Add exercises from section 2.3.4

chapter-2/ex-2.67.scm

@@ -0,0 +1,67 @@
+#lang sicp
+
+(define (make-leaf symbol weight) (list 'leaf symbol weight))
+(define (leaf? object) (eq? (car object) 'leaf))
+(define (symbol-leaf x) (cadr x))
+(define (weight-leaf x) (caddr x))
+
+(define (make-code-tree left right)
+  (list left
+        right
+        (append (symbols left) (symbols right))
+        (+ (weight left) (weight right))))
+
+(define (left-branch tree) (car tree))
+(define (right-branch tree) (cadr tree))
+(define (symbols tree)
+  (if (leaf? tree)
+      (list (symbol-leaf tree))
+      (caddr tree)))
+(define (weight tree)
+  (if (leaf? tree)
+      (weight-leaf tree)
+      (cadddr tree)))
+
+(define (decode bits tree)
+  (define (decode-1 bits current-branch)
+    (if (null? bits)
+        '()
+        (let ((next-branch
+               (choose-branch (car bits) current-branch)))
+          (if (leaf? next-branch)
+              (cons (symbol-leaf next-branch)
+                    (decode-1 (cdr bits) tree))
+              (decode-1 (cdr bits) next-branch)))))
+  (decode-1 bits tree))
+
+(define (choose-branch bit branch)
+  (cond ((= bit 0) (left-branch branch))
+        ((= bit 1) (right-branch branch))
+        (else (error "bad bit: CHOOSE-BRANCH" bit))))
+
+(define (adjoin-set x set)
+  (cond ((null? set) (list x))
+        ((< (weight x) (weight (car set))) (cons x set))
+        (else (cons (car set)
+                    (adjoin-set x (cdr set))))))
+
+(define (make-leaf-set pairs)
+  (if (null? pairs)
+      '()
+      (let ((pair (car pairs)))
+        (adjoin-set (make-leaf (car pair)   ; symbol
+                               (cadr pair)) ; frequency
+                    (make-leaf-set (cdr pairs))))))
+
+(define sample-tree
+  (make-code-tree (make-leaf 'A 4)
+                  (make-code-tree
+                   (make-leaf 'B 2)
+                   (make-code-tree
+                    (make-leaf 'D 1)
+                    (make-leaf 'C 1)))))
+
+(define sample-message '(0 1 1 0 0 1 0 1 0 1 1 1 0))
+
+; (decode sample-message sample-tree)
+; (A D A B B C A)

chapter-2/ex-2.68.scm

@@ -0,0 +1,99 @@
+#lang sicp
+
+(define (make-leaf symbol weight) (list 'leaf symbol weight))
+(define (leaf? object) (eq? (car object) 'leaf))
+(define (symbol-leaf x) (cadr x))
+(define (weight-leaf x) (caddr x))
+
+(define (make-code-tree left right)
+  (list left
+        right
+        (append (symbols left) (symbols right))
+        (+ (weight left) (weight right))))
+
+(define (left-branch tree) (car tree))
+(define (right-branch tree) (cadr tree))
+(define (symbols tree)
+  (if (leaf? tree)
+      (list (symbol-leaf tree))
+      (caddr tree)))
+(define (weight tree)
+  (if (leaf? tree)
+      (weight-leaf tree)
+      (cadddr tree)))
+
+(define (decode bits tree)
+  (define (decode-1 bits current-branch)
+    (if (null? bits)
+        '()
+        (let ((next-branch
+               (choose-branch (car bits) current-branch)))
+          (if (leaf? next-branch)
+              (cons (symbol-leaf next-branch)
+                    (decode-1 (cdr bits) tree))
+              (decode-1 (cdr bits) next-branch)))))
+  (decode-1 bits tree))
+
+(define (choose-branch bit branch)
+  (cond ((= bit 0) (left-branch branch))
+        ((= bit 1) (right-branch branch))
+        (else (error "bad bit: CHOOSE-BRANCH" bit))))
+
+(define (adjoin-set x set)
+  (cond ((null? set) (list x))
+        ((< (weight x) (weight (car set))) (cons x set))
+        (else (cons (car set)
+                    (adjoin-set x (cdr set))))))
+
+(define (make-leaf-set pairs)
+  (if (null? pairs)
+      '()
+      (let ((pair (car pairs)))
+        (adjoin-set (make-leaf (car pair)   ; symbol
+                               (cadr pair)) ; frequency
+                    (make-leaf-set (cdr pairs))))))
+
+(define sample-tree
+  (make-code-tree (make-leaf 'A 4)
+                  (make-code-tree
+                   (make-leaf 'B 2)
+                   (make-code-tree
+                    (make-leaf 'D 1)
+                    (make-leaf 'C 1)))))
+
+(define sample-message '(0 1 1 0 0 1 0 1 0 1 1 1 0))
+
+; (decode sample-message sample-tree)
+; (A D A B B C A)
+
+; until now previous exercise
+
+; setup for the current one
+(define (encode message tree)
+  (if (null? message)
+      '()
+      (append (encode-symbol (car message) tree)
+              (encode (cdr message) tree))))
+
+; my procedure
+; assumes that a tree is well formed, ie. that if a symbol appears in a
+; parent node's list it must be in one of the child branches
+; not the most efficient since it checks symbol lists at almost every level
+; twice
+(define (encode-symbol symbol tree)
+  (cond ((not (member-set symbol (symbols tree)))
+         (error "encode-symbol: no seq for the symbol in tree" symbol tree))
+        ((leaf? tree)
+         '())
+        ((member-set symbol (symbols (left-branch tree)))
+         (cons 0 (encode-symbol symbol (left-branch tree))))
+        ((member-set symbol (symbols (right-branch tree)))
+         (cons 1 (encode-symbol symbol (right-branch tree))))))
+
+(define member-set member)
+; since a set is represented as a list, we can simply use the member procedure
+
+;test:
+; input: (A D A B B C A)
+; output:(0 1 1 0 0 1 0 1 0 1 1 1 0)
+; it works!

