Iterative Deepening Research Articles

Flexible Multi-Agent Decision Making Under Time Pressure

Autonomous agents need considerable computational resources to perform rational decision making. These demands are even more severe when other agents are present in the environment. In these settings, the quality of an agent's alternative behaviors depends not only on the state of the environment, but also on the actions of other agents, which in turn depend on the others' beliefs about the world, their preferences, and further on the other agents' beliefs about others, and so on. The complexity becomes prohibitive when large number of agents are present and when decisions have to be made under time pressure. In this paper, we investigate strategies intended to tame the computational burden by using offline computation in conjunction with online reasoning. We investigate two approaches. First, we use rules compiled offline to constrain alternative actions considered during online reasoning. This method minimizes overhead, but is not sensitive to changes in real-time demands of the situation at hand. Second, we use performance profiles computed offline and the notion of urgency (i.e., the value of time) computed online to choose the amount of information to be included during online deliberation. This method can adjust to various levels of real-time demands, but incurs some overhead associated with iterative deepening. We test our framework with experiments in a simulated anti-air defense domain. The experiments show that both procedures are effective in reducing computation time while offering good performance under time pressure.

Solutions for product configuration management: An empirical study

Customers can directly express their preferences on many options when ordering products today. Mass customization manufacturing thus has emerged as a new trend for its aiming to satisfy the needs of individual customers. This process of offering a wide product variety often induces an exponential growth in the volume of information and redundancy for data storage. Thus, a technique for managing product configuration is necessary, on the one hand, to provide customers faster configured and lower priced products, and on the other hand, to translate customers' needs into the product information needed for tendering and manufacturing. This paper presents a decision-making scheme through constructing a product family model (PFM) first, in which the relationship between product, modules, and components are defined. The PFM is then transformed into a product configuration network. A product configuration problem assuming that customers would like to have a minimum-cost and customized product can be easily solved by finding the shortest path in the corresponding product configuration network. Genetic algorithms (GAs), mathematical programming, and tree-searching methods such as uniform-cost search and iterative deepening A* are applied to obtain solutions to this problem. An empirical case is studied in this work as an example. Computational results show that the solution quality of GAs retains 93.89% for a complicated configuration problem. However, the running time of GAs outperforms the running time of other methods with a minimum speed factor of 25. This feature is very useful for a real-time system.

Performance evaluation of parallel iterative deepening A * on clusters of workstations

In this paper we investigate the performance of distributed heuristic search methods based on a well-known heuristic search algorithm, the iterative deepening A * (IDA *). The contribution of this paper includes proposing and assessing a distributed algorithm for IDA *. The assessment is based on space, time and solution quality that are quantified in terms of several performance parameters such as generated search space and real execution time among others. The experiments are conducted on a cluster computer system consisting of 16 hosts built around a general-purpose network. The objective of this research is to investigate the feasibility of cluster computing as an alternative for hosting applications requiring intensive graph search. The results reveal that cluster computing improves on the performance of IDA * at a reasonable cost.

The [formula omitted]-search algorithm: Application to tsume-shogi

This paper proposes a new search algorithm, denoted PN ∗ , for AND/OR tree search. The algorithm is based on proof-number (PN) search, a best-first search algorithm, proposed by Allis et al. [Artificial Intelligence 66 (1) (1994) 91–124], and on Korf's RBFS algorithm [Artificial Intelligence 62 (1) (1993) 41–78]. PN ∗ combines several existing ideas. It transforms a best-first PN-search algorithm into an iterative-deepening depth-first approach. Moreover, it is enhanced by methods such as recursive iterative deepening, dynamic evaluation, efficient successor ordering, and pruning by dependency relations. The resulting algorithm turns out to be highly efficient as witnessed by the experimental results. The PN ∗ algorithm is implemented in a tsume-shogi (Japanese-chess mating-problem) program, and evaluated by testing it on 295 notoriously difficult tsume-shogi problems (one problem has a depth of search of over 1500 plies). The experimental results are compared with those of other programs. The PN ∗ program shows by far the best results, solving all problems but one. Needless to say, it outperforms the best human tsume-shogi problem solvers by far.

