annual ACM Symposium on Principles of Distributed Computing, PODC '05, pp.
An illustrative example of the ST problem
An illustrative example of the PCST problem
An example of the discovery step in the distributed K-SPH
An illustrative example of the SPF algorithm
An illustrative example of the DST algorithm
A state before merging of two shortest path trees in the DST algorithm
An illustrative example of the STCCM-A algorithm
A case of merging of two components in the modified D-PCST algorithm
An illustrative example of the D-PCST algorithm: merge operation
An illustrative example of the D-PCST algorithm: deactivation
An illustrative example of the D-PCST algorithm: proceed operation
An illustrative example of the D-PCST algorithm: pruning operation and the
![Figure 2.1: An example of the discovery step in the distributed K-SPH due to Bauer and Varma [12]](https://thumb-ap.123doks.com/thumbv2/azpdfnet/10554900.0/39.892.228.670.213.461/figure-example-discovery-step-distributed-sph-bauer-varma.webp)
![Table 2.5: Summary of related work of the PCST problem in sequential setting. LP: linear pro- pro-gramming, ILP: integer LP, MA: memetic algorithm, DP: dynamic propro-gramming, ES: empirical study, GW-clustering: Goemans and Williamson clustering [58], “*”](https://thumb-ap.123doks.com/thumbv2/azpdfnet/10554900.0/42.892.97.749.306.772/summary-sequential-gramming-algorithm-empirical-clustering-williamson-clustering.webp)
