Deriving divide-and-conquer dynamic programming algorithms using solver-aided transformations

Abstract

We introduce a framework allowing domain experts to manipulate computational terms in the interest of deriving better, more efficient implementations. It employs deductive reasoning to generate provably correct efficient implementations from a very high-level specification of an algorithm, and inductive constraint-based synthesis to improve automation. Semantic information is encoded into program terms through the use of refinement types.In this paper, we develop the technique in the context of a system called Bellmania that uses solver-aided tactics to derive parallel divide-and-conquer implementations of dynamic programming algorithms that have better locality and are significantly more efficient than traditional loop-based implementations. Bellmania includes a high-level language for specifying dynamic programming algorithms and a calculus that facilitates gradual transformation of these specifications into efficient implementations. These transformations formalize the divide-and-conquer technique; a visualization interface helps users to interactively guide the process, while an SMT-based back-end verifies each step and takes care of low-level reasoning required for parallelism.We have used the system to generate provably correct implementations of several algorithms, including some important algorithms from computational biology, and show that the performance is comparable to that of the best manually optimized code.

Solving the 0/1 Knapsack Problem Using Metaheuristic and Neural Networks for the Virtual Machine Placement Process in Cloud Computing Environment

“…Bellman and allows us to apprehend a problem in a diferent way than we could imagine at frst sight. Te basic concept is simple: an optimal solution is the sum of subproblems solved in an optimal way [30]. Terefore, a given problem must be Te particle swarm optimization (PSO) algorithm, the genetic algorithm, and a combination of these two algorithms called genetic-based PSO (gbPSO) are used as optimizers in the proposed Q-learning method.…”

Phased synthesis of divide and conquer programs

“…The literature on divide-and-conquer synthesis can be divided into two categories based on the class of input computations targeted: (1) those with list homomorphisms as input, with the aim of synthesizing efficient map-reduce [7] programs [1,14,21,23], (2) those that go beyond list homomorphisms [8,9,11,15,19,22], and target code with more dependencies. In category (2), the techniques in [8,9,11] synthesize list homomorphisms through some variation of lifting, the approach in [22] uses symbolic execution at runtime and to identify and defer dependencies, and Bellmania [15] targets input programs in the style of dynamic programming and orchestrates an efficient execution schedule to accommodate the dependencies. A direct comparison with work in [1,9,11,23] with respect to the class of input programs appears in Section 8.…”

Counterexample-Guided Partial Bounding for Recursive Function Synthesis

“…We defer the reasoning about the operations on the data to an SMT solver, and therefore need not rely on such rules. Techniques based on program transformation have been applied to the synthesis of special classes of recursive programs before [13,15]. For example, the work in [1] focuses on tail recursion and a lot of attention has been given to producing divide-and-conquer recursions in the way of automated parallelization [2,8,23].…”