1 / 55

Dynamically Extensible Data Structures

Dynamically Extensible Data Structures. Discrete Mathematics and Its Applications Baojian Hua bjhua@ustc.edu.cn. Information Association. In processing data structures, one often need to associate information with some kind of items Examples from graphs:

fiorentino
Download Presentation

Dynamically Extensible Data Structures

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Dynamically Extensible Data Structures Discrete Mathematics and Its Applications Baojian Hua bjhua@ustc.edu.cn

  2. Information Association • In processing data structures, one often need to associate information with some kind of items • Examples from graphs: • Vertex: data, value, degree (in, out), … • Edge: weight, … • BFS: visited, distance, …, • DFS: visited, discover, finish, … • MST: tree edges, … • Dijkstra: tree edges, … • …

  3. Several Methods • Monomorphic data in item • Polymorphic data in item • Auxiliary table (dictionary) • Dynamically extensible space in item Next, I’ll take DFS as a running example

  4. 0 0->1 0->2 0->3 1 2 3 Graph Representation: Adjacency List #include “linkedList.h” #include “graph.h” struct graph { linkedList vertices; }; typedef struct vertex *vertex; typedef struct edge *edge; struct vertex { poly data; linkedList edges; }; struct edge { vertex from; vertex to; }

  5. DFS Algorithm dfs (vertex start, tyVisit visit) { visit (start); markVisited (start); for (each adjacent vertex u of “start”) if (not visited u) // but how do we know? dfs (u, visit); }

  6. DFS Algorithm void dfsMain (graph g, poly start, tyVisit visit) { vertex startV = searchVertex (g, start); dfs (startV, visit); for (each vertex u in graph g) if (not visited u) // but how do we know? dfs (u, visit); }

  7. 0 0->1 0->2 0->3 1 2 3 Method #1: Monomorphic Data in Vertex #include “linkedList.h” #include “graph.h” struct graph { linkedList vertices; }; typedef struct vertex *vertex; typedef struct edge *edge; struct vertex { poly data; int visited; // a flag linkedList edges; }; struct edge { vertex from; vertex to; }

  8. Adjacency List-based:Creating New Vertex vertex newVertex (poly data) { vertex v = malloc (sizeof (*v)); v->data = data; v->visited = 0; v->edges = newLinkedList (); return v; } data v data visited=0 edges ### /\

  9. DFS Algorithm dfs (vertex start, tyVisit visit) { visit (start); start->visited = 1; for (each adjacent vertex u of “start”) if (0==u->visited) // Easy! :-) dfs (u, visit); }

  10. DFS Algorithm void dfsMain (graph g, poly start, tyVisit visit) { vertex startV = searchVertex (g, start); dfs (startV, visit); for (each vertex u in graph g) if (0==u->visited) dfs (u, visit); }

  11. Sample Graph DFS dfs (g, “a”, strOutput); a 0 b 0 c 0 d 0 e 0 f 0

  12. Sample Graph DFS dfs (g, “a”, strOutput); print a; a 1 b 0 c 0 d 0 e 0 f 0

  13. Sample Graph DFS dfs (g, “a”, strOutput); print a; // a choice print b; a 1 b 1 c 0 d 0 e 0 f 0

  14. Sample Graph DFS dfs (g, “a”, strOutput); print a; // a choice print b; print e; a 1 b 1 c 0 d 0 e 1 f 0

  15. Sample Graph DFS dfs (g, “a”, strOutput); print a; // a choice print b; print e; print d; a 1 b 1 c 0 d 1 e 1 f 0

  16. Sample Graph DFS dfs (g, “a”, strOutput); print a; // a choice print b; print e; print d; // a choice print c; a 1 b 1 c 1 d 1 e 1 f 0

  17. Sample Graph DFS dfs (g, “a”, strOutput); print a; // a choice print b; print e; print d; // a choice print c; print f; a 1 b 1 c 1 d 1 e 1 f 1

  18. 0 0->1 0->2 0->3 1 2 3 Method #2: Polymorphic Data in Vertex #include “linkedList.h” #include “graph.h” #include “poly.h” struct graph { linkedList vertices; }; typedef struct vertex *vertex; typedef struct edge *edge; struct vertex { poly data; poly visited; // a hook linkedList edges; }; struct edge { vertex from; vertex to;}

