/* -*- mode: C++; c-basic-offset: 2; indent-tabs-mode: nil -*- */
/*
 *  Main authors:
 *     Patrick Pekczynski <pekczynski@ps.uni-sb.de>
 *
 *  Copyright:
 *     Patrick Pekczynski, 2004
 *
 *  Last modified:
 *     $Date: 2007-08-09 15:30:21 +0200 (Thu, 09 Aug 2007) $ by $Author: tack $
 *     $Revision: 4790 $
 *
 *  This file is part of Gecode, the generic constraint
 *  development environment:
 *     http://www.gecode.org
 *
 *  Permission is hereby granted, free of charge, to any person obtaining
 *  a copy of this software and associated documentation files (the
 *  "Software"), to deal in the Software without restriction, including
 *  without limitation the rights to use, copy, modify, merge, publish,
 *  distribute, sublicense, and/or sell copies of the Software, and to
 *  permit persons to whom the Software is furnished to do so, subject to
 *  the following conditions:
 *
 *  The above copyright notice and this permission notice shall be
 *  included in all copies or substantial portions of the Software.
 *
 *  THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
 *  EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
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 */

namespace Gecode { namespace Int { namespace GCC {

  /**
   * \brief Lower Bounds constraint (LBC) stating
   * \f$ \forall j \in \{0, \dots, |k|-1\}:
   * \#\{i\in\{0, \dots, |x| - 1\} | x_i = card(k_j)\} \geq min(k_j)\f$
   * Hence the lbc constraints the variables such that every value occurs
   * at least as often as specified by its lower cardinality bound.
   * \param home current space
   * \param x  the problem variables
   * \param nb denotes number of unique bounds
   * \param hall contains information about the hall structure of the problem
   *        (cf. HallInfo)
   * \param rank ranking information about the variable bounds (cf. Rank)
   * \param lps partial sum structure for the lower cardinality bounds (cf. PartialSum)
   * \param mu permutation \f$ \mu \f$ such that
   *        \f$ \forall i\in \{0, \dots, |x|-2\}:
   *        max(x_{\mu(i)}) \leq max(x_{\mu(i+1)})\f$
   * \param nu permutation \f$ \nu \f$ such that
   *        \f$ \forall i\in \{0, \dots, |x|-2\}:
   *        min(x_{\mu(i)}) \leq min(x_{\mu(i+1)})\f$
   */

  template <class View, class Card, bool shared>
  inline ExecStatus
  lbc(Space* home, ViewArray<View>& x, int& nb,
      HallInfo hall[], Rank rank[],
      PartialSum<Card>* lps,
      int mu[], int nu[]){

    ExecStatus es = ES_FIX;
    int n = x.size();

    /*
     *  Let I(S) denote the number of variables whose domain intersects
     *  the set S and C(S) the number of variables whose domain is containded
     *  in S. Let further  min_cap(S) be the minimal number of variables
     *  that must be assigned to values, that is
     *  min_cap(S) is the sum over all l[i] for a value v_i that is an
     *  element of S.
     *
     *  A failure set is a set F if
     *       I(F) < min_cap(F)
     *  An unstable set is a set U if
     *       I(U) = min_cap(U)
     *  A stable set is a set S if
     *      C(S) > min_cap(S) and S intersetcs nor
     *      any failure set nor any unstable set
     *      forall unstable and failure sets
     *
     *  failure sets determine the satisfiability of the LBC
     *  unstable sets have to be pruned
     *  stable set do not have to be pruned
     *
     * hall[].ps ~ stores the unstable
     *             sets that have to be pruned
     * hall[].s  ~ stores sets that must not be pruned
     * hall[].h  ~ contains stable and unstable sets
     * hall[].d  ~ contains the difference between interval bounds, i.e.
     *             the minimal capacity of the interval
     * hall[].t  ~ contains the critical capacity pointer, pointing to the
     *             values
     */

    // LBC lower bounds

    int i = 0;
    int j = 0;
    int w = 0;
    int z = 0;
    int v = 0;

    //initialization of the tree structure
    int rightmost = nb + 1; // rightmost accesible value in bounds
    int bsize     = nb + 2;
    w = rightmost;

    // test
    // unused but uninitialized
    hall[0].d = 0;
    hall[0].s = 0;
    hall[0].ps = 0;

    for (i = bsize; --i; ) { // i must not be zero
      int pred   = i - 1;
      hall[i].s  = pred;
      hall[i].ps = pred;
      hall[i].d  = lps->sumup(hall[pred].bounds, hall[i].bounds - 1);

