Richard M. Karp and Yanjun Zhang. 1993. Randomized parallel algorithms for backtrack search and branch-and-bound computation. Journal of the ACM, 40(3):765--789, July.  Home/Search   Document Not in Database   Summary   ACM   TOC   Related Articles   Check

....of processors used. Consequently, since AABBs are very fast to test for overlap, we only got very limited speedup 30 with 4 processors. For OBBs, however, the speedup peaks at 3.2 as can be seen in Figure 1. d) 6 Related Work and Discussion Several older parallel branch and bound techniques [2, 5, 7, 9, 10, 19] and depthfirst search algorithms like backtracking [11 13] seem at a first glance to be applicable to the applications we have at hand. Our results indicate, however, that the load distribution strategies in these algorithms do not apply very well to tree traversals found in VFC and collision ....

Richard M. Karp, Yanjun Zhang, "Randomized Parallel Algorithms for Backtrack Search and Branch-and-Bound Computation", Journal of the ACM, Volume 40, Pages 765-789, Issue 3, 1993.

....space requirements and communication costs. Related work in this area includes a randomized work stealing algorithm for load balancing independent jobs [89] algorithms for dynamically embedding trees in fixedconnection networks [5, 71] and algorithms for backtrack search and branch and bound [61, 65, 75, 86, 109]. Backtrack search can be viewed as a multithreaded computation with no synchronization, and in the work just cited, the only algorithm with reasonable space bounds is the random work stealing algorithm of Karp and Zhang [65, 109] though they did not make this observation until the later work of ....

.... and algorithms for backtrack search and branch and bound [61, 65, 75, 86, 109] Backtrack search can be viewed as a multithreaded computation with no synchronization, and in the work just cited, the only algorithm with reasonable space bounds is the random work stealing algorithm of Karp and Zhang [65, 109], though they did not make this observation until the later work of Zhang and Ortynski [108] Our results specialize to match theirs. Likewise, most of the systems oriented work on multithreading has ignored the space issue. Notable exceptions include the k bounded loops of Culler and Arvind [34, ....

[Article contains additional citation context not shown here]

Richard M. Karp and Yanjun Zhang. Randomized parallel algorithms for backtrack search and branch-and-bound computation. Journal of the ACM, 40(3):765--789, July 1993.

....processor gets approximately an equal amount of computation load and communicating processes are assigned to processors that are as close as possible. This problem is known as the dynamic tree embedding problem. It is an important problem in parallel computing and has attracted various efforts [1 3, 5 10]. Let tree T (V T , E T ) represent a computation (nodes for processes, edges for communications) and graph G (V G , EG ) represent a parallel computer (nodes for processors, edges for communication links) For simplicity, it is usually assumed that all the tree nodes carry an equal amount ....

....algorithms. There have been various randomized algorithms proposed for dynamic tree embedding in hypercubes by Aiello and Leighton [1] Bhatt et al. 3] Leighton et al. 6] and Bhatt and Cai [2] in butterfly networks by Ranade [10] and in completely connected networks by Karp and Zhang [5] and Li [7] A common technique used in many randomized algorithms for dynamic tree embedding [2, 3, 6] is a random walk: when a new node u # is created by tree node u, u # is assigned to a randomly chosen neighbor of #(u) with an equal probability among all #(u) s neighbors 146 SHEN et al. in ....

R. M. Karp and Y. Zhang, Randomized parallel algorithms for backtrack search and branch-and-bound computation, J. Assoc. Comput. Mach. 40, 3 (1993), 765--789.

....be addressed: a) How often the load balancing algorithm is invoked by the algorithm (b) Is it necessary to balance the loads of the sub pools completely perhaps it would save some time to stop when a partial balance is achieved. The only results in this direction that we are aware of are [KZ93, BL94] Chapter 5 Conclusions and Future Directions In Section 5.1, we summarize the chief contributions of this thesis, placing them in the context of past work. In Section 5.2, we point out directions for further research in other areas of parallel operating systems. Naturally, the ....

R.M. Karp and Y. Zhang. Randomized parallel algorithms for backtrack search and branch-and-bound computation. Journal of the Association of Computing Machinery, 40(3):765--789, 1993.