  19. Adjacency List-based:Creating New Vertex vertex newVertex (poly data) { vertex v = malloc (sizeof (*v)); v->data = data; v->visited = newNat (0); v->edges = newLinkedList (); return v; } data v data 0 visited edges ### /\

  20. DFS Algorithm dfs (vertex start, tyVisit visit) { visit (start); start->visited = newNat (1); for (each adjacent vertex u of “start”) if (natIsZero ((nat)(u->visited))) dfs (u, visit); }

  21. DFS Algorithm void dfsMain (graph g, poly start, tyVisit visit) { vertex startV = searchVertex (g, start); dfs (startV, visit); for (each vertex u in graph g) if (natIsZero ((nat)(u->visited))) dfs (u, visit); }

  22. Sample Graph DFS dfs (g, “a”, strOutput); a b c 0 0 0 d e f 0 0 0

  23. A garbage! A style of functional programming! Sample Graph DFS dfs (g, “a”, strOutput); print a; a b c 0 0 0 1 d e f 0 0 0

  24. Sample Graph DFS 1 dfs (g, “a”, strOutput); print a; // a choice print b; a b c 0 0 0 1 d e f 0 0 0 The rest is left as an exercise!

  25. 0 0->1 0->2 0->3 1 2 3 Method #3: Auxiliary Table (Memoization) #include “linkedList.h” #include “graph.h” struct graph { linkedList vertices; }; // Data structure definitions unchanged! :-) typedef struct vertex *vertex; typedef struct edge *edge; struct vertex { poly data; linkedList edges; }; struct edge { vertex from; vertex to; }

  26. Interface for the “table” ADT // We need a table to memoize the data items that // satisfy some conditions. #ifndef TABLE_H #define TABLE_H typedef struct tableStruct *table; table newTable (); void tableInsert (table t, poly key, poly value); void tableLookup (table t, poly key); #endif // Implementation is any dictionary-like data // structure, such as linkedList, bst, hash, etc.

  27. DFS Algorithm dfs (vertex start, tyVisit visit, table tb) { visit (start); tableInsert (tb, start, 1); for (each adjacent vertex u of “start”) if (0==tableLookup (tb, u)) dfs (u, visit, tb); }

  28. DFS Algorithm void dfsMain (graph g, poly start, tyVisit visit) { vertex startV = searchVertex (g, start); table tb = newTable (); dfs (startV, visit, tb); for (each vertex u in graph g) if (!tableLookup (tb, u)) dfs (u, visit, tb); }

  29. Sample Graph DFS dfs (g, “a”, strOutput); a b c d e f tb = newTable(); tb

  30. Sample Graph DFS dfs (g, “a”, strOutput); print a; a b c d e f tableEnter(tb, “a”, newNat(1)); tb

  31. Sample Graph DFS dfs (g, “a”, strOutput); print a; a b c d e f tableLookup (tb, “b”); // ==NULL tb

  32. Sample Graph DFS dfs (g, “a”, strOutput); print a; // a choice print b; a b c d e f tableEnter (tb, “b”, newNat(1)); tb The rest left to you!

  33. 0 0->1 0->2 0->3 1 2 3 Method #4: Dynamically Extensible Space (DES) in Vertex #include “linkedList.h” #include “graph.h” #include “plist.h” struct graph { linkedList vertices; }; typedef struct vertex *vertex; typedef struct edge *edge; struct vertex { poly data; plist list; // a list of hooks linkedList edges; }; struct edge { vertex from; vertex to;}

  34. What’s a “plist”? • A “plist” stands for “property list” • Property: some kind of value we care • A generalization of polymorphic data fields data v data ### plist /\ edges ### /\

  35. What’s a “plist”? • A “plist” stands for “property list” • it’s just a list of hooks • A hook holds some property • Dynamically extensible data v1 v2 v3 v data … ### plist edges ### /\

