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Maximal sum in a matrix


Authors: Jean-Christophe Filliâtre

Topics: Matrices

Tools: Why3

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(* Given a nxn matrix m of nonnegative integers, we want to pick up one element
   in each row and each column, so that their sum is maximal.

   We generalize the problem as follows: f(i,c) is the maximum for rows >= i
   and columns in set c. Thus the solution is f(0,{0,1,...,n-1}).

   f is easily defined recursively, as we have

      f(i,c) = max{j in c} m[i][j] + f(i+1, C\{j})

   As such, it would still be a brute force approach (of complexity n!)
   but we can memoize f and then the search space decreases to n*2^n.

   The following code implements such a solution. Sets of integers are
   provided in theory Bitset. Hash tables for memoization are provided
   in module HashTable (see file hash_tables.mlw for an implementation).
   Code for f is in module MaxMatrixMemo (mutually recursive functions
   maximum and memo).
*)

theory Bitset

  use int.Int

  constant size : int (* elements belong to 0..size-1 *)

  type set

  (* membership
     [mem i s] can be implemented as [s land (1 lsl i) <> 0] *)
  val predicate mem int set

  (* removal
     [remove i s] can be implemented as [s - (1 lsl i)] *)
  val function remove (x: int) (s: set): set
    ensures { forall y: int. mem y result <-> y <> x /\ mem y s }

  (* the set {0,1,...,n-1}
     [below n] can be implemented as [1 lsl n - 1] *)
  val function below (n: int): set
    requires { 0 <= n <= size }
    ensures { forall x: int. mem x result <-> 0 <= x < n }

  val function cardinal set: int

  axiom cardinal_empty:
    forall s: set. cardinal s = 0 <-> (forall x: int. not (mem x s))

  axiom cardinal_remove:
    forall x: int. forall s: set.
    mem x s -> cardinal s = 1 + cardinal (remove x s)

  axiom cardinal_below:
    forall n: int.  0 <= n <= size ->
    cardinal (below n) = if n >= 0 then n else 0

end

module HashTable

  use option.Option
  use int.Int
  use map.Map

  type t 'a 'b = private { ghost mutable contents: map 'a (option 'b) }

  function ([]) (h: t 'a 'b) (k: 'a) : option 'b = Map.get h.contents k

  val create (n:int) : t 'a 'b
    requires { 0 < n } ensures { forall k: 'a. result[k] = None }

  val clear (h: t 'a 'b) : unit writes {h}
    ensures { forall k: 'a. h[k] = None }

  val add (h: t 'a 'b) (k: 'a) (v: 'b) : unit writes {h}
    ensures { h[k] = Some v /\ forall k': 'a. k' <> k -> h[k'] = (old h)[k'] }

  exception Not_found

  val find (h: t 'a 'b) (k: 'a) : 'b
    ensures { h[k] = Some result } raises { Not_found -> h[k] = None }

end

module MaxMatrixMemo

  use int.Int
  use int.MinMax
  use ref.Ref
  use Bitset
  use map.Map

  clone appmap.Appmap with type key = int, axiom .

  val constant n : int
    ensures { 0 <= result <= size }

  val constant m : t (t int)
    ensures { forall i j: int. 0 <= i < n -> 0 <= j < n -> 0 <= result[i][j] }

  type mapii = Map.map int int

  predicate solution (s: mapii) (i: int) =
    (forall k: int. i <= k < n -> 0 <= Map.get s k < n) /\
    (forall k1 k2: int. i <= k1 < k2 < n -> Map.get s k1 <> Map.get s k2)

  predicate permutation (s: mapii) = solution s 0

  function f (s: mapii) (i: int) : int = m[i][Map.get s i]
  clone sum.Sum with type container = mapii, function f = f, axiom .

  lemma sum_ind:
    forall i: int. i < n -> forall j: int.
    forall s: mapii. sum (Map.set s i j) i n = m[i][j] + sum s (i+1) n

  use option.Option
  use HashTable as H

  type key = (int, set)
  type value = (int, t int)

  predicate pre (k: key) =
    let (i, c) = k in
    0 <= i <= n /\ cardinal c = n-i /\ (forall k: int. mem k c -> 0 <= k < n)

  predicate post (k: key) (v: value) =
    let (i, c) = k in
    let (r, sol) = v in
    0 <= r /\ solution sol.contents i /\
    (forall k: int. i <= k < n -> mem sol[k] c) /\
    r = sum sol.contents i n /\
    (forall s: mapii.
       solution s i -> (forall k: int. i <= k < n -> mem (Map.get s k) c) ->
       r >= sum s i n)

  type table = H.t key value

  val table: table

  predicate inv (t: table) =
    forall k: key, v: value. H.([]) t k = Some v -> post k v

  let rec maximum (i:int) (c: set) : (int, t int) variant {2*n-2*i}
    requires { pre (i, c) /\ inv table }
    ensures { post (i,c) result /\ inv table }
  = if i = n then
      (0, create 0)
    else begin
      let r = ref (-1) in
      let sol = ref (create 0) in
      for j = 0 to n-1 do
        invariant {
          inv table /\
          (  (!r = -1 /\ forall k: int. 0 <= k < j -> not (mem k c))
          \/
            (0 <= !r /\ solution !sol.contents i /\
              (forall k: int. i <= k < n -> mem !sol[k] c) /\
              !r = sum !sol.contents i n /\
              (forall s: mapii.
                 solution s i -> (forall k: int. i <= k < n -> mem (Map.get s k) c) ->
                 mem (Map.get s i) c -> Map.get s i < j -> !r >= sum s i n)))
        }
        if mem j c then
          let (r', sol') = memo (i+1) (remove j c) in
          let x = m[i][j] + r' in
          if x > !r then begin r := x; sol := sol'[i <- j] end
      done;
      assert { 0 <= !r };
      (!r, !sol)
    end

  with memo (i:int) (c: set) : (int, t int) variant {2*n-2*i+1}
    requires { pre (i,c) /\ inv table }
    ensures { post (i,c) result /\ inv table }
  = try  H.find table (i,c)
    with H.Not_found -> let r = maximum i c in H.add table (i,c) r; r end

  let maxmat ()
    ensures { exists s: mapii. permutation s /\ result =  sum s 0 n }
    ensures { forall s: mapii. permutation s -> result >= sum s 0 n }
  = H.clear table;
    assert { inv table };
    let (r, _) = maximum 0 (below n) in r

end

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Why3 Proof Results for Project "max_matrix"

Theory "max_matrix.MaxMatrixMemo": fully verified

ObligationsAlt-Ergo 1.30CVC4 1.4CVC4 1.5Z3 4.5.0
VC n0.00---------
VC m0.00---------
sum_ind0.03---------
VC maximum------------
split_goal_right
VC maximum.00.01---------
VC maximum.10.00---------
VC maximum.20.00---------
VC maximum.30.01---------
VC maximum.4------------
split_vc
VC maximum.4.00.01---------
VC maximum.4.1------------
right
VC maximum.4.1.0------------
split_goal_right
VC maximum.4.1.0.0------0.04---
VC maximum.4.1.0.1---------0.10
VC maximum.4.1.0.20.03---------
VC maximum.4.1.0.3------0.06---
VC maximum.4.1.0.44.08---------
VC maximum.5---2.49------
VC maximum.60.04---------
VC maximum.70.01---------
VC maximum.80.08---------
VC maximum.90.00---------
VC memo0.02---------
VC maxmat0.12---------