Create rod_cutting.py (#373)

dynamic_programming/rod_cutting.py

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+### PROBLEM ###
+"""
+We are given a rod of length n and we are given the array of prices, also of
+length n. This array contains the price for selling a rod at a certain length.
+For example, prices[5] shows the price we can sell a rod of length 5.
+Generalising, prices[x] shows the price a rod of length x can be sold.
+We are tasked to find the optimal solution to sell the given rod.
+"""
+
+### SOLUTION ###
+"""
+Profit(n) = max(1<i<n){Price(n),Price(i)+Profit(n-i)}
+
+When we receive a rod, we have two options:
+a) Don't cut it and sell it as is (receiving prices[length])
+b) Cut it and sell it in two parts. The length we cut it and the rod we are
+left with, which we have to try and sell separately in an efficient way.
+Choose the maximum price we can get.
+"""
+
+def CutRod(n):
+    if(n == 1):
+        #Cannot cut rod any further
+        return prices[1]
+
+    noCut = prices[n] #The price you get when you don't cut the rod
+    yesCut = [-1 for x in range(n)] #The prices for the different cutting options
+
+    for i in range(1,n):
+        if(solutions[i] == -1):
+            #We haven't calulated solution for length i yet.
+            #We know we sell the part of length i so we get prices[i].
+            #We just need to know how to sell rod of length n-i
+            yesCut[i] = prices[i] + CutRod(n-i)
+        else:
+            #We have calculated solution for length i.
+            #We add the two prices.
+            yesCut[i] = prices[i] + solutions[n-i]
+
+    #We need to find the highest price in order to sell more efficiently.
+    #We have to choose between noCut and the prices in yesCut.
+    m = noCut #Initialize max to noCut
+    for i in range(n):
+        if(yesCut[i] > m):
+            m = yesCut[i]
+
+    solutions[n] = m
+    return m
+
+
+
+### EXAMPLE ###
+length = 5
+#The first price, 0, is for when we have no rod.
+prices = [0, 1, 3, 7, 9, 11, 13, 17, 21, 21, 30]
+solutions = [-1 for x in range(length+1)]
+
+print(CutRod(length))