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Summary of FCS Lecture - Revision 3

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  • Shlomi Fish
    Flame away! I fixed it according to Guy s and Nadav s input. Regards, Shlomi Fish ... * There are 8 stacks of cards, 4 freecells, and 4 foundations. The game
    Message 1 of 1 , Nov 1, 2001
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      Flame away!

      I fixed it according to Guy's and Nadav's input.

      Regards,

      Shlomi Fish

      Freecell's Rules:
      -----------------

      * There are 8 stacks of cards, 4 freecells, and 4 foundations. The game
      is played with one deck.

      * At the beginning of play, the cards are dealt to the stacks (4*7+4*6).
      The faces of all the cards are visible at the beginning of play.

      * A freecell can hold one _single_ card.

      * The foundations are built from ace to king and the object of the
      game is to move all the cards to the foundations.

      * A card can be placed on top of a card that is of the opposite colour,
      and is higher in rank by 1, assuming that the latter is at the top of a
      stack.

      * An empty stack may be filled with any card.

      * An atomic move consists of:
      - moving a card from a stack to a freecell.
      - moving a card from a freecell to a parent card on a stack.
      - moving a card from the top of a stack on top of a parent card on a
      different stack
      - moving a card from the top of a stack or from a freecell to the
      foundation.

      Strategies:
      -----------

      * A sequence of more than one card can be moved by moving the top cards to
      the freecells or to unoccupied stacks. The later is commonly called a
      "supermove"

      * Sometimes, it is useful to move cards to the freecells, so the card
      below them can serve as a basis for a sequence.

      The Freecell Solver 0.2 Architecture:
      -------------------------------------

      * A DFS scan:

      Solve-State(state, prev_states, ret)
      if (state == empty_state)
      Push(ret, state)
      return SOLVED
      for each move possible on state
      new_state <- state
      apply(new_state, move)
      if (new_state in prev_states)
      ; Do nothing
      else
      add new_state to prev_states
      if (Solve-State(new_state, prev_states, ret) == SOLVED)
      Push(ret, state)
      return SOLVED
      return UNSOLVED

      Freecell-Solver(init_state)
      if (Solve-State(init_state, Null, ret) == SOLVED)
      print "State was solved"
      while (state <- Pop(ret))
      print state
      else
      print "I could not solve this board";


      * Several tests to get to subsequent moves - give some examples:
      I called them "multi-move" to indicate they may include one or more
      moves or "supermoves". By coding them I tried to emulate the kind of
      moves a human player would try to do.

      Examples:
      ---------

      * Put top stacks cards in the foundations.
      * Put Freecell cards in the foundations.
      * Put non-top stack cards in the foundations. (by moving cards above them
      to freecells or vacant stacks).
      * Move stack cards to different stacks.
      * Move sequences of cards onto free stacks.
      * etc.

      I later improved some of these moves and added an extra one.

      Due to their multi-moves heuristic nature, it is possible that some boards
      are solveable, yet cannot be solved by FCS.

      * Once the program finds a solution it outputs the intermediate boards to
      the screen.

      * Each state was represented as a relatively large data structure containing
      other data structures.
      - A card was { short, short}
      - A stack was { int, card[] },
      - A state was { stack[] }.

      * The State collection was implemented as a sorted vector of whole state data
      structures.
      - It had a sort margin, so not every new element would require moving
      many states aside.
      - After several elements were collected the array and its sort margin
      were qsort()ed.

      The State Collection Representation Improvements:
      -------------------------------------------------

      B-search based merge to add the sort margin instead of qsort:
      -------------------------------------------------------------

      * The sort margin is kept outside the main array.

      * It is added to the main array by using a binary search-based merge.

      - The reason why it was preferred over a normal linear merge
      was because there are usually much more elements in the main
      array so a lot of time would be spent on comparisons.

      * The merge was done from end to start, so there was not any need in
      allocating an extra space.

      Pointers to structs instead of a vector of whole structs:
      ---------------------------------------------------------

      * I converted the vector to be a vector of pointer to structs, instead
      of vector of whole structs.

      * There was a great speed improvements, because only pointers were
      manipulated and not entire structs, whose movement in memory requires
      a lot of time.

      A balanced binary tree:
      -----------------------

      * I don't really know how to program my own balanced binary tree
      implementation, but I found some on the Net:

      1. libavl - contains implementations of AVL and Red-Black trees.
      2. libredblack - contains an implementation of a Red-Black tree.
      3. glib - contains an implementation of a balanced binary tree.