      /* Let [hall[i].bounds,hall[i-1].bounds]=:I
       * If the capacity is zero => min_cap(I) = 0
       * => I cannot be a failure set
       * => I is an unstable set
       */
      if (hall[i].d == 0) {
        hall[pred].h = w;
      } else {
        hall[w].h = pred;
        w         = pred;
      }
    }

    w = rightmost;
    for (i = bsize; i--; ) { // i can be zero
      hall[i].t = i - 1;
      if (hall[i].d == 0) {
        hall[i].t = w;
      } else {
        hall[w].t = i;
        w = i;
      }
    }

    /*
     * The algorithm assigns to each value v in bounds
     * empty buckets corresponding to the minimal capacity l[i] to be
     * filled for v. (the buckets correspond to hall[].d containing the
     * difference between the interval bounds) Processing it
     * searches for the smallest value v in dom(x_i) that has an
     * empty bucket, i.e. if all buckets are filled it is guaranteed
     * that there are at least l[i] variables that will be
     * instantiated to v. Since the buckets are initially empty,
     * they are considered as FAILURE SETS
     */

    for (i = 0; i < n; i++) {
      // visit intervals in increasing max order
      int x0   = rank[mu[i]].min;
      int y    = rank[mu[i]].max;
      int succ = x0 + 1;
      z        = pathmax_t(hall, succ);
      j        = hall[z].t;

      /*
       * POTENTIALLY STABLE SET:
       *  z \neq succ \Leftrigharrow z>succ, i.e.
       *  min(D_{\mu(i)}) is guaranteed to occur min(K_i) times
       *  \Rightarrow [x0, min(y,z)] is potentially stable
       */

      if (z != succ) {
        w = pathmax_ps(hall, succ);
        v = hall[w].ps;
        pathset_ps(hall, succ, w, w);
        w = std::min(y, z);
        pathset_ps(hall, hall[w].ps, v, w);
        hall[w].ps = v;
      }

      /*
       * STABLE SET:
       *   being stable implies being potentially stable, i.e.
       *   [hall[y].ps, hall[y].bounds-1] is the largest stable subset of
       *   [hall[j].bounds, hall[y].bounds-1].
       */

      if (hall[z].d <= lps->sumup(hall[y].bounds, hall[z].bounds - 1)) {
        w = pathmax_s(hall, hall[y].ps);
        pathset_s(hall, hall[y].ps, w, w);
        // Path compression
        v = hall[w].s;
        pathset_s(hall, hall[y].s, v, y);
        hall[y].s = v;
      } else {
        /*
         * FAILURE SET:
         * If the considered interval [x0,y] is neither POTENTIALLY STABLE
         * nor STABLE there are still buckets that can be filled,
         * therefore d can be decreased. If d equals zero the intervals
         * minimum capacity is met and thepath can be compressed to the
         * next value having an empty bucket.
         * see DOMINATION in "gcc/ubc.icc"
         */
        if (--hall[z].d == 0) {
          hall[z].t = z + 1;
          z         = pathmax_t(hall, hall[z].t);
          hall[z].t = j;
        }

        /*
         * FINDING NEW LOWER BOUND:
         * If the lower bound belongs to an unstable or a stable set,
         * remind the new value we might assigned to the lower bound
         * in case the variable doesn't belong to a stable set.
         */
        if (hall[x0].h > x0) {
          hall[i].newBound = pathmax_h(hall, x0);
          w                = hall[i].newBound;
          pathset_h(hall, x0, w, w); // path compression
        } else {
          // Do not shrink the variable: take old min as new min
          hall[i].newBound = x0;
        }

        /* UNSTABLE SET
         * If an unstable set is discovered
         * the difference between the interval bounds is equal to the
         * number of variables whose domain intersect the interval
         * (see ZEROTEST in "gcc/ubc.icc")
         */
        // CLEARLY THIS WAS NOT STABLE == UNSTABLE
        if (hall[z].d == lps->sumup(hall[y].bounds, hall[z].bounds - 1)) {
          if (hall[y].h > y)
            /*
             * y is not the end of the potentially stable set
             * thus ensure that the potentially stable superset is marked
             */
            y = hall[y].h;
          // Equivalent to pathmax since the path is fully compressed
          int predj = j - 1;
          pathset_h(hall, hall[y].h, predj, y);
          // mark the new unstable set [j,y]
          hall[y].h = predj;
        }
      }
      pathset_t(hall, succ, z, z); // path compression
    }

