Problem 42779. GJam March 2016 IOW: Polynesiaglot Large

Created by Richard Zapor

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This Challenge is derived from <a href = "http://code.google.com/codejam/contest/8274486/dashboard#s=p2">GJam March 2016 Annual I/O for Polynesiaglot</a>. This is the Large input set. The max Qraw is 100^500, (V+C)^L.The GJam story goes that words are such that consonants are always followed by a vowel. Determine the number of possible words of length L using C consonants and V vowels. The final Q is to be modulo of the prime 1E9+7.Input: [C V L] , C[1,50], V[1,50], 1&lt;=L&lt;=500Output: [Q] max Qraw is 100^500; Q=mod(Qraw,1E9+7)Examples: [C V L] [Q]<pre class="language-matlab">[1 1 4] [5] {aaaa,aaba,abaa,baaa,baba} invalid are {bbaa, aaab} 
[1 2 2] [6] {aa,ae,ba,be,ee,ea} invalid are {ab,eb,bb}
</pre><a href = "http://code.google.com/codejam">Google Code Jam 2016 Open Qualifier: April 8, 2016</a>Theory: This is a huge value problem, on the order of (C+V)^L, thus brute force will not work. This is also a probability tree type problem. Tree calculations can be reduced to a linear in L evaluation. Inspection shows Q(1)=V, Q(2)=V^2, L=3 Q(3)=V^3+V*C*V+C*V^2 = V*Q(L-1)+V*C*Q(L-2)+C*Q(L-1). There are no Cs at the Q1 level since can not end in a C. Qnext=f(Q(-1),Q(-2)). Qfinal=Q+C*Q(L-1)<pre class="language-matlab">Q3 V C
Q2 V C V
Q1 V V V
</pre>One method to succeed in this problem is to use the java capability of Matlab. <a href = "http://www.mathworks.com/matlabcentral/cody/problems/1833-usage-of-java-math-add-multiply-pow">Cody Java Challenge</a>. The primary reference sites are <a href = "http://docs.oracle.com/javase/1.5.0/docs/api/java/lang/Number.html">Java Math</a>, <a href = "http://docs.oracle.com/javase/1.5.0/docs/api/java/math/BigDecimal.html">Java BigDecimal</a>, and <a href = "http://docs.oracle.com/javase/1.5.0/docs/api/java/math/BigInteger.html">Java BigInteger</a>. The Java solution by the winner Stacy is included at the bottom of the test suite.

This Challenge is derived from <http://code.google.com/codejam/contest/8274486/dashboard#s=p2 GJam March 2016 Annual I/O for Polynesiaglot>. This is the Large input set. The max Qraw is 100^500, (V+C)^L.

The GJam story goes that words are such that consonants are always followed by a vowel. Determine the number of possible words of length L using C consonants and V vowels. The final Q is to be modulo of the prime 1E9+7.

*Input:* [C V L] , C[1,50], V[1,50], 1<=L<=500

*Output:* [Q] max Qraw is 100^500; Q=mod(Qraw,1E9+7)

*Examples:* [C V L] [Q]

[1 1 4] [5] {aaaa,aaba,abaa,baaa,baba}  invalid are {bbaa, aaab} 
  [1 2 2] [6] {aa,ae,ba,be,ee,ea} invalid are {ab,eb,bb}

*<http://code.google.com/codejam Google Code Jam 2016 Open Qualifier: April 8, 2016>*

*Theory:* This is a huge value problem, on the order of (C+V)^L, thus brute force will not work. This is also a probability tree type problem. Tree calculations can be reduced to a linear in L evaluation. Inspection shows Q(1)=V, Q(2)=V^2, L=3 Q(3)=V^3+V*C*V+C*V^2 = V*Q(L-1)+V*C*Q(L-2)+C*Q(L-1).  There are no Cs at the Q1 level since can not end in a C. Qnext=f(Q(-1),Q(-2)). Qfinal=Q+C*Q(L-1)

Q3    V          C
  Q2  V   C       V
  Q1 V   V       V

One method to succeed in this problem is to use the java capability of Matlab.
<http://www.mathworks.com/matlabcentral/cody/problems/1833-usage-of-java-math-add-multiply-pow Cody Java Challenge>. The primary reference sites are <http://docs.oracle.com/javase/1.5.0/docs/api/java/lang/Number.html Java Math>, <http://docs.oracle.com/javase/1.5.0/docs/api/java/math/BigDecimal.html Java BigDecimal>, and <http://docs.oracle.com/javase/1.5.0/docs/api/java/math/BigInteger.html Java BigInteger>. The Java solution by the winner Stacy is included at the bottom of the test suite.

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