chapter-2/ex-2.69.scm

@@ -0,0 +1,66 @@
+#lang sicp
+
+(define (make-leaf symbol weight) (list 'leaf symbol weight))
+(define (leaf? object) (eq? (car object) 'leaf))
+(define (symbol-leaf x) (cadr x))
+(define (weight-leaf x) (caddr x))
+
+(define (make-code-tree left right)
+  (list left
+        right
+        (append (symbols left) (symbols right))
+        (+ (weight left) (weight right))))
+
+(define (left-branch tree) (car tree))
+(define (right-branch tree) (cadr tree))
+(define (symbols tree)
+  (if (leaf? tree)
+      (list (symbol-leaf tree))
+      (caddr tree)))
+(define (weight tree)
+  (if (leaf? tree)
+      (weight-leaf tree)
+      (cadddr tree)))
+
+(define (decode bits tree)
+  (define (decode-1 bits current-branch)
+    (if (null? bits)
+        '()
+        (let ((next-branch
+               (choose-branch (car bits) current-branch)))
+          (if (leaf? next-branch)
+              (cons (symbol-leaf next-branch)
+                    (decode-1 (cdr bits) tree))
+              (decode-1 (cdr bits) next-branch)))))
+  (decode-1 bits tree))
+
+(define (choose-branch bit branch)
+  (cond ((= bit 0) (left-branch branch))
+        ((= bit 1) (right-branch branch))
+        (else (error "bad bit: CHOOSE-BRANCH" bit))))
+
+(define (adjoin-set x set)
+  (cond ((null? set) (list x))
+        ((< (weight x) (weight (car set))) (cons x set))
+        (else (cons (car set)
+                    (adjoin-set x (cdr set))))))
+
+(define (make-leaf-set pairs)
+  (if (null? pairs)
+      '()
+      (let ((pair (car pairs)))
+        (adjoin-set (make-leaf (car pair)   ; symbol
+                               (cadr pair)) ; frequency
+                    (make-leaf-set (cdr pairs))))))
+
+; actual exercise
+(define (generate-huffman-tree pairs)
+  (successive-merge (make-leaf-set pairs)))
+
+(define (successive-merge tree-set)
+  (if (null? (cdr tree-set)) ; (= (length tree-list) 1)
+      (car tree-set)
+      (let ((tree1 (car tree-set))
+            (tree2 (cadr tree-set)))
+        (successive-merge (adjoin-set (make-code-tree tree1 tree2)
+                                      (cddr tree-set))))))