  36. Sample Vertex After DFS visited==1, discover==3, finish=6; // Suppose the function adding property p to // vertex v is: attach (vertex v, poly p); // which is roughly equivalent to: linkedListInsertHead (v->plist, p); “a” 1 3 6 v data ### plist /\ edges ###

  37. Sample Vertex After DFS visited==1, discover==3, finish=6; linkedListInsertHead (v->plist, p); // the calls: attach (“a”, newNat (6)); attach (“a”, newNat (3)); attach (“a”, newNat (1)); “a” 1 3 6 v data ### plist /\ edges ###

  38. 6 1 data v data ### plist /\ edges ### /\ What’s a “plist”? data v Suppose we have two calls: attach (v, newNat (1)); // discover attach (v, newNat (6)); // finish data ### plist /\ edges ### /\

  39. 6 1 data v data ### plist /\ edges ### /\ How to Find a Property? data v Suppose we have two functions: attach (v, newNat (1)); // discover attach (v, newNat (6)); // finish // How to find v’s finish time? findPlist (v->plist);??? data ### plist /\ edges ### /\ Associate every data item with a tag (think a key).

  40. How to Find a Property? Modifications to attach functions: k1 = attach (v, newNat (1)); // discover k2 = attach (v, newNat (6)); // finish // How to find v’s finish time? findPlist (v, k1); // return 1 findPlist (v, k2); // return 6 data v data ### plist /\ edges ### /\ k2 6 k1 1 data v data Associate every data item with a tag (think of a key). (k1, 1), (k2, 6) All ks are unique. ### plist /\ edges ### /\

  41. Property List Interface // In file “plist.h” #ifndef PLIST_H #define PLIST_H #include “key.h” typedef struct plistStruct *plist; // essentially a list of tuples: (key, data) void plistInsert (plist list, key k, poly data); poly plistLookup (plist list, key k); #endif // Read the offered code for implementation.

  42. 0 0->1 0->2 0->3 1 2 3 Representation Revisited #include “linkedList.h” #include “graph.h” #include “plist.h” struct graph { linkedList vertices; }; typedef struct vertex *vertex; typedef struct edge *edge; struct vertex { poly data; plist list; // a list of hooks linkedList edges; }; struct edge { vertex from; vertex to;}

  43. DFS with DES dfsMain (graph g, vertex start, tyVisit visit) { key visited = newKey (); dfs (start, visit, visited); for (each vertex u in graph g) if (!(find (u, visited)) dfs (u, visit, visited); }

  44. DFS with DES dfs (vertex v, tyVisit visit, key visited) { visit (v); attach (v, visited, newNat (1)); for (each successor t of v) if (!(find (t, visited))) dfs (t, visit, visited); }

  45. A Case Study: Order Statistics on Red-Black Tree

  46. Red-Black Tree in C // In file “rbTree.h” #ifndef RED_BLACK_TREE #define RED_BLACK_TREE typedef struct rbTree *rbTree; rbTree newRbTree (); void rbTreeInsert (rbTree t, poly data); void rbTreeDelete (rbTree t, poly data); #endif

  47. Red-Black Tree in C // In file “rbTree.c” #include “rbTree.h” struct rbTree { poly data; int color; // 0 for black, 1 for red rbTree left; rbTree right; }; // functions left to you

  48. Rank • We want to add a “rank” field (property) to the red-black tree: • to maintain the tree vertex’s order statistics • may after the red-black tree and its operations have been implemented • after the fact • How to implement this?

  49. Red-Black Tree with Rank // In file “rbTree.h” #ifndef RED_BLACK_TREE #define RED_BLACK_TREE typedef struct rbTree *rbTree; rbTree newRbTree (); void rbTreeInsert (rbTree t, poly data); void rbTreeDelete (rbTree t, poly data); void rbTreeInsertRank(rbTree t, poly data, key k); void rbTreeDeleteRank(rbTree t, poly data, key k); int rbTreeRank (rbTree t, poly data, key k); #endif

  50. Red-Black Tree in C // In file “rbTree.c” #include “plist.h” #include “rbTree.h” struct rbTree { poly data; enum {RED, BLACK} color; plist plist; rbTree left; rbTree right; }; // functions left to you

More Related