....shuffle exchange networks, and de Bruijn graphs [24] Embedding dynamic trees has also been treated from other points of view. Karp and Zhang presented randomized parallel backtrack and branch and bound algorithms on completely connected networks that run in optimal time with high probability [11]. Ranade strengthened the results by considering butterflies which are bounded degree networks [28] Instead of embedding trees with lower dilation and even load distribution, the aim of their research is to minimize the parallel execution time. These pioneer studies have been followed by a number ....

R. M. Karp and Y. Zhang, Randomized parallel algorithms for backtrack search and branch-andbound computation, J. Assoc. Comput. Mach. 40(3) (1993), 765#789.

....of the tasks themselves. The problem is more complex if the tasks have explicit dependencies; for instance, the tasks may represent a multi threaded computation modeled as a directed acyclic graph [BL94, ABP98] or the tasks may be generated by a backtrack search or branch and bound algorithm [KZ93] The situation where the tasks are independent is somewhat simpler and several models for generating and consuming independent tasks are considered in the literature [RSAU91, BFM98, BFS99] In the random load model each processor in each step generates a task with a xed probability and ....

R. M. Karp and Y. Zhang. Randomized parallel algorithms for backtrack search and branch-and-bound computation. Journal of the ACM, 40(3):765-789, July 1993.

....of the tasks themselves. The problem is more complex if the tasks have explicit dependencies; for instance, the tasks may represent a multi threaded computation modeled as a directed acyclic graph [BL94, ABP98] or the tasks may be generated by a backtrack search or branch and bound algorithm [KZ93] The situation where the tasks are independent is somewhat simpler and several models for generating and consuming independent tasks are considered in the literature [RSAU91, BFM98, BFS99] In the random load model each processor in each step generates a task with a fixed probability and ....

R. M. Karp and Y. Zhang. Randomized parallel algorithms for backtrack search and branch-and-bound computation. Journal of the ACM, 40(3):765--789, July 1993.

....to result in significant load imbalance among processors. The core of parallel formulations of DFS algorithms is thus a dynamic load balancing technique that minimizes inter processor communication and processor idling. A number of load distribution techniques have been developed for parallel DFS [25, 41, 22, 17, 21]. Load balancing techniques may be receiver initiated or sender initiated. In receiver initiated techniques, when a processor becomes idle, it requests a selected processor for work. Many different selection policies have been proposed such as use of a centralized server, random polling [12] ....

R. Karp and Y. Zhang. Randomized parallel algorithms for backtrack search and branch-and-bound computation. Journal of ACM, 40:765--789, 1993.

....multiplicative factor in the running time of Theorem 1 captures the average overhead per step required to ensure that the workload is equitably distributed among the processors. Efficient backtrack search algorithms have been obtained previously for the completely connected network of processors [KZ93, LAB93], the butterfly network [Ran94] and the two dimensional mesh [KP94] under a variety of cost models. A number of related problems such as branch and bound [Ran90, KZ93, LAB93, KP94] and dynamic tree embedding [LNRS92, AL91, BGLL91] have also been addressed in the literature. Both Ranade s ....

....Efficient backtrack search algorithms have been obtained previously for the completely connected network of processors [KZ93, LAB93] the butterfly network [Ran94] and the two dimensional mesh [KP94] under a variety of cost models. A number of related problems such as branch and bound [Ran90, KZ93, LAB93, KP94] and dynamic tree embedding [LNRS92, AL91, BGLL91] have also been addressed in the literature. Both Ranade s butterfly algorithm [Ran94] and the Karp Zhang algorithm [Ran90, KZ93] are randomized and explore the tree in O(n=p h) steps, with high probability. It should be noted that the ....

[Article contains additional citation context not shown here]

R.M. Karp and Y. Zhang. Randomized parallel algorithms for backtrack search and branch and bound computation. Journal of the ACM, 40:765-- 789, 1993.

