Representing Tables

When we studied various ways of representing sets in chapter 2, we mentioned in section 2.3.3 the task of maintaining a table of records indexed by identifying keys. In the implementation of data-directed programming in section 2.4.3, we made extensive use of two-dimensional tables, in which information is stored and retrieved using two keys. Here we see how to build tables as mutable list structures.

We first consider a one-dimensional table, in which each value is stored under a single key. We implement the table as a list of records, each of which is implemented as a pair consisting of a key and the associated value. The records are glued together to form a list by pairs whose cars point to successive records. These gluing pairs are called the backbone of the table. In order to have a place that we can change when we add a new record to the table, we build the table as a headed list. A headed list has a special backbone pair at the beginning, which holds a dummy "record" -- in this case the arbitrarily chosen symbol *table*. Figure 3.22 shows the box-and-pointer diagram for the table

a:  1
b:  2
c:  3

Figure 3.22:  A table represented as a headed list.

To extract information from a table we use the lookup procedure, which takes a key as argument and returns the associated value (or false if there is no value stored under that key). Lookup is defined in terms of the assoc operation, which expects a key and a list of records as arguments. Note that assoc never sees the dummy record. Assoc returns the record that has the given key as its car.^{noteBecause assoc uses equal?, it can recognize keys that are symbols, numbers, or list structure.} Lookup then checks to see that the resulting record returned by assoc is not false, and returns the value (the cdr) of the record.

(define (lookup key table)
  (let ((record (assoc key (cdr table))))
    (if record
        (cdr record)
        false)))
(define (assoc key records)
  (cond ((null? records) false)
        ((equal? key (caar records)) (car records))
        (else (assoc key (cdr records)))))

To insert a value in a table under a specified key, we first use assoc to see if there is already a record in the table with this key. If not, we form a new record by consing the key with the value, and insert this at the head of the table's list of records, after the dummy record. If there already is a record with this key, we set the cdr of this record to the designated new value. The header of the table provides us with a fixed location to modify in order to insert the new record.^{noteThus, the first backbone pair is the object that represents the table "itself"; that is, a pointer to the table is a pointer to this pair. This same backbone pair always starts the table. If we did not arrange things in this way, insert! would have to return a new value for the start of the table when it added a new record.}

(define (insert! key value table)
  (let ((record (assoc key (cdr table))))
    (if record
        (set-cdr! record value)
        (set-cdr! table
                  (cons (cons key value) (cdr table)))))
  'ok)

To construct a new table, we simply create a list containing the symbol *table*:

(define (make-table)
  (list '*table*))

Two-dimensional tables

In a two-dimensional table, each value is indexed by two keys. We can construct such a table as a one-dimensional table in which each key identifies a subtable. Figure 3.23 shows the box-and-pointer diagram for the table

math:
    +:  43
    -:  45
    *:  42
letters:
    a:  97
    b:  98

which has two subtables. (The subtables don't need a special header symbol, since the key that identifies the subtable serves this purpose.)

Figure 3.23:  A two-dimensional table.

When we look up an item, we use the first key to identify the correct subtable. Then we use the second key to identify the record within the subtable.

(define (lookup key-1 key-2 table)
  (let ((subtable (assoc key-1 (cdr table))))
    (if subtable
        (let ((record (assoc key-2 (cdr subtable))))
          (if record
              (cdr record)
              false))
        false)))

To insert a new item under a pair of keys, we use assoc to see if there is a subtable stored under the first key. If not, we build a new subtable containing the single record (key-2, value) and insert it into the table under the first key. If a subtable already exists for the first key, we insert the new record into this subtable, using the insertion method for one-dimensional tables described above:

(define (insert! key-1 key-2 value table)
  (let ((subtable (assoc key-1 (cdr table))))
    (if subtable
        (let ((record (assoc key-2 (cdr subtable))))
          (if record
              (set-cdr! record value)
              (set-cdr! subtable
                        (cons (cons key-2 value)
                              (cdr subtable)))))
        (set-cdr! table
                  (cons (list key-1
                              (cons key-2 value))
                        (cdr table)))))
  'ok)

Creating local tables

The lookup and insert! operations defined above take the table as an argument. This enables us to use programs that access more than one table. Another way to deal with multiple tables is to have separate lookup and insert! procedures for each table. We can do this by representing a table procedurally, as an object that maintains an internal table as part of its local state. When sent an appropriate message, this "table object" supplies the procedure with which to operate on the internal table. Here is a generator for two-dimensional tables represented in this fashion:

(define (make-table)
  (let ((local-table (list '*table*)))
    (define (lookup key-1 key-2)
      (let ((subtable (assoc key-1 (cdr local-table))))
        (if subtable
            (let ((record (assoc key-2 (cdr subtable))))
              (if record
                  (cdr record)
                  false))
            false)))
    (define (insert! key-1 key-2 value)
      (let ((subtable (assoc key-1 (cdr local-table))))
        (if subtable
            (let ((record (assoc key-2 (cdr subtable))))
              (if record
                  (set-cdr! record value)
                  (set-cdr! subtable
                            (cons (cons key-2 value)
                                  (cdr subtable)))))
            (set-cdr! local-table
                      (cons (list key-1
                                  (cons key-2 value))
                            (cdr local-table)))))
      'ok)    
    (define (dispatch m)
      (cond ((eq? m 'lookup-proc) lookup)
            ((eq? m 'insert-proc!) insert!)
            (else (error "Unknown operation -- TABLE" m))))
    dispatch))

Using make-table, we could implement the get and put operations used in section 2.4.3 for data-directed programming, as follows:

(define operation-table (make-table))
(define get (operation-table 'lookup-proc))
(define put (operation-table 'insert-proc!))

Get takes as arguments two keys, and put takes as arguments two keys and a value. Both operations access the same local table, which is encapsulated within the object created by the call to make-table.

Structure and Interpretation of Computer Programs by Harold Abelson and Gerald Jay Sussman with Julie Sussman is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License by the MIT Press. Modifications to the material were made by syntaxhelper.com.