chapter-2/ex-2.70.scm

@@ -0,0 +1,109 @@
+#lang sicp
+
+(define (make-leaf symbol weight) (list 'leaf symbol weight))
+(define (leaf? object) (eq? (car object) 'leaf))
+(define (symbol-leaf x) (cadr x))
+(define (weight-leaf x) (caddr x))
+
+(define (make-code-tree left right)
+  (list left
+        right
+        (append (symbols left) (symbols right))
+        (+ (weight left) (weight right))))
+
+(define (left-branch tree) (car tree))
+(define (right-branch tree) (cadr tree))
+(define (symbols tree)
+  (if (leaf? tree)
+      (list (symbol-leaf tree))
+      (caddr tree)))
+(define (weight tree)
+  (if (leaf? tree)
+      (weight-leaf tree)
+      (cadddr tree)))
+
+(define (decode bits tree)
+  (define (decode-1 bits current-branch)
+    (if (null? bits)
+        '()
+        (let ((next-branch
+               (choose-branch (car bits) current-branch)))
+          (if (leaf? next-branch)
+              (cons (symbol-leaf next-branch)
+                    (decode-1 (cdr bits) tree))
+              (decode-1 (cdr bits) next-branch)))))
+  (decode-1 bits tree))
+
+(define (choose-branch bit branch)
+  (cond ((= bit 0) (left-branch branch))
+        ((= bit 1) (right-branch branch))
+        (else (error "bad bit: CHOOSE-BRANCH" bit))))
+
+(define (adjoin-set x set)
+  (cond ((null? set) (list x))
+        ((< (weight x) (weight (car set))) (cons x set))
+        (else (cons (car set)
+                    (adjoin-set x (cdr set))))))
+
+(define (make-leaf-set pairs)
+  (if (null? pairs)
+      '()
+      (let ((pair (car pairs)))
+        (adjoin-set (make-leaf (car pair)   ; symbol
+                               (cadr pair)) ; frequency
+                    (make-leaf-set (cdr pairs))))))
+
+(define (generate-huffman-tree pairs)
+  (successive-merge (make-leaf-set pairs)))
+
+(define (successive-merge tree-set)
+  (if (null? (cdr tree-set)) ; (= (length tree-list) 1)
+      (car tree-set)
+      (let ((tree1 (car tree-set))
+            (tree2 (cadr tree-set)))
+        (successive-merge (adjoin-set (make-code-tree tree1 tree2)
+                                      (cddr tree-set))))))
+
+; the encode functionality from 2.68
+(define (encode message tree)
+  (if (null? message)
+      '()
+      (append (encode-symbol (car message) tree)
+              (encode (cdr message) tree))))
+
+(define (encode-symbol symbol tree)
+  (cond ((not (member-set symbol (symbols tree)))
+         (error "encode-symbol: no seq for the symbol in tree" symbol tree))
+        ((leaf? tree)
+         '())
+        ((member-set symbol (symbols (left-branch tree)))
+         (cons 0 (encode-symbol symbol (left-branch tree))))
+        ((member-set symbol (symbols (right-branch tree)))
+         (cons 1 (encode-symbol symbol (right-branch tree))))))
+
+(define member-set member)
+; since a set is represented as a list, we can simply use the member procedure
+
+; actual exercise 2.70
+
+(define sample-tree
+  (generate-huffman-tree '((a 2)
+                           (get 2)
+                           (sha 3)
+                           (wah 1)
+                           (boom 1)
+                           (job 2)
+                           (na 16)
+                           (yip 9))))
+
+(define sample-message
+  '(get a job
+	sha na na na na na na na na
+        get a job
+	sha na na na na na na na na
+        wah yip yip yip yip yip yip yip yip yip
+        sha boom))
+
+; (length (encode sample-message sample-tree)) -> 84
+; if we used a fixed length code, it would have to have length of >=3
+; since there are 36 words here, the answer would be 108

chapter-2/ex-2.71.txt

@@ -0,0 +1,36 @@
+(A 1), (B 2), (C 4), ...
+Tree sketch (n=5):
+
+      *
+   /     \
+(E 16)    *
+       /     \
+    (D 8)     *
+           /     \
+        (C 4)     *
+               /     \
+            (B 2)   (A 1)
+
+tree sketch (n=10):
+       *
+    /     \
+(J 512)   *
+       /     \
+   (I 256)    *
+           /     \
+       (H 128)    *
+               /     \
+            (G 64)    *
+                   /     \
+                (F 32)    *
+                       /     \
+                    (E 16)    *
+                           /     \
+                        (D 8)     *
+                               /     \
+                            (C 4)     *
+                                   /     \
+                                (B 2)   (A 1)
+
+The most frequent symbol always requires exactly 1 bit.
+The two least frequent symbols will always require exactly n-1 bits.

chapter-2/ex-2.72.txt

@@ -0,0 +1,12 @@
+Searching a single node requires on the order of k operations, where
+k is the number of symbols given at that node.
+If the tree is roughly balanced, then the number of symbols listed in
+each node on the way down should decrease exponentially, and the number
+of these searches would be on the order of O(log(n)), where n is the
+number of symbols generally available for encoding.
+
+If this is true, than each encoding of a symbol would take on the order
+of O(n*log(n)) operations, however if the tree is very unbalanced,
+which is the case in exercise 2.71, the order of growth is roughly
+O(n^2), for the least frequent symbols, and O(n), for the most common
+symbol.