      There are others, but I decided to stick to those three.

      * By using a balanced binary tree I managed to increase the brute speed
      by %33. (and the net speed times 2 or so).

      A Hash Table:
      -------------

      Description: What is a Hash Table?
      ----------------------------------

      * A hash table is a vector of linked lists. Each linked list contains
      a set of keys (or key-value pairs).

      * The index of the list in which the key is placed is determined by
      a hash function. Usually, the hash function generates an integer
      in a very large range, and the index is its modulo with the number
      of elements in the hash table.

      Rehashing:
      ----------

      Sometimes, when a hash becomes two crowded, the number of elements
      in it is increased and all of its existing elements are placed in
      their new place.


      * I first tried using glib's hash implementation with a 4-byte wide XOR as
      the hash function. This generated very awful results.

      * Then I tried using the MD5 hash function, and got excellent results
      similar to what I encountered when using a balanced binary tree.

      * I coded my own stripped-down hash implementation in my attempt to figure
      out what went wrong with using a hash. It turned out to be a bit faster
      than glib's hash.

      Hash Optimizations:
      -------------------

      * Storing the hash values along with the keys. That way, the hash
      values can be first compared before comparing the entire keys.

      * Moving elements that were hit to the start of their chains.

      * When re-hashing, using the same malloc'ed elements, only re-linking
      them to their new siblings.

      The State Representation:
      -------------------------

      Reducing the Data Type Width:
      -----------------------------

      * At first FCS represented cards as a struct of two shorts where one was
      the rank and the other the suit.

      * Having seen that it consumed quite a lot of memory, I decided to try to
      reduce the size. I made a card as a char whose first 4 bits were the
      rank and the two more upper bits the rank.

      * Surprisingly, this made the game even faster.

      * I consulted a mailing list of mine with this anomaly and reached the
      conclusion that it happened because there were fewer cache misses
      this way.

      Pointers to stacks instead of a vector of stacks:
      -------------------------------------------------

      * I later encountered two variants of Freecell (dubbed Der Katzenschwanz
      and Die Schlange) whose stacks could be extremely long (roughly as
      long as the number of cards in two decks).

      * This made representing the states as a direct vector of constant-sized
      stacks very wasteful.

      * What I did was store pointers for the stacks in each state, and store
      the stacks in their separate cache.

      * If two states had an identical stack, this stack would be stored in
      memory only once.

      * With this technique, I managed to scale the number of states on a 48 MB
      + 1 GB swap machine to 1,000,000 and more.

      Remembering the Original Stack and Freecell Locations:
      ------------------------------------------------------

      * To avoid the possibility of having two states, which differ only in
      the order of their stacks or freecells, FCS 0.2 sorted the stacks in
      a canonical order, after every multi-move.

      * It displayed them this way, too, making the solution harder to follow.

      * In order to solve this problem, I wanted to keep the original order
      somehow.

      * If I sticked to the original order and sorted the stacks before every
      comparison, it would have costed dearly, because every insert requires
      more than one comparison.

      * The solution: "Indexes out of Bounds of Main Data Structure"<tm> -
      anti-patent pending technology.

      * The original indexes of the stacks and freecells are kept outside the
      main data structure:

      struct state
      {
      /* ... */
      };

      struct state_with_locations
      {
      struct state s;
      int fc_locs[4];
      int stack_locs[8];
      };

      * A comparison is made only on the first sizeof(struct state) bytes.

      Scans:
      ------

      * The original Hard-DFS scan.

      * A*

      * Soft-DFS rev. 1.

      * Soft-DFS rev. 2.

      * The BFS optimization scan.

      Move Stacks:
      ------------

      * In the beginning, the user had to deduce what had happened between two
      subsequent states.

      * That was:
      A. Not comfortable.
      B. Even harder for a Freecell implementation to do on its own.

      * I created the concept of move stacks, which each test loaded with the moves
      that were it performed.

      * Those stacks were later collected into one stack, to get the total move
      stack from the initial board to the final solution.




      ----------------------------------------------------------------------
      Shlomi Fish shlomif@...
      Home Page: http://t2.technion.ac.il/~shlomif/
      Home E-mail: shlomif@...

      If:
      1. A is A
      2. A is not not-A
      does it imply that
      1. B is B
      2. B is not not-B
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