Code accompanying a forthcoming paper of Yu-Han Lyu, Akimasa Morihata, and me: Constructing Datatype-Generic Fully Polynomial-Time Approximation Schemes Using Generalised Thinning.

The file fptas.zip consists of the following Haskell modules:

• KnapsackSpec: specification of the 0-1 knapsack problem.
• Knapsack: a thinning algorithm solving knapsack (knapsack), and an approximation algorithm (knapsack_apx).
• KnapsackTest: some QuickCheck properties to test the code.
• PartTreesSpec: specification of the maximal tree partition problem.
• PartTrees: a thinning algorithm solving the tree partition problem (mtp), and an approximation algorithm (mtp_apx).
• PartTreesTest: some QuickCheck properties to test the code.
• Utilities: some utilities used by PartTreesSpec.
• Merging: generalised merging and bumping functions for both programs.

The problem instances generated by QuickCheck very rapidly get too large in size. The function smallCheck defined in both *Test modules restricts the sizes of instances generated.

Quick links: [expository program | linear time program | proofs (late update: 2010.04.17)].

Expository Code

The expository program [download here] intends to be an executable implementation of the code in the paper. For clarity we use Haskell lists for all sequences, and do not implement optimisations such as storing the areas and breadths of segments, thus the program is not linear time yet. A linear time implementation will follow soon.

The code is mostly consistent with the paper, with some minor differences: many functions take an additional argument of type BreadthSpec = (Breadth, Maybe Breadth). The first component is the lower bound, while the second component is an optional upperbound. The main function is:

mds :: BreadthSpec ->  [Elem] -> Maybe [Elem]

which takes a BreadthSpec and switches between the modes with or without an upper bound.

To try the code, you may either load the module Main into ghci and invoke the function mds:

mds (lb, Just ub) [x1, x2, x3, ... ]

or load the module Test and run some QuickCheck properties:

Test.QuickCheck.quickCheck (prop_mds_correct bb (lb, Just ub))

where lb and ub are the lower and upper bounds, and bb is the bound on breadths of generated test elements. The property prop_mds_correct asserts that mds (lb,ub) x =d mds_spec (lb,ub) x for all x.

The gzipped file consists of the following Haskell modules:

• Main: containing the main program mds, our variant of zygomorphism zh, wp2, smsp2, maxChop, trim, etc.
• Spec: containing a specification of the maximally dense segment problem:

  mds_spec :: BreadthSpec -> [Elem] -> Maybe [Elem]
  mds_spec bs = maxlistM d . filter (bounds bs) . nonEmptySegs

  Many types having area, breadth, and density defined are collected into a type class Block, and functions like maxChop are define on the type class.

• DRSP: specification of right-skew segments and DRSP, with functions including rightSkew, sdars, lrsp, and drsp.
• DPTrees: defining DTrees and PTrees, and functions like addD and prependD allowing one to construct DRSPs in a fold.
• Utilities: some utility functions processing lists.
• Test: a module defining some QuickCheck properties to test the code.

Linear Time Implementation

A linear time implementation can be downloaded here. The program uses Data.Sequence to represent the compulsory part and the first level of the DForest and the PForest of the window, as well as annotating them with areas and breadths. The subtrees of a DTree and a PTree, however, can be represented simply by snoc-lists and cons-lists respectively.

Organisation of the code is the same as the first program.

Proofs

Proofs accompanying the paper [PDF]. Theorems and lemmas are labelled with both their own numbers as well as the numbers in the paper, if any. For example, Lemma A.1 (3.1) is Lemma 3.1 in the paper.

An Agda library accompanying the paper Algebra of Programming in Agda: Dependent Types for Relational Program Derivation, developed in co-operation with Hsiang-Shang Ko and Patrik Jansson.

Dependent type theory is rich enough to express that a program satisfies an input/output relational specification, but it could be hard to construct the proof term. On the other hand, squiggolists know very well how to show that one relation is included in another by algebraic reasoning. The AoPA library allows one to encode Algebra of Programming style program derivation, both functional and relational, in Agda.

Example

The following is a derivation of insertion sort in progress:

isort-der : ∃ (\f → ordered? ○ permute ⊒ fun f )
isort-der = (_ , (
  ⊒-begin
      ordered? ○ permute
  ⊒⟨ (\vs -> ·-monotonic ordered? (permute-is-fold vs)) ⟩
      ordered? ○ foldR combine nil
  ⊒⟨ foldR-fusion ordered? ins-step ins-base ⟩
      foldR (fun (uncurry insert)) nil
  ⊒⟨ { foldR-to-foldr insert []}0 ⟩
      { fun (foldr insert [])
  ⊒∎ }1))

isort : [ Val ] -> [ Val ]
isort = proj₁ isort-der


The type of isort-der is a proposition that there exists a function f that is contained in ordered ? ◦ permute , a relation mapping a list to one of its ordered permutations. The proof proceeds by derivation from the specification towards the algorithm. The first step exploits monotonicity of ◦ and that permute can be expressed as a fold. The second step makes use of relational fold fusion. The shaded areas denote interaction points — fragments of (proof ) code to be completed. The programmer can query Agda for the expected type and the context of the shaded expression. When the proof is completed, an algorithm isort is obtained by extracting the witness of the proposition. It is an executable program that is backed by the type system to meet the specification.

The complete program is in the Example directory of the code.

The Code

The code consists of the following files and folders:

• AlgebraicReasoning: a number of modules supporting algebraic reasoning. At present we implement our own because the PreorderReasoning module in earlier versions of the Standard Library was not expressive enough for our need. We may adapt to the new Standard Library later.
• Data: defining relational fold, unfold, hylomorphism (using well-founded recursion), the greedy theorem, and the converse-of-a-function theorem, etc, for list and binary tree.
• Examples: currently we have prepared four examples: a functional derivation of the maximum segment sum problem, a relational derivation of insertion sort and quicksort (following the paper Functional Algorithm Design by Richard Bird), and solving an optimisation problem using the greedy theorem.
• Relations: modules defining various properties of relations.
• Sets: a simple encoding of sets, upon with Relations are built.

Download

To grab the latest code, install darcs and check our the code from the repository:

darcs get http://pc-scm.iis.sinica.edu.tw/repos/AoPA


AoPA makes use of the Standard Library, to install which you will need darcs.

While studying XML stream processing, we noticed that programmers need concurrency to save space, especially in a lazy language. We propose the idea of pushing datatypes — when a pushing closure is demanded, all expressions referring to it are evaluated concurrently to weak head normal forms. The closure is no more alive and may thus be garbage collected.

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1. mssu.lhs: an amortised linear-time algorithm computing the maximum sum of segments not longer than an upper-bound;
2. msdlb.lhs: an O(n log L) algorithm computing the maximum density (average) of segments not shorter than a lower-bound;
3. msdll.lhs: computing the maximum density (average) of segments not shorter than a lower-bound. With the discovery of Glodwasser et al. we are able to refine the algorithm to amortised linear time again.

2007/06/26 Update: fixed binary search.
2007/11/04 Update: linear time algorithm for MSDL.

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