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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 2^n-1.

   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 "sets of small integers"

  use import 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] *)
  predicate mem int set

  (* removal
     [remove i s] can be implemented as [s - (1 lsl i)] *)
  function remove int set : set

  axiom remove_def1:
    forall x y: int, s: set.
    mem x (remove y s) <-> x <> y /\ mem x s

  (* the set {0,1,...,n-1}
     [below n] can be implemented as [1 lsl n - 1] *)
  function below int : set

  axiom below_def:
    forall x n: int. 0 <= n <= size ->
    mem x (below n) <-> 0 <= x < n

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

  type t 'a 'b model { 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 import int.Int
  use import int.MinMax
  use import map.Map
  use map.Const
  use import ref.Ref

  constant n : int
  axiom n_nonneg: 0 <= n

  use import Bitset
  axiom integer_size: n <= size

  constant m : map int (map int int)
  axiom m_pos: forall i j: int. 0 <= i < n -> 0 <= j < n -> 0 <= m[i][j]

  predicate solution (s: map int int) (i: int) =
    (forall k: int. i <= k < n -> 0 <= s[k] < n) /\
    (forall k1 k2: int. i <= k1 < k2 < n -> s[k1] <> s[k2])

  predicate permutation (s: map int int) = solution s 0

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

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

  use import option.Option
  use HashTable as H

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

  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 i /\
    (forall k: int. i <= k < n -> mem sol[k] c) /\
    r = sum sol i n /\
    (forall s: map int int.
       solution s i -> (forall k: int. i <= k < n -> mem 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, map int int) variant {2*n-2*i}
    requires { pre (i, c) /\ inv table }
    ensures { post (i,c) result /\ inv table }
  = if i = n then
      (0, Const.const 0)
    else begin
      let r = ref (-1) in
      let sol = ref (Const.const 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 i /\
              (forall k: int. i <= k < n -> mem !sol[k] c) /\
              !r = sum !sol i n /\
              (forall s: map int int.
                 solution s i -> (forall k: int. i <= k < n -> mem s[k] c) ->
                 mem s[i] c -> 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, map int 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: map int int. permutation s /\ result =  sum s 0 n }
    ensures { forall s: map int int. 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 in 6.36 s

ObligationsAlt-Ergo (0.99.1)CVC3 (2.4.1)Z3 (3.2)
sum_ind0.15------
VC for maximum---------
split_goal_wp
  1. postcondition0.02------
2. assertion0.01------
3. postcondition0.01------
4. loop invariant init0.00------
5. variant decrease0.02------
6. precondition0.03------
7. loop invariant preservation---------
split_goal_wp
  1. VC for maximum0.01------
2. VC for maximum------1.28
8. loop invariant preservation---4.41---
9. loop invariant preservation0.07------
10. assertion0.01------
11. postcondition---0.08---
VC for memo---------
split_goal_wp
  1. postcondition---------
split_goal_wp
  1. VC for memo0.02------
2. VC for memo0.00------
2. variant decrease0.010.010.02
3. precondition0.00------
4. postcondition0.04------
VC for maxmat---------
split_goal_wp
  1. assertion0.01------
2. precondition---------
inline_goal
  1. precondition---------
split_goal_wp
  1. VC for maxmat0.00------
2. VC for maxmat0.01------
3. VC for maxmat0.01------
4. VC for maxmat0.01------
5. VC for maxmat0.00------
6. VC for maxmat0.01------
3. postcondition---------
split_goal_wp
  1. postcondition---------
inline_goal
  1. postcondition0.10------
4. postcondition---------
inline_goal
  1. postcondition0.01------