Finished Chapter 1

Foundations/overview/README.md

@@ -70,3 +70,5 @@ for n in range(1, 100):
 
 ### Problems 1-1
 > For each function `f(n)` and time `t` in the following table, determine the largest size `n` of a problem that can be solved in time `t`, assuming that the algorithm to solve the problem takes `f(n)` microseconds.
+
+check and run [calc.py](calc.py)

Foundations/overview/calc.py

@@ -0,0 +1,109 @@
+import math
+from decimal import *
+
+sec = 1000000
+min = 60 * sec
+hour = 60 * min
+day = 24 * hour
+month = 30 * day
+year = 365 * day
+century = 100 * year
+
+# n * lg(n) == c
+# ==> 2 ** (lg(n)) * lg (n) == c
+# ==> lg(n) == W(c)
+# ==> n == exp(W(c))
+# see http://en.wikipedia.org/wiki/Lambert_W_function#Example_2
+
+def nlgn(c):
+	lower = 0.0
+	upper = 10e13
+	while True:
+		middle = (lower + upper) / 2
+		if lower == middle or middle == upper:
+			return middle
+		if middle * math.log(middle, 2) > c:
+			upper = middle
+		else:
+			lower = middle
+
+# n!
+def defactorial(c):
+	n = 0
+	while True:
+		if math.factorial(n) >= c:
+			return n - 1;
+		else:
+			n += 1
+
+# lg(n)
+print("2 ^ 10 ^ 6")
+print("2 ^ ( %d * 10 ^ 6 )" % (min/sec))
+print("2 ^ ( %d * 10 ^ 6 )" % (hour/sec))
+print("2 ^ ( %d * 10 ^ 6 )" % (day/sec))
+print("2 ^ ( %d * 10 ^ 6 )" % (month/sec))
+print("2 ^ ( %d * 10 ^ 6 )" % (year/sec))
+print("2 ^ ( %d * 10 ^ 6 )" % (century/sec))
+print("")
+# sqrt(n)
+print("{:.1e}".format(Decimal(sec ** 2)))
+print("{:.1e}".format(Decimal(min ** 2)))
+print("{:.3e}".format(Decimal(hour ** 2)))
+print("{:.5e}".format(Decimal(day ** 2)))
+print("{:.6e}".format(Decimal(month ** 2)))
+print("{:.9e}".format(Decimal(year ** 2)))
+print("{:.9e}".format(Decimal(century ** 2)))
+print("")
+# n
+print("{:.1e}".format(Decimal(sec)))
+print("{:.1e}".format(Decimal(min)))
+print("{:.1e}".format(Decimal(hour)))
+print("{:.2e}".format(Decimal(day)))
+print("{:.3e}".format(Decimal(month)))
+print("{:.4e}".format(Decimal(year)))
+print("{:.4e}".format(Decimal(century)))
+print("")
+# n lg(n)
+print(nlgn(sec))
+print(nlgn(min))
+print(nlgn(hour))
+print(nlgn(day))
+print(nlgn(month))
+print(nlgn(year))
+print(nlgn(century))
+print("")
+# n^2
+print(math.sqrt(sec))
+print(math.sqrt(min))
+print(math.sqrt(hour))
+print(math.sqrt(day))
+print(math.sqrt(month))
+print(math.sqrt(year))
+print(math.sqrt(century))
+print("")
+# n^3
+print(sec ** (1/3.0))
+print(min ** (1/3.0))
+print(hour ** (1/3.0))
+print(day ** (1/3.0))
+print(month ** (1/3.0))
+print(year ** (1/3.0))
+print(century ** (1/3.0))
+print("")
+# 2^n
+print(math.log(sec, 2))
+print(math.log(min, 2))
+print(math.log(hour, 2))
+print(math.log(day, 2))
+print(math.log(month, 2))
+print(math.log(year, 2))
+print(math.log(century, 2))
+print("")
+# n!
+print(defactorial(sec))
+print(defactorial(min))
+print(defactorial(hour))
+print(defactorial(day))
+print(defactorial(month))
+print(defactorial(year))
+print(defactorial(century))

Foundations/overview/calc_time.py

@@ -1,11 +0,0 @@
-import math
-
-sec  	= 1000000
-min  	= 60 * sec
-hour	= 60 * min
-day	= 24 * hour
-month	= 30 * day
-year	= 365 * day
-century = 100 * year
-
-print("%3e" % 2 ** sec)