    /* If there is a FAILURE SET left the minimum occurences of the values
     * are not guaranteed. In order to satisfy the LBC the last value
     * in the stable and unstable datastructure hall[].h must point to
     * the sentinel at the beginning of bounds.
     */
    if (hall[nb].h != 0) {
      return ES_FAILED; // no solution
    }

    // Perform path compression over all elements in
    // the stable interval data structure. This data
    // structure will no longer be modified and will be
    // accessed n or 2n times. Therefore, we can afford
    // a linear time compression.
    for (i = bsize; --i;) {
      if (hall[i].s > i) {
        hall[i].s = w;
      } else {
        w = i;
      }
    }

    /*
     * UPDATING LOWER BOUND:
     * For all variables that are not a subset of a stable set,
     * shrink the lower bound, i.e. forall stable sets S we have:
     * x0 < S_min <= y <=S_max  or S_min <= x0 <= S_max < y
     * that is [x0,y] is NOT a proper subset of any stable set S
     */
    for (i = n; i--; ) {
      int x0 = rank[mu[i]].min;
      int y  = rank[mu[i]].max;
      // update only those variables that are not contained in a stable set
      if ((hall[x0].s <= x0) || (y > hall[x0].s)) {
        //still have to check this out, how skipping works (consider dominated indices)
        int m = lps->skipNonNullElementsRight(hall[hall[i].newBound].bounds);
        ModEvent me = x[mu[i]].gq(home, m);
        GECODE_ME_CHECK(me);
        if (me_modified(me) && m != x[mu[i]].min()) {
          es = ES_NOFIX;
        }
        if (shared && me_modified(me)) {
          es = ES_NOFIX;
        }
      }
    }

    //LBC narrow upper bounds

    w = 0;
    for (i = 0; i <= nb; i++) {
      hall[i].d = lps->sumup(hall[i].bounds, hall[i + 1].bounds - 1);
      if (hall[i].d == 0) {
        hall[i].t = w;
      } else {
        hall[w].t = i;
        w         = i;
      }
    }
    hall[w].t = i;

    w         = 0;
    for (i = 1; i <= nb; i++) {
      if (hall[i - 1].d == 0) {
        hall[i].h = w;
      } else {
         hall[w].h = i;
         w         = i;
      }
    }
    hall[w].h = i;

    for (i = n; i--; ) {
      // visit intervals in decreasing min order
      // i.e. minsorted from right to left
      int x0   = rank[nu[i]].max;
      int y    = rank[nu[i]].min;
      int pred = x0 - 1; // predecessor of x0 in the indices
      z        = pathmin_t(hall, pred);
      j        = hall[z].t;

      /* If the variable is not in a discovered stable set
       * (see above condition for STABLE SET)
       */
      if (hall[z].d > lps->sumup(hall[z].bounds, hall[y].bounds - 1)) {
        //FAILURE SET
        if (--hall[z].d == 0) {
          hall[z].t = z - 1;
          z         = pathmin_t(hall, hall[z].t);
          hall[z].t = j;
        }
        //FINDING NEW UPPER BOUND
        if (hall[x0].h < x0) {
          w                = pathmin_h(hall, hall[x0].h);
          hall[i].newBound = w;
          pathset_h(hall, x0, w, w); // path compression
        } else {
          hall[i].newBound = x0;
        }
        //UNSTABLE SET
        if (hall[z].d == lps->sumup(hall[z].bounds, hall[y].bounds - 1)) {
          if (hall[y].h < y) {
            y = hall[y].h;
          }
          int succj = j + 1;
          //mark new unstable set [y,j]
          pathset_h(hall, hall[y].h, succj, y);
          hall[y].h = succj;
        }
      }
      pathset_t(hall, pred, z, z);
    }

    // UPDATING UPPER BOUND
    for (i = n; i--; ) {
      int x0 =  rank[nu[i]].min;
      int y  =  rank[nu[i]].max;
      if ((hall[x0].s <= x0) || (y > hall[x0].s)){
        int m = lps->skipNonNullElementsLeft(hall[hall[i].newBound].bounds - 1);
        ModEvent me = x[nu[i]].lq(home, m);
        GECODE_ME_CHECK(me);
        if (me_modified(me) && m != x[nu[i]].max()) {
          es = ES_NOFIX;
        }
        if (shared && me_modified(me)) {
          es = ES_NOFIX;
        }
      }
    }
    return es;
  }

}}}

// STATISTICS: int-prop