Stochastic node caching for memory-bounded search

Linear-space search algorithms such as IDA* (Iterative Deepening A*) cache only those nodes on the current search path, but may revisit the same node again and again. This causes IDA* to take an impractically long time to find a solution. In this paper, we propose a simple and effective algorithm called Stochastic Node Caching (SNC) for reducing the number of revisits. SNC caches a node with the best estimate, which is currently known of the minimum estimated cost from the node to the goal node. Unlike previous related research such as MREC, SNC caches nodes selectively, based on a fixed probability. We demonstrate that SNC can effectively reduce the number of revisits compared to MREC, especially when the state-space forms a lattice.

Stochastic node caching for memory‐bounded search

Linear-space search algorithms such as IDA* (Iterative Deepening A*) cache only those nodes on the current search path, but may revisit the same node again and again. This causes IDA* to take an impractically long time to find a solution. In this paper, we propose a simple and effective algorithm called Stochastic Node Caching (SNC) for reducing the number of revisits. SNC caches a node with the best estimate, which is currently known of the minimum estimated cost from the node to the goal node. Unlike previous related research such as MREC, SNC caches nodes selectively, based on a fixed probability. We demonstrate that SNC can effectively reduce the number of revisits compared to MREC, especially when the state-space forms a lattice.

Performance of linear-space search algorithms

Search algorithms that use space linear in the search depth are widely employed in practice to solve difficult problems optimally, such as planning and scheduling. In this paper, we study the average-case performance of linear-space search algorithms, including depth-first branch-and-bound (DFBnB), iterative-deepening (ID), and recursive best-first search (RBFS). To facilitate our analyses, we use a random tree T(b,d) that has mean branching factor b, depth d, and node costs that are the sum of the costs of the edges from the root to the nodes. We prove that the expected number of nodes expanded by DFBnB on a random tree is no more than bd times the expected number of nodes expanded by best-first search (BFS) on the same tree, which usually requires space that is exponential in depth d. We also show that DFBnB is asymptotically optimal when BFS runs in exponential time, and ID and RBFS are asymptotically optimal when the edge costs of T( b, d) are integers. If bp 0 is the expected number of children of a node whose costs are the same as that of their parent, then the expected number of nodes expanded by these three linear-space algorithms is exponential when bp o < 1, at most O( d 4) when bp 0 = 1, and at most quadratic when bp 0 > 1. In addition, we study the heuristic branching factor of T( b, d) and the effective branching factor of BFS, DFBnB, ID, and RBFS on T( b, d). Furthermore, we use our analytic results to explain a surprising anomaly in the performance of these algorithms, and to predict the existence of a complexity transition in the Asymmetric Traveling Salesman Problem.

Agent Searching in a Tree and the Optimality of Iterative Deepening

The agent searching framework models the effort of a search strategy in terms of the distance traversed by an agent while exploring the search space. The framework has been found to be useful in modeling search problems where the cost of backtracking and retracing search paths is important in determining search complexity. In this paper we show that depth-first iterative deepening (DFID) strategies are optimal for an agent searching in a line, in m concurrent rays, and in uniform b-ary trees. In the conventional search model it is known that DFID is asymptotically optimal for uninformed search of uniform b-ary trees. In this paper we prove the stronger result that for agent searching in uniform b-ary trees, iterative deepening is optimal up to lower-order terms. We also discuss the problems involved in optimally performing agent search in a graph.

Agent searching in a tree and the optimality of iterative deepening

The agent searching framework models the effort of a search strategy in terms of the distance traversed by an agent while exploring the search space. The framework has been found to be useful in modeling search problems where the cost of backtracking and retracing search paths is important in determining search complexity. In this paper we show that depth-first iterative deepening (DFID) strategies are optimal for an agent searching in a line, in m concurrent rays, and in uniform b-ary trees. In the conventional search model it is known that DFID is asymptotically optimal for uninformed search of uniform b-ary trees. In this paper we prove the stronger result that for agent searching in uniform b-ary trees, iterative deepening is optimal up to lower-order terms. We also discuss the problems involved in optimally performing agent search in a graph.