....the first stage will select list elements so that g, the length of longest sublist, is sufficiently small. Once the list has been split in this way, in the second stage we invoke a straightforward adaptation to the CRCW PRAM of the well known randomized backtrack search algorithm of Karp and Zhang [6] that marks all the list nodes while balancing the work among the processors in O( p g) time, with high probability. Variants of this splitting technique are employed by Greene and Knuth [4] for graph traversal, and by Ullman and Yannakakis [10] for graph searching. By interleaving the above ....

Karp, R. M., and Zhang, Y. Randomized parallel algorithms for backtrack search and branch and bound computation J. ACM 40, 3 (July 1993), 765--789.

....algorithms that were on line (b) On line arrivals, minimizing makespan [SWW95] c) On line arrivals, minimizing # w j C j [HSSW97, CPS 96] 6. On line Load Balancing [AAF 93] maybe more 7. Shop Scheduling Models 8. Scheduling Jobs in Parallel Environments (a) Branch and Bound [KZ93] b) Packet Routing and scheduling [LMR94] c) Network Scheduling [PSW98] d) Communication Delays [PY90] 9. Optional Topics More scheduling models and problems 2 ....

Richard M. Karp and Yanjun Zhang. Randomized parallel algorithms for backtrack search and branch-and-bound computation. Journal of the ACM, 40(3):765--789, July 1993.

....very difficult when the program has complicated data access patterns. Perhaps the earliest class of techniques was to attempt to execute threads that are close in the computation graph on the same processor [1, 9, 20, 23, 26, 28] The work stealing algorithm is the most studied of these techniques [9, 11, 19, 20, 24, 36, 37]. Blumofe et al. showed that fully strict computations achieve a provably good data locality [7] when executed with the work stealing algorithm on a dag consistent distributed shared memory systems. In recent work, Narlikar showed that work stealing improves the performance of space efficient ....

....as race free computations. In this paper we consider only race free computations. The work stealing algorithm is a thread scheduling algorithm for multithreaded computations. The idea of work stealing dates back to the research of Burton and Sleep [11] and has been studied extensively since then [2, 9, 19, 20, 24, 36, 37]. In the work stealing algorithm, each process maintains a pool of ready threads and obtains work from its pool. When a process spawns a new thread the process adds the thread into its pool. When a process runs out of work and finds its pool empty, it chooses a random process as its victim and ....

Richard M. Karp and Yanjun Zhang. Randomized parallel algorithms for backtrack search and branch-and-bound computation. Journal of the ACM, 40(3):765--789, July 1993.

....associated with the nodes are generated dynamically in an irregular and unpredictable fashion. A suitable abstract framework for studying the balancing and communication issues involved in the parallel implementation of branch and bound is provided by the branch and boundproblem, introduced in [KZ93], which can be specified as follows. Let T be an arbitrary tree of finite size. Initially, a pointer to the root is available, while pointers to children are revealed only after their parent is visited. A node can be visited only if a pointer to it is available, and it is assumed that the visit ....

....algorithm to select the k th smallest item in an infinite heap in O (k) time. The algorithm can be easily adapted to yield an optimal O (n) sequential algorithm for branch and bound. In parallel computation, the branch and bound problem has been studied on a variety of machine models. In [KZ93], Karp and Zhang show, by a simple work diameter argument, that any algorithm for the problem requires at least Omega (n=p h) time on any p processor machine, and devise a general randomized algorithm, running in O (n=p h) steps, with high probability. Each step of the algorithm entails a ....

R.M. Karp and Y. Zhang. Randomized Parallel Algorithms for Backtrack Search and Branch and Bound Computation. Journal of the ACM, 40:765--789, 1993.

....and that communication overhead is kept at the minimum. It turns out that randomization is necessary if both the maximum load and dilation are to be minimized [8] Randomized tree embedding algorithms and their performance analysis have been investigated by many researchers in recently years [1, 2, 3, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23]. In this paper, we provide a unified framework for studying the performance of randomized tree embedding algorithms which allow a newly created tree node to take a random walk of a short distance to reach a processor nearby. In particular, we propose simple randomized algorithms on several most ....

R.M. Karp and Y. Zhang, "Randomized parallel algorithms for backtrack search and branch-and-bound computation," Journal of the ACM, vol.40, no.3, pp.765-789, 1993.

.... h) time, since (a) at least Omega Gamma n) work is needed to visit n nodes and (b) there exists a path of h nodes whose exploration times form a strictly increasing sequence. A number of earlier studies have investigated this problem [Ran91, KP95] and the related problems of branch and bound [KZ93, KP95] and dynamic tree embedding [BGLL91] and motivate the significance of these problems by indicating their application to various tree structured combinatorial optimization problems. Optimal randomized algorithms requiring time O(n=p h) with high probability are given for the p processor clique ....

....tree embedding [BGLL91] and motivate the significance of these problems by indicating their application to various tree structured combinatorial optimization problems. Optimal randomized algorithms requiring time O(n=p h) with high probability are given for the p processor clique architecture [KZ93] and the butterfly interconnection [Ran91] The paper of Kaklamanis and Persiano [KP95] describes a deterministic algorithm with a running time of O( p ph log p) on the p node two dimensional mesh and a lower bound showing that this algorithm is within a polylogarithmic factor of optimality for ....

R.M. Karp and Y. Zhang. Randomized parallel algorithms for backtrack search and branch and bound computation. Journal of the ACM, 40:765--789, 1993.

.... scheduling a computation whose ultimate shape is unknown (precluding offline planning as in [3, 4, 7] on an architecture that has a large diameter (precluding efficient random placements as in [6] and small bisection bandwidth (precluding efficient massive data transmission as in [5]) We empirically analyze and compare two distributed, low overhead dynamic scheduling policies. Our simpler policy called KOSO, for keep one send one has each PE keep one child of a spawning task and pass the other to its clockwise neighbor in the ring; our more sophisticated ....

R.M. Karp and Y. Zhang (1988): Randomized parallel algorithms for backtrack search and branch-and-boundcomputation. J. ACM 40, 765-789.

....sequence, it follows that any algorithm for the backtrack search problem requires n=p h) time on a p processor machine. A number of works on parallel backtrack search have appeared in the literature. Randomized algorithms have been developed for the completely connected network of processors [KZ93,LAB93] and the butter y network [Ran94] which run, optimally, in O(n=p h) steps, with high probability. It should be noted, however, that the butter y algorithm focuses on the number of node visiting steps and does not fully account for overhead due to manipulations of local data structures. A ....

....to work for larger tree sizes. The relationship between computation and communication for the exploration of trees arising from irregular divide and conquer computations has been studied in [WK91] A number of related problems have also been addressed in the literature, such as branch and bound [Ran90,KZ93,LAB93,KP94] and dynamic tree embeddings [AL91,BGLL91,LNRS92] In this paper, we present a deterministic PRAM algorithm for backtrack search whose running time is within a triply logarithmic factor of the natural lower bound discussed above. Our main result is summarized in the following theorem. Theorem 1 ....

[Article contains additional citation context not shown here]

R.M. Karp and Y. Zhang. Randomized parallel algorithms for backtrack search and branch and bound computation. Journal of the ACM, 40:765{ 789, 1993.

....(G) In the rest of this paper, we use the notation h instead of h(G) that is, we omit G whenever it is clear from the context. In this work, we concentrate on multithreaded computations with h 0. The case of strict computations with no dependency edges (where h = 0) has already been studied in [12, 34, 57]. Such computations are, e.g. backtrack search computations. Our example computation of Figure 1 is a strict computation. However, the dashed dependency edges of this computation are not fully strict edges. If we remove the dashed edges from the multithreaded computation of Figure 1, it becomes a ....

....stealing. Moreover, since each dependency accounts for at most a constant number of messages, the communication cost of resolving any dependency is amortized by the cost n d S max . Thus, Proposition 4.4 immediately implies the stated claims. 2 42 5 Related Work Substantial research (see e.g. [1, 34, 50, 56]) has been reported in the literature concerning the scheduling of multithreaded computations, ignoring though space requirements and communication costs. The work stealing paradigm dates back at least as far as Burton and Sleep s research [16] on parallel execution of functional programs and ....

R. M. Karp and Y. Zhang, \Randomized parallel algorithms for backtrack search and branch-andbound computation," Journal of the ACM, 40(3):765-789, July 1993.