Articles Published Elsewhere Enhanced Iterative-Deepening Search

Articles Published Elsewhere Enhanced Iterative-Deepening Search

IDNAF prolog

The relationship between resolution theorem proving and graph search is complicated by control statements that affect the program's meaning. An interesting case arises when combining Prolog interpreters implementing the cut and iterative deepening search. Because these features are dependent, an interpreter that naively combines both features gives unintended answers to queries. In particular, negation as failure breaks down. One solution is to use sound negation, but many certain AI problems use both sound negation and negation as failure. As well, when debugging large Prolog programs, it is useful to be able to execute the same program with and without iterative deepening, even though the program contains cuts. Besides generating unintended answers, the Prolog interpreter naively combining cuts and iterative deepening also generates redundant answers. Here, we show how to avoid both.

Proof-number search

Proof-number search (pn-search) is designed for finding the game-theoretical value in game trees. It is based on ideas derived from conspiracy-number search and its variants, such as applied cn-search and αβ-cn search. While in cn-search the purpose is to continue searching until it is unlikely that the minimax value of the root will change, pn-search aims at proving the true value of the root. Therefore, pn-search does not consider interim minimax values. Pn-search selects the next node to be expanded using two criteria: the potential range of subtree values and the number of nodes which must conspire to prove or disprove that range of potential values. These two criteria enable pn-search to treat efficiently game trees with a non-uniform branching factor. It is shown that in non-uniform trees pn-search outperforms other types of search, such as α-β iterative-deepening search, even when enhanced with transposition tables, move ordering for the full principal variation, etc. Pn-search has been used to establish the game-theoretical values of Connect-Four, Qubic, and Go-Moku. There pn-search was able to find a forced win for the player to move first. The experiments described here are in the domain of Awari, a game which has not yet been solved. The experiments are repeatable for other games with a non-uniform branching factor. This article describes the underlying principles of pn-search, presents an appropriate implementation, and provides an analysis of its strengths and weaknesses.

Linear-space best-first search

Best-first search is a general heuristic search algorithm that always expands next a frontier node of lowest cost. It includes as special cases breadth-first search, Dijkstra's single-source shortest-path algorithm, and the A∗ algorithm. Its applicability, however, is limited by its exponential memory requirement. Previous approaches to this problem, such as iterative deepening, do not expand nodes in best-first order if the cost function can decrease along a path. We present a linear-space best-first search algorithm (RBFS) that always explores new nodes in best-first order, regardless of the cost function, and expands fewer nodes than iterative deepening with a nondecreasing cost function. On the sliding-tile puzzles, RBFS with a nonmonotonic weighted evaluation function dramatically reduces computation time with only a small penalty in solution cost. In general, RBFS reduces the space complexity of best-first search from exponential to linear, at the cost of only a constant factor in time complexity in our experiments.

A Prolog technology theorem prover: a new exposition and implementation in Prolog

A Prolog technology theorem prover (PTTP) is an extension of Prolog that is complete for the full first-order predicate calculus. It differs from Prolog in its use of unification with the occurs check for soundness, depth-first iterative-deepening search instead of unbounded depth-first search to make the search strategy complete, and the model elimination reduction rule that is added to Prolog inferences to make the inference system complete. This paper describes a new Prolog-based implementation of PTTP. It uses three compile-time transformations to translate formulas into Prolog clauses that directly execute, with the support of a few run-time predicates, the model elimination procedure with depth-first iterative-deepening search and unification with the occurs check. Its high performance exceeds that of Prolog-based PTTP interpreters, and it is more concise and readable than the earlier Lisp-based compiler, which makes it superior for expository purposes. Examples of inputs and outputs of the compile-time transformations provide an easy and precise way to explain how PTTP works. This Prolog-based version makes it easier to incorporate PTTP theorem-proving ideas into Prolog programs. Some suggestions are made on extensions to Prolog that could be used to improve PTTP's performance.