....prove that for any ffl 0, with probablity at least 1 Gamma ffl the algorithm s execution time is O(T 1 =P hT1 log P log(1=ffl) while its communication complexity is O(P (hT1 log(1=ffl) 1 n d )S max ) with probability again at least 1 Gamma ffl. Substantial research (see e.g. [1, 20, 24, 28]) has been reported in the literature concerning the scheduling of multithreaded computations, ignoring though space requirements and communication costs. Burton shows in [10] how to limit space in certain parallel computations without causing deadlock. More recently, Burton [9] has developed and ....

R. M. Karp and Y. Zhang, "Randomized parallel algorithms for backtrack search and branch-andbound computation," Journal of the ACM, Vol. 40, No. 3, pp. 765--789, July 1993.

....of the processors memories. Load balancing processes are required which produce additional simulation overhead. Thus, new algorithms for distributed memory were investigated. These algorithms use local sequential priority queues on each processor to maintain a global parallel priority queue [23] Early attempts in this direction proposed algorithms that do not guarantee the selection of the element with global maximum priority. The semantics of a priority queue changed in a way not acceptable for many applications. Recently published algorithms provide real parallel priority queues for ....

R.M. Karp and Y. Zhang. Randomized parallel algorithms for backtrack search and branch-andbound computation. Journal of the ACM, 40(3):765--789, 1993.

....associated with the nodes are generated dynamically in an irregular and unpredictable fashion. A suitable abstract framework for studying the balancing and communication issues involved in the parallel implementation of branch and bound is provided by the branch and bound problem, introduced in [8], which can be specified as follows. Let T be an arbitrary tree of finite size. Initially, a pointer to the root is available, while pointers to children are revealed only after their parent is visited. A node can be visited only if a pointer to it is available, and it is assumed that the visit ....

....algorithm to select the k th smallest item in an infinite heap in O (k) time. The algorithm can be easily adapted to yield an optimal O (n) sequential algorithm for branch and bound. In parallel computation, the branch and bound problem has been studied on a variety of machine models. In [8], Karp and Zhang show, by a simple work diameter argument, that any algorithm for the problem requires at least Omega (n=p h) time on any p processor machine, and devise a general randomized algorithm, running in O (n=p h) steps, with high probability. Each step of the algorithm entails a ....

[Article contains additional citation context not shown here]

R.M. Karp and Y. Zhang, "Randomized parallel algorithms for backtrack search and branch and bound computation," J. of the ACM 40 (1993) 765--789.

....which takes one unit of time to transfer messages from send buffers to receive buffers at the destination. We believe this is reasonable in applications that involve the atomic transfer of large data sets. Unit delay assumptions are also made in the literature on PRAMs and complete networks [28, 29]. Unlike these models however, we explicitly account for message contention and do not allow multiple messages to be received in one step by a processor. The issue of contention at the receiving module is also addressed in models for optical communication [21] and module parallel computers [27, ....

....leaf is h, it is easy to see that the time to examine all leaves is at least Omega Gamma n=p h) where p denotes the number of processors. Branch and bound search is similar to backtrack search, except that only a subtree of the search tree must necessarily be explored. Following Karp and Zhang [28, 29], we model a branch and bound tree as a binary search tree, each of whose vertices has an associated cost. The cost of each vertex is strictly less than the cost of each of its children (for simplicity we assume that all vertex costs are distinct) The problem is to find the leaf with minimum cost ....

R.M. Karp and Y. Zhang. Randomized parallel algorithms for backtrack search and branch-and-bound computations. Journal of the ACM, 40, 1993.

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Richard M. Karp and Yanjun Zhang. 1993. Randomized parallel algorithms for backtrack search and branch-and-bound computation. Journal of the ACM, 40(3):765--789, July.

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R. Karp and Y. Zhang. Randomized parallel algorithms for backtrack search and branch-andbound computation. Journal of the Association for Computing Machinery, 40(3):765--789, July 1993.

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Karp R.M.; Zhang Y.: Randomized Parallel Algorithms for Backtrack Search and Branch-and-Bound Computation; Journal of the ACM 40 (1993); pp. 765-789.

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