Effective use of memory in iterative deepening search

The Iterative Deepening A ∗ (IDA ∗) (R.E. Korf, Artificial Intelligence 27 (1985)) algorithm often reexpands too many nodes while solving certain combinatorial problems. Algorithm IDA ∗ _CR (U.K. Sarkar, Artificial Intelligence 50 (1991)) attempted to remedy this drawback. These algorithms require very little memory although much more is available in practice. This paper introduces an algorithm IDA ∗ _CRM which shows how the available memory can be advantageously utilized in IDA ast; _CR in order to reduce the number of expanded nodes. IDA ∗ _CRAM, an approximation scheme derived from IDA ∗ _CRM, is also presented. Computational results are shown for the Flow-Shop Scheduling Problem.

Reducing reexpansions in iterative-deepening search by controlling cutoff bounds

It is known that a best-first search algorithm like A∗ [5, 6] requires too much space (which often renders it unusable) and a depth-first search strategy does not guarantee an optimal cost solution. The iterative-deepening algorithm IDA∗ [4] achieves both space and cost optimality for a class of tree searching problems. However, for many other problems, it takes too much of computation time due to excessive reexpansion of nodes. This paper presents a modification of IDA∗ to an admissible iterative depth-first branch and bound algorithm IDA∗_CR for trees which overcomes this drawback of IDA∗ and operates much faster using the same amount of storage. Algorithm IDA∗_CRA, a bounded suboptimal cost variation of IDA∗_CR is also presented in order to reduce the execution time still further. Results with the 0/1 Knapsack Problem, Traveling Salesman Problem, and the Flow Shop Scheduling Problem are shown.

PARALLEL IMPRECISE ITERATIVE DEEPENING FOR COMBINATORIAL OPTIMIZATION

In this paper we present a new parallel Iterative Deepening A* (IDA*) algorithm, the imprecise IDA* algorithm, for combinatorial optimization. This algorithm employs inadmissible heuristics. But approximate solutions which are arbitrarily close to the optimal ones can be obtained. In addition a linear speedup can be achieved by our algorithm. Experimental study has been carried out and the results are also reported here. Two computationally difficult problems, the quadratic assignment and the generalized assignment, are used in our experimental study. Our algorithm is effective for these problems of reasonable size.

Depth-limited search for real-time problem solving

We propose depth-limited heuristic search as a general paradigm for real-time problem solving in a dynamic environment. When combined with iterative-deepening, it provides the ability to commit to an action almost instantaneously, but allows the quality of that decision to improve as long as time is available. Once a deadline is reached, the best decision arrived at is executed. We illustrate the paradigm in three different settings, corresponding to single-agent search, two-player games, and multi-agent problem solving. First we review two-player minimax search with alpha-beta pruning. Minimax can be extended to themaxn algorithm for more than two players, which admits a much weaker form of alpha-beta pruning. Finally, we explore real-time search algorithms for single-agent problems. Minimax is specialized tominimin, which allows a very powerfulalpha pruning algorithm. In addition,real-time-A * allows backtracking while still guaranteeing a solution and making locally optimal decisions.

Refinements to depth-first iterative-deepening search in automatic theorem proving

This paper will discuss two refinements to the depth-first iterative-deepening search strategy. The first refinement, the priority system, is an attempt to simulate best-first search using depth-first iterative-deepening search. A new data structure, the priority list, is introduced into depth-first iterative-deepening search by the refinement. Some complexity results about the priority system are also given. The second refinement is based on a syntactic viewpoint of proof development, which views the process of finding proofs as an incremental process of constructing instances with a certain property. We quantify this process to control the depth-first iterative-deepening search. Both refinements have been implemented in a sequent-style back chaining theorem prover and tested on a large number of problems and have been shown to be effective.

A prolog technology theorem prover: Implementation by an extended prolog compiler

A Prolog technology theorem prover (PTTP) is an extension of Prolog that is complete for the full first-order predicate calculus. It differs from Prolog in its use of unification with the occurs check for soundness, the model-elimination reduction rule that is added to Prolog inferences to make the inference system complete, and depth-first iterative-deepening search instead of unbounded depthfirst search to make the search strategy complete. A Prolog technology theorem prover has been implemented by an extended Prolog-to-LISP compiler that supports these additional features. It is capable of proving theorems in the full first-order predicate calculus at a rate of thousands